Prime Chains

Having written about prime chains of the Cunningham variety in my previous post, I find that today's prime number (24499) also forms the start of a prime chain. Specifically, the number is part of OEIS A023315: numbers \(n\) such that \(n\) remains prime through four iterations of function \(f(x)=5x+6\). The numbers listed are:

79, 401, 1259, 2477, 3019, 4409, 10303, 15679, 20509, 24499, 34127, 43987, 44389, 53101, 66359, 71287, 74857, 81097, 85903, 90803, 93053, 102811, 103231, 104999, 112601, 125453, 132533, 144731, 156347, 157793, 160817, 161839, 163981, 170641

Confirming this, we have: 

• 24499 x 5+6 = 122501 (prime)
• 122501 x 5+6 = 612511 (prime)
• 612511 x 5+6 = 3062561 (prime)
• 3062561 x 5+6 = 15312811 (prime)
• 15312811 x 5+6 = 76564061 (factors to 7×10937723)

Cunningham Chains Revisited

I've already discussed Cunningham chains in an earlier post but today I was 24496 days old and an analysis of the this number led me to a record-breaking example of such a chain. The factors or 24496 are 2^4×1531 and 1531 turns out to be the smallest prime that leads to a Cunningham chain of the second kind (2p-1) with a length of 5: 1531, 3061, 6121, 12241, 24481

• 2131 holds the record for length 4 (2131, 4261, 8521, 17041)
• 2 holds the record for length 3 (2, 3, 5)
• 7 holds the record for length 2 (7, 13)
• 11 holds the record for length 1 (11 only)

These numbers (11, 7, 2, 2131 and 1531) form the first five terms of OEIS A109828. The subsequent five terms are 385591, 16651, 15514861, 857095381 and 205528443121.

Taylor Series

My diurnal age today is 24491 and this number features in OEIS A077946:  

A077946

Expansion of \( \dfrac{1}{1 - x - 2x^2 - 2x^3} \)

The number arises as a coefficient of \(x\) in the Taylor Series expansion of the function at \(x=0\). The coefficients listed up to 24491 are:

1, 1, 3, 7, 15, 35, 79, 179, 407, 923, 2095, 4755, 10791, 24491

This means that the function can be expressed as:

\(1 + x + 3x^2 + 7x^3 + 15x^4 + 35x^5 + \dots \)

The Taylor Series for any continuous function \( f(x)\)at a point \(x=a\) is given by the following expression:$$f(x)=f(a)+\frac{f'(a)}{1!} (x-a) + \frac{f''(a)}{a!} (x-a)^2+ \frac{f'''(a)}{3!} (x-a)^3 + \dots$$A Maclaurin Series is a Taylor Series where \(a=0\).

Here is a permalink that will display these results using SageMathCell. I have a later post about Taylor Series that I created on April 27th 2018.

REFURBISHED on Saturday November 26th 2022

Repunits and Smith Numbers

Today's number 24490 is a member of OEIS A104167: numbers n which when multiplied by any repunit prime Rp give a Smith number. The first few such numbers are:

1540, 1720, 2170, 2440, 5590, 6040, 7930, 8344, 8470, 8920, 23590, 24490, 25228, 29080, 31528, 31780, 33544, 34390, 35380, 39970, 40870, 42490, 42598, 43480, 44380, 45955, 46270, 46810, 46990, 47908, 48790, 49960, 51490, 51625, 52345, 52570, 53290, 57070

The OEIS gives an example for 1720:

1720 is a number in the sequence because 1720*Rp is always a Smith number, where Rp is a Repunit prime. Let Rp=11, so 1720*11=18920 which is a Smith number as sum of digits of 18920 is 1+8+9+2+0=20 and sum of digits of prime factors of 18920 (i.e., 2*2*2*5*11*43) is also 20 (i.e., 2+2+2+5+1+1+4+3).

A repunit is defined by Wikipedia as a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler in his book "Recreations in the Theory of Numbers". A repunit prime is a repunit that is also a prime number.

A Smith number is defined by Wikipedia as a composite number for which, in a given base (in base 10 by default), the sum of its digits is equal to the sum of the digits in its prime factorization. For example, 378 = 2 × 3 × 3 × 3 × 7 is a Smith number since 3 + 7 + 8 = 2 + 3 + 3 + 3 + 7. In this definition the factors are treated as digits: for example, 22 factors to 2 × 11 and yields three digits: 2, 1, 1. Therefore 22 is a Smith number because 2 + 2 = 2 + 1 + 1.

The first few Smith numbers are:

4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438, 454, 483, 517,526, 535, 562, 576, 588, 627, 634, 636, 645, 648, 654, 663, 666, 690, 706, 728, 729, 762, 778, 825, 852, 861, 895, 913, 915, 922, 958, 985, 1086 … (sequence A006753 in OEIS)

Smith numbers were named by Albert Wilansky of Lehigh University. He noticed the property in the phone number (493-7775) of his brother-in-law Harold Smith:

4937775 = 3 × 5 × 5 × 65837, while 4 + 9 + 3 + 7 + 7 + 7 + 5 = 3 + 5 + 5 + 6 + 5 + 8 + 3 + 7 = 42.

In the case of 24490 and the repunit prime 11, the product is 269390 and the factorisation is 2×5×11×31×79. The sum of the digits of 269390 is 29 while the sum of the digits of its prime factorisation is 29 also.

Prime Number Chains

Now that I have regular Internet access I can resume scrutiny of my numbered days and today's number happens to be 24481 and prime. OEIS A110059 states that 24481 is the member of a sequence such that it is the smallest prime ending a complete Cunningham Chain of the second kind (2x-1) of length n. See my earlier post about Cunningham Chains. The sequence (up to n=13) is:

1. 11
2. 13
3. 5
4. 17041
5. 24481
6. 12338881
7. 1065601
8. 1985902081
9. 219416417281
10. 105230562877441
11. 1422461638625281
12. 444124661486837761
13. 3105111850422067201

Now n=5 for 24481 and so adding 1 and dividing by 2 successively yields 12241, 6121, 3061 and 1531. So the complete chain is:

1531, 3061, 6121, 12241, 24481

The number is also a prime of the form 1+2n+3n^2 (OEIS A122430) and for 24481 the value on n is 90.

This prime number also has the property that it is a number n such that n remains prime through 5 iterations of the function f(x)=3x+10 (OEIS A023338). Applying this function rule yields progressively 73453, 220369, 661117, 1983361 and 5950093 and thus we have the prime number sequence:

24481, 73453, 220369, 661117, 1983361, 5950093

It also turns out that 24481 is a lucky number. To remind myself what that means, I've attached this definition from WolframAlpha:

Write out all odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, .... The first odd number >1 is 3, so strike out every third number from the list: 1, 3, 7, 9, 13, 15, 19, .... The first odd number greater than 3 in the list is 7, so strike out every seventh number: 1, 3, 7, 9, 13, 15, 21, 25, 31, .... 

Numbers remaining after this procedure has been carried out completely are called lucky numbers. The first few are 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, ... (OEIS A000959). Many asymptotic properties of the prime numbers are shared by the lucky numbers. The asymptotic density is 1/lnN, just as the prime number theorem, and the frequency of twin primes and twin lucky numbers are similar. A version of the Goldbach conjecture also seems to hold.

So it would seem that 24481 is indeed an interesting number.