Let Us Count the Ways

I was curious as to how to count the number of ways that a given number can be represented in Loeschian form based on its prime factorisation. I knew there was such a method for counting how many ways a number can be represented as a sum of two squares. I got Gemini to investigate, compare and summarise the results. Notice the use of the ceiling function when determining the number of essentially distinct pairs. Remember also that the invalid primes become valid if raised to an even power but they don't contribute to the count. The \(b_{\textit{i}}\) in the table refers to the exponents of the generating primes.

The algebraic frameworks for expressing integers as \(x^2 + xy + y^2\) and \(x^2 + y^2\) mirror each other directly through their respective complex integer rings.

Characteristic	Loeschian Form \(x^2 + xy + y^2\)	Sum of Two Squares \(x^2 + y^2\)
Complex Ring	Eisenstein Integers \(\mathbb{Z}[e^{2\pi i/3}]\)	Gaussian Integers \(\mathbb{Z}[i]\)
Invalid Primes (Require Even Exponents)	\(p \equiv 2 \pmod 3\)	\(p \equiv 3 \pmod 4\)
Generating Primes (Exponents \(b_i\))	\(p \equiv 1 \pmod 3\)	\(p \equiv 1 \pmod 4\)
Neutral Prime	\(p = 3\)	\(p = 2\)
Unit Symmetries (Multiplier)	6	4
Total Ordered Pairs \((x,y)\)	\(6 \prod (b_i + 1)\)	\(4 \prod (b_i + 1)\)
Essentially Distinct Pairs \(\{x,y\}\)	\(\lceil \frac{1}{2} \prod (b_i + 1) \rceil\)	\(\lceil \frac{1}{2} \prod (b_i + 1) \rceil\)

Factorization Conditions

Both systems demand that primes which cannot be represented natively by the quadratic form must be "squared away" with even exponents in the prime factorization. For Gaussian integers, primes congruent to \(3 \pmod 4\) (such as 3, 7, 11) cannot form sums of two squares. For Eisenstein integers, primes congruent to \(2 \pmod 3\) (such as 2, 5, 11) cannot be Loeschian. If any of these restricted primes have an odd exponent, the integer cannot be represented in that form, and the total count is zero.

Symmetry and Distinct Ways

The difference in the total number of ordered coordinate pairs stems entirely from the geometry of the complex rings. Gaussian integers form a square lattice with 4 units (\(\pm 1, \pm i\)), leading to a rotational symmetry multiplier of 4. Eisenstein integers form a hexagonal lattice with 6 units (\(\pm 1, \pm e^{2\pi i/3}, \pm e^{4\pi i/3}\)), leading to a multiplier of 6.

Once these geometric symmetries are factored out by dividing by the respective unit count, both systems calculate the number of essentially distinct, non-negative integer pairs using the exact same ceiling function formula. The generating primes dictate this final count, while the unique neutral prime in each system scales the base integer without creating new algebraic variations.

Binomial Expansion Using Modular Arithmetic

 1

1    1

1    2    1

1    3    3    1

1    4    6    4    1

1    5    10    10    5    1

1    6    15    20    15    6    1

1    7    21    35    35    21    7    1

Above we see the beginning of Pascal's Triangle, that is of great assistance is working out the coefficients of the terms in the binomial expansion of \( (a+b)^n \) for various positive integer values of \(n\). The first line corresponds to \(n=0\), then \(n=1\), \(n=2\) and so on. For example:$$(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4$$A common mistake that's made by beginner's is to write the following:$$(a+b)^n=a^n+b^n$$What is regarded as a mistake actually turns out to true for certain values of \(n\) when modular arithmetic is involved.

Let's consider mod 2 where only two numbers exist: 0 and 1. Any numbers larger than 1 reduce to either 0 or 1. Thus \(2 \equiv 0 \, mod \, 2\) and \(3 \equiv 1 \, mod \, 2\) etc. Let's look at the expansion of \( (a+b)^2 \) in this light. We have:$$ \begin{align} (a+b)^2 &=a^2+2ab+b^2\\ &=a^2+b^2 \end{align}$$In the case of mod 2, the \(2ab\) term must equal zero What about mod 3? The only numbers here are 0, 1 and 2 with \(3 \equiv 0 \,mod \,3 \). The binomial expansion for the case of \(n=3\) becomes:$$ \begin{align} (a+b)^3 &=a^3+3a^3b+3ab^2+b^3\\ &=a^3+b^3 \end{align}$$Here the \(3a^2b\) and \(3ab^2\) terms equal zero. We seem to on a roll. Let's consider the case of \(n=4\) or mod 4. Here the only numbers are 0, 1, 2 and 3 with \(4 \equiv 0 \, mod \, 4\), \(5 \equiv 1 \, mod \, 4\), \(6 \equiv 2 \, mod \, 4\) and so on. Thus we have:$$ \begin{align} (a+b)^4 &= a^4 + 4a^3b+6a^2b^2+4ab^3+b^4\\ &=a^4 + 2a^2b^2+b^4 \end{align}$$In this case, things don't quite work out because the middle term \(6a^2b^2\) after conversion to \(2a^2b^2\).

What about the case of \(n=5\) or mod 5? Here the numbers are 0, 1, 2, 3 and 4 and the expansion of \( (a+b)^5\) works out as follows:$$ \begin{align} (a+b)^5 &= a^5 + 5a^4b + 10a^3b^2+10a^2b^3+5ab^4+b^5 \\ &=a^5+b^5 \end{align}$$The central terms all disappear because \(5 \equiv 0 \, mod \,5\) and \(10 \equiv 0 \, mod \, 5\). If we consider the case of mod 6, the simplification fails but works again for mod 7. The generalisation from all of this is that if \(p\) is prime and we are working with mod \(p\) then the following always holds true:$$(a+b)^p=a^p+b^p$$I'm thankful to this YouTube video for informing me of this interesting twist on the binomial expansion.

Modest Numbers

Modest numbers are a category of numbers that I haven't paid a great deal of attention to. They are defined by Numbers Aplenty in the following manner:

A number \(n\) is called modest if its digits can be separated into two numbers \(a\) and \(b\) such that \(n\) divided by \(b\) gives \(a\) as remainder.

The reason its came to my attention is that Julian Day Number 2459999 is approaching and it has the property of being a modest number because 9999 divides 2459999 to give 245 or to put it in another format:$$2459999 \! \! \mod 99999 \equiv 245$$I developed an algorithm in SageMath to determine all the modest numbers in a given range. Here are the modest numbers in the range up to 40,000 (permalink), constituting 2.25% of the range:

13, 19, 23, 26, 29, 39, 46, 49, 59, 69, 79, 89, 103, 109, 111, 133, 199, 203, 206, 209, 211, 218, 222, 233, 266, 299, 309, 311, 327, 333, 399, 406, 409, 411, 412, 418, 422, 433, 436, 444, 466, 499, 509, 511, 515, 533, 545, 555, 599, 609, 611, 618, 622, 627, 633, 654, 666, 699, 709, 711, 721, 733, 763, 777, 799, 809, 811, 812, 818, 822, 824, 833, 836, 844, 866, 872, 888, 899, 911, 927, 933, 981, 999, 1003, 1009, 1011, 1015, 1018, 1022, 1027, 1030, 1033, 1037, 1045, 1055, 1066, 1090, 1099, 1111, 1133, 1199, 1218, 1222, 1227, 1233, 1236, 1244, 1254, 1266, 1299, 1333, 1339, 1399, 1418, 1421, 1422, 1433, 1442, 1463, 1466, 1477, 1499, 1527, 1533, 1545, 1555, 1599, 1618, 1622, 1624, 1633, 1636, 1644, 1648, 1666, 1672, 1688, 1699, 1733, 1751, 1799, 1822, 1827, 1833, 1854, 1866, 1881, 1899, 1933, 1957, 1999, 2003, 2006, 2009, 2018, 2022, 2027, 2030, 2033, 2036, 2037, 2044, 2045, 2054, 2055, 2060, 2066, 2074, 2090, 2099, 2111, 2127, 2133, 2163, 2177, 2199, 2222, 2233, 2266, 2299, 2333, 2369, 2399, 2427, 2433, 2436, 2444, 2454, 2466, 2472, 2488, 2499, 2533, 2545, 2555, 2575, 2599, 2633, 2639, 2666, 2678, 2699, 2733, 2781, 2799, 2833, 2836, 2842, 2844, 2863, 2866, 2877, 2884, 2899, 2933, 2987, 2999, 3009, 3027, 3033, 3037, 3045, 3054, 3055, 3066, 3081, 3090, 3099, 3111, 3133, 3193, 3199, 3233, 3236, 3244, 3248, 3266, 3272, 3288, 3296, 3299, 3333, 3399, 3451, 3466, 3499, 3545, 3555, 3563, 3577, 3599, 3644, 3654, 3666, 3681, 3699, 3799, 3857, 3866, 3899, 3999, 4006, 4009, 4012, 4018, 4027, 4036, 4037, 4044, 4045, 4054, 4055, 4060, 4066, 4072, 4074, 4088, 4090, 4099, 4108, 4111, 4148, 4199, 4222, 4254, 4263, 4266, 4277, 4299, 4333, 4399, 4444, 4466, 4499, 4555, 4581, 4599, 4666, 4669, 4699, 4799, 4854, 4866, 4872, 4888, 4899, 4963, 4977, 4999, 5009, 5015, 5027, 5037, 5045, 5055, 5066, 5075, 5090, 5099, 5111, 5135, 5185, 5199, 5266, 5278, 5299, 5333, 5399, 5466, 5481, 5499, 5555, 5599, 5663, 5666, 5672, 5677, 5684, 5688, 5699, 5799, 5866, 5887, 5899, 5999, 6009, 6018, 6027, 6037, 6054, 6066, 6074, 6081, 6090, 6099, 6111, 6162, 6199, 6222, 6266, 6293, 6299, 6333, 6377, 6381, 6399, 6466, 6472, 6488, 6496, 6499, 6599, 6666, 6699, 6799, 6899, 6999, 7009, 7021, 7027, 7037, 7063, 7077, 7090, 7099, 7111, 7189, 7199, 7259, 7281, 7288, 7299, 7333, 7399, 7499, 7599, 7699, 7777, 7799, 7899, 7999, 8009, 8012, 8018, 8024, 8027, 8036, 8037, 8054, 8072, 8074, 8088, 8090, 8099, 8108, 8111, 8148, 8199, 8216, 8222, 8296, 8299, 8333, 8399, 8444, 8499, 8599, 8666, 8699, 8799, 8888, 8899, 8999, 9027, 9037, 9081, 9099, 9111, 9199, 9243, 9299, 9333, 9399, 9499, 9599, 9699, 9799, 9899, 9999, 10003, 10009, 10011, 10015, 10018, 10027, 10030, 10033, 10037, 10045, 10054, 10074, 10090, 10099, 10101, 10111, 10135, 10185, 10222, 10270, 10303, 10333, 10370, 10555, 10666, 10909, 10999, 11027, 11033, 11037, 11099, 11111, 11297, 11333, 11407, 11999, 12018, 12027, 12036, 12037, 12054, 12074, 12081, 12108, 12111, 12148, 12162, 12222, 12324, 12333, 12444, 12666, 12999, 13027, 13037, 13039, 13111, 13117, 13333, 13351, 13481, 13999, 14018, 14021, 14027, 14037, 14042, 14054, 14063, 14074, 14111, 14126, 14189, 14222, 14259, 14333, 14378, 14518, 14666, 14777, 14999, 15027, 15037, 15045, 15081, 15111, 15135, 15185, 15333, 15405, 15555, 15999, 16018, 16024, 16027, 16036, 16037, 16048, 16054, 16072, 16074, 16108, 16111, 16144, 16148, 16216, 16222, 16296, 16333, 16432, 16444, 16592, 16666, 16888, 16999, 17027, 17037, 17051, 17111, 17153, 17333, 17459, 17629, 17999, 18027, 18037, 18054, 18074, 18081, 18111, 18162, 18222, 18243, 18333, 18486, 18666, 18999, 19027, 19037, 19057, 19111, 19171, 19333, 19513, 19703, 19999, 20003, 20006, 20009, 20011, 20018, 20022, 20027, 20030, 20033, 20036, 20037, 20045, 20054, 20060, 20066, 20074, 20090, 20099, 20101, 20108, 20111, 20135, 20148, 20180, 20185, 20198, 20202, 20222, 20270, 20303, 20333, 20370, 20444, 20540, 20555, 20606, 20666, 20740, 20909, 20999, 21027, 21037, 21063, 21081, 21111, 21189, 21259, 21333, 21567, 21777, 21818, 21999, 22027, 22033, 22037, 22054, 22066, 22074, 22099, 22111, 22198, 22222, 22297, 22333, 22407, 22594, 22666, 22814, 22999, 23027, 23037, 23069, 23111, 23207, 23333, 23621, 23851, 23999, 24027, 24036, 24037, 24054, 24072, 24074, 24081, 24108, 24111, 24148, 24162, 24216, 24222, 24296, 24324, 24333, 24444, 24648, 24666, 24888, 24999, 25027, 25037, 25045, 25075, 25111, 25135, 25185, 25225, 25333, 25555, 25675, 25925, 25999, 26027, 26037, 26039, 26054, 26074, 26078, 26111, 26117, 26222, 26234, 26333, 26351, 26481, 26666, 26702, 26962, 26999, 27037, 27081, 27111, 27243, 27333, 27729, 27999, 28036, 28037, 28042, 28054, 28063, 28074, 28084, 28108, 28111, 28126, 28148, 28189, 28222, 28252, 28259, 28333, 28378, 28444, 28518, 28666, 28756, 28777, 28999, 29037, 29087, 29111, 29261, 29333, 29783, 29999, 30009, 30011, 30027, 30033, 30037, 30045, 30054, 30074, 30081, 30090, 30099, 30101, 30111, 30135, 30162, 30185, 30222, 30270, 30297, 30303, 30333, 30370, 30405, 30555, 30666, 30810, 30909, 30999, 31037, 31093, 31111, 31279, 31333, 31837, 31999, 32036, 32037, 32048, 32054, 32072, 32074, 32096, 32108, 32111, 32144, 32148, 32216, 32222, 32288, 32296, 32333, 32432, 32444, 32592, 32666, 32727, 32864, 32888, 32999, 33037, 33081, 33099, 33111, 33297, 33333, 33407, 33891, 33999, 34037, 34051, 34054, 34074, 34102, 34111, 34153, 34222, 34306, 34333, 34459, 34629, 34666, 34918, 34999, 35037, 35045, 35063, 35105, 35111, 35135, 35185, 35189, 35259, 35315, 35333, 35555, 35777, 35945, 35999, 36037, 36054, 36074, 36081, 36108, 36111, 36148, 36162, 36222, 36243, 36324, 36333, 36444, 36486, 36666, 36972, 36999, 37111, 37333, 37999, 38054, 38057, 38074, 38111, 38114, 38171, 38222, 38333, 38342, 38513, 38666, 38703, 38999, 39081, 39111, 39117, 39333, 39351, 39481, 39999

Even though the first three modest numbers are prime, the prime modest numbers occupy only 0.27% of the range (108 numbers). Consecutive modest numbers are few and far between. Here are the pairs of such numbers up to 40,000:

411 and 412
811 and 812
1421 and 1422
2036 and 2037
2044 and 2045
2054 and 2055
3054 and 3055
4036 and 4037
4044 and 4045
4054 and 4055
8036 and 8037
12036 and 12037
16036 and 16037
20036 and 20037
24036 and 24037
28036 and 28037
32036 and 32037

No triplets have been found as far as I know.

It's of interest to consider the frequency of final digits of modest numbers.

Here is the breakdown of final digits in the range up to 100000 (permalink):

There are a total of 1728 modest numbers:

0 --> 67
1 --> 218
2 --> 151
3 --> 219
4 --> 161
5 --> 128
6 --> 172
7 --> 159
8 --> 144
9 --> 309

Clearly numbers ending in the digit 9 predominate. 

CONTINUED IN NEXT POST DUE TO TECHNICAL DIFFICULTIES

Sums of Squares Pattern

I watched an interesting video that describes a pattern involving sums of squares. The property can be stated as follows:$$n^2+(n+1)^2+n^2 \cdot (n+1)^2 =(n \cdot (n+1)+1)^2\\ \text{where both sides are equal to } n^4 + 2n^3 + 3n^2 + 2n + 1$$The author of the video arrives at the pattern by considering specific numerical examples and doesn't look at the underlying algebra.

The pattern is more meaningful when we look at some specific examples:$$ \begin{align} 1^2+2^2+2^2&=3^2\\2^2+3^2+6^2&=7^2\\3^2+4^2+12^2&=13^2\\4^2+5^2+20^2&=21^2\\5^2+6^2+30^2&=31^2 \end{align}$$Looking at this property the other way, we can say that for any integer \(a\):$$a^2=(a-1)^2+x^2+(x+1)^2$$where \(x\) is an integer such that \(x \cdot (x+1)=a-1\) if such integers exist, which of course will not often be the cause.

However, number of the form \(x \cdot (x+1) \) are pronic and members of OEIS A002378:

A002378

Oblong (or promic, pronic, or heteromecic) numbers: \(a(n) = n \cdot (n+1) \).

0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2070, 2162, 2256, 2352, 2450, 2550, ...

Thus for every number that is one more than a pronic number, there will always be an integer value of \(x\) to satisfy \(a^2=(a-1)^2+x^2+(x+1)^2\). Looking at the list of pronic numbers above, let's take 2550 as an example. We will then have:$$ \begin{align} 2551^2&=2550^2+x^2+(x+1)^2\\x^2+(x+1)^2+2550^2-2551^2 &=0\\x^2+x^2+2x+1 +5101 \cdot (-1) &=0\\2x^2+2x-5100 &=0\\(2x-100)(x+51) &=0 \\x&=50 \text{ or }-51\end{align}$$Thus, choosing the positive solution, we have \(2551^2=2550^2+51^2+50^2\).

Of course, only numbers that are not equal to 7 mod 8 can be represented as a sum of three squares and we find that \( 2551^2 \! \! \mod 8 \equiv 1\). In fact, this value of 1 holds for all such square numbers whose square roots are one more than a pronic number. Thus we can say that if \(n\) is a pronic number then  \( (n+1)^2 \! \! \mod 8 \equiv 1\). 

There's nothing earth shattering in these observations. The connection with pronic numbers caught my attention after watching the video and I thought I'd follow up on it. Of course, numbers like \(2551^2=6507601\) can be expressed as a sum of three squares in probably thousands of ways and the pattern described earlier is just one these.

The 4k+1 versus 4k+3 Composite Race

It was only recently that I posted about the \(4k-1\) versus \(4k+1\) Prime Race in connection with my diurnal age of 26833 and today, aged 26835 days, I find myself involved in another race, this time the \(4k+1\) versus \(4k+3\) Composite Race. Of course \(4k+3\) is the same as \(4k-1\) but I'll stick with the former as that's the way the OEIS entry is described:

A093180

Odd composites (including 1 in the count) where the number 1 mod 4 equals the number 3 mod 4.

The sequence runs:

26835, 26851, 26855, 26859, 26867, 26871, 26875, 26883, 26887, 26895, 26899, 26923, 616771, 616775, 616779, 616795, 616831, 616835, 616839, 616847, 616853, 616857, 616861, 616865, 616869, 616875, 616881, 616885, 616889, 616893, 617013, 617017, 617021, 617025, 617031, 617035, 617259, 617263, 617267, 617329, 617361, 617367, 617373, 617377, 617381, 617385, 617391, 617395, 617399, 617405, 617409, 617415, 617419, 617423, 617427, 617433, 617437, 617441, 617445, 617451, 617457, 617461, 617465, 617475, 617511, 617515, 617519, 617525, 617529, 617535, 617541, 617545, 617549, 617553, 617557, 617561, 617565, 617569, 617573, 617577, 617583, 617683, 617687, 617701, 617705, 617711, 617715, 618523, 618527, 618531, 618535, 618539, 618543, 618595, 618599, 618603, 618607, 618611, 618615, 618639, 622795, 622799, 622803, 622807, 622811, 622817, 622821, 622825, 622829, 622833, 622837, 622841, 622845, 622989, 622993, 623005, 623109, 623113, 623117, 623121, 623125, 623129, 623133, 623137, 623141, 623145, 623149, 623153, 623157, 623161, 623165, 623169, 623175, 623179, 623183, 623187, 623191, 623195, 623199, 623203, 623207, 623213, 623217, 623301, 623307, 623311, 623315, 623319, 623325, 623331, 623335, 623339, 623345, 623349, 623357, 623361, 623365, 623369, 623373, 623377, 623381, 623395, 623399, 623405, 623409, 623413, 623425, 623429, 623435, 623441, 623445, 623449, 623453, 623457, 623461, 623465, 623469, 623473, 623593, 623597, 623601, 623605, 623609, 623613, 623805, 623809, 623813, 623817, 623821, 623825, 623829, 623833, 623837, 623843, 623847, 623895, 623899, 623903, 623907, 623911, 623915, 623919, 623931, 623937, 623941, 623945, 623951, 623955, 623961, 623967, 623971, 623975, 623981, 623987, 623993, 623997, 624001, 624005, 624011, 624015, 624019, 624023, 624027, 624039, 624043, 624051, 624055, 624059, 624063, 624091, 624095, 624101, 624105, 624109, 624113, 624117, 624123, 624127, 624131, 624137, 624143, 624147, 624153, 624157, 624161, 624167, 624171, 624175, 624179, 624183, 624187, 624235, 624239, 624245, 624249, 624255, 626923, 626927, 626933, 626937, 626941, 626945, 626951, 626957, 627043, 627047, 627051, 627055, 627351, 627357, 627361, 627365, 627369, 627373, 627381, 627387, 627435, 627439, 627443, 627447, 627453, 627457, 627461, 627465, 627469, 627473, 627477, 627485, 627489, 627495, 627499, 627503, 627507, 627543, 627723, 627727, 627731, 627737, 627741, 627745, 627793, 627801, 627805, 627809, 627815, 627819, 627823, 627827, 627831, 627835, 627839, 627845, 627849, 627853, 627857, 627863, 627867, 627871, 627875, 627879, 627883, 627887, 627891, 627895, 627899, 627905, 627909, 627915, 627975, 627979, 627983, 627987, 627991, 627995, 627999, 628003, 628007, 628011, 628017, 628185, 628401, 628405, 628409, 628413, 628417, 628421, 628923, 628927, 628931, 628935, 628943, 628947, 628951, 628955, 628959, 628963, 628967, 628971, 628977, 628981, 628985, 628989, 629025, 629061, 629065, 629069, 629073, 629077, 629245, 629265, 629269, 629273, 629277, 629313, 629317, 629321, 629325, 629329, 629333, 629337, 629345, 629349, 629355, 629359, 629363, 629367, 629403, 629407, 629419, 629423, 629427, 629433, 629437, 629441, 629445, 629469, 629473, 629477, 629481, 629487, 629511, 629517, 629521, 629525, 629529, 629533, 629725, 629729, 629733, 629745, 629751, 629755, 629759, 629763, 629775, 630715, 630735, 630741, 630745, 630749, 630753, 630757, 630761, 630765, 630769, 630773, 630777, 630781, 630785, 630789, 630793, 630805, 630809, 630813, 630817, 630821, 630843, 630847, 630851, 630855, 630859, 632223, 632563, 632567, 632571, 632575, 632579, 632583, 632587, 632611, 632615, 632619, 632679, 632719, 632723, 632727, 632731, 632735, 632739, 632779, 632783, 632787, 632791, 632795, 632799, 632803, 632807, 632811, 632817, 632821, 632825, 632829, 632833, 632837, 632883, 632887, 632891, 632895, 632901, 632905, 632909, 632915, 632919, 633003, 633007, 633011, 633017, 633021, 633025, 633029, 633033, 633081, 633085, 633089, 633095, 633099, 633103, 633107, 633111, 633115, 633119, 633123, 633127, 633131, 633137, 633141, 633145, 633149, 633155, 633159, 633165, 633169, 633173, 633177, 633181, 633185, 633191, 633195, 633201, 633205, 633309, 633313, 633385, 633389, 633393, 633397, 633409, 633413, 633417, 633421, 633425, 633431, 633435, 633439, 633443, 633447, 633453, 633457, 633465, 633489, 633493, 633561, 633565, 633573, 633577, 633581, 633587, 633591, 633595, 633615, 633619, 633631, 633635, 633639, 633643, 633647, 633669, 633673, 633677, 633681, 633685, 633689, 633693, 633697, 633701, 633705, 633709, 633713, 633717, 633721, 633725, 633729, 633733, 633737, 633743, 633747, 633759, 633763, 633783, 633787, 633795, 633879

Here is a permalink to the algorithm that I used to generate these numbers. Interestingly 26835 is the very first member of the sequence while 633879 is the last in the range up to one million. The graph of the count looks pretty much the same as the one in the Prime Race post. See Figure 1 where is the plot is from 1 to one million with one million marked as 1.0 and 0.6 representing 600,000 etc.

Figure 1: permalink