Proth Numbers

My first dedicated post about Proth numbers dates back to January of 2020. I also mention them in Sierpinski Numbers (June 2026) and Riesel and Sierpinski Numbers (April 2021). Let's review the definition of a Proth number:

A Proth number is any whole number written in the form \(k \cdot 2^n + 1\), where \(k\) is an odd number, \(n\) is a positive whole number, and \(2^{n}\) is greater than \(k\).

I've never listed all the Proth numbers in the range up to 40000 so I got Gemini to write a program to do this and here is the result (permalink):

Proth Numbers:

3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, 129, 145, 161, 177, 193, 209, 225, 241, 257, 289, 321, 353, 385, 417, 449, 481, 513, 545, 577, 609, 641, 673, 705, 737, 769, 801, 833, 865, 897, 929, 961, 993, 1025, 1089, 1153, 1217, 1281, 1345, 1409, 1473, 1537, 1601, 1665, 1729, 1793, 1857, 1921, 1985, 2049, 2113, 2177, 2241, 2305, 2369, 2433, 2497, 2561, 2625, 2689, 2753, 2817, 2881, 2945, 3009, 3073, 3137, 3201, 3265, 3329, 3393, 3457, 3521, 3585, 3649, 3713, 3777, 3841, 3905, 3969, 4033, 4097, 4225, 4353, 4481, 4609, 4737, 4865, 4993, 5121, 5249, 5377, 5505, 5633, 5761, 5889, 6017, 6145, 6273, 6401, 6529, 6657, 6785, 6913, 7041, 7169, 7297, 7425, 7553, 7681, 7809, 7937, 8065, 8193, 8321, 8449, 8577, 8705, 8833, 8961, 9089, 9217, 9345, 9473, 9601, 9729, 9857, 9985, 10113, 10241, 10369, 10497, 10625, 10753, 10881, 11009, 11137, 11265, 11393, 11521, 11649, 11777, 11905, 12033, 12161, 12289, 12417, 12545, 12673, 12801, 12929, 13057, 13185, 13313, 13441, 13569, 13697, 13825, 13953, 14081, 14209, 14337, 14465, 14593, 14721, 14849, 14977, 15105, 15233, 15361, 15489, 15617, 15745, 15873, 16001, 16129, 16257, 16385, 16641, 16897, 17153, 17409, 17665, 17921, 18177, 18433, 18689, 18945, 19201, 19457, 19713, 19969, 20225, 20481, 20737, 20993, 21249, 21505, 21761, 22017, 22273, 22529, 22785, 23041, 23297, 23553, 23809, 24065, 24321, 24577, 24833, 25089, 25345, 25601, 25857, 26113, 26369, 26625, 26881, 27137, 27393, 27649, 27905, 28161, 28417, 28673, 28929, 29185, 29441, 29697, 29953, 30209, 30465, 30721, 30977, 31233, 31489, 31745, 32001, 32257, 32513, 32769, 33025, 33281, 33537, 33793, 34049, 34305, 34561, 34817, 35073, 35329, 35585, 35841, 36097, 36353, 36609, 36865, 37121, 37377, 37633, 37889, 38145, 38401, 38657, 38913, 39169, 39425, 39681, 39937

Total Proth numbers between 1 and 40000: 283

Percentage of total range: 0.7075%

I asked Gemini whether there was any special name given to Proth-like numbers of the form \(k \cdot 2^n - 1\) and here was its response:

Yes, numbers of the form $k \cdot 2^n - 1$ (where $k$ is an odd positive integer and $2^n > k$) are formally known as Proth numbers of the second kind.

If a number in this sequence is prime, it is appropriately called a Proth prime of the second kind.

Depending on the specific value assigned to $k$, these numbers also branch into several famously named subsets in number theory: 

• Mersenne numbers: when $k = 1$, the formula simplifies to the widely studied $2^n - 1$. 

• Thâbit (or Thâbit ibn Kurrah) numbers: when $k = 3$, yielding $3 \cdot 2^n - 1$. 

• Woodall numbers: When $k$ is equal to the exponent $n$, generating numbers of the form $n \cdot 2^n - 1$. Note that this category drops the strict $2^n > k$ inequality rule.

This $k \cdot 2^n - 1$ formula is also the foundational equation for Riesel numbers. A Riesel number is not the output of the formula, but rather a specific odd integer $k$ that guarantees $k \cdot 2^n - 1$ will be composite for every possible integer $n$. This makes Riesel numbers the direct mathematical counterpart to Sierpiński numbers, which apply the exact same always-composite rule to the standard Proth sequence ($k \cdot 2^n + 1$).

Of course the Proth numbers of the second kind will be 2 less the Proth numbers listed earlier but I'll list them here for completeness (permalink):

Proth Numbers of the Second Kind:

1, 3, 7, 11, 15, 23, 31, 39, 47, 55, 63, 79, 95, 111, 127, 143, 159, 175, 191, 207, 223, 239, 255, 287, 319, 351, 383, 415, 447, 479, 511, 543, 575, 607, 639, 671, 703, 735, 767, 799, 831, 863, 895, 927, 959, 991, 1023, 1087, 1151, 1215, 1279, 1343, 1407, 1471, 1535, 1599, 1663, 1727, 1791, 1855, 1919, 1983, 2047, 2111, 2175, 2239, 2303, 2367, 2431, 2495, 2559, 2623, 2687, 2751, 2815, 2879, 2943, 3007, 3071, 3135, 3199, 3263, 3327, 3391, 3455, 3519, 3583, 3647, 3711, 3775, 3839, 3903, 3967, 4031, 4095, 4223, 4351, 4479, 4607, 4735, 4863, 4991, 5119, 5247, 5375, 5503, 5631, 5759, 5887, 6015, 6143, 6271, 6399, 6527, 6655, 6783, 6911, 7039, 7167, 7295, 7423, 7551, 7679, 7807, 7935, 8063, 8191, 8319, 8447, 8575, 8703, 8831, 8959, 9087, 9215, 9343, 9471, 9599, 9727, 9855, 9983, 10111, 10239, 10367, 10495, 10623, 10751, 10879, 11007, 11135, 11263, 11391, 11519, 11647, 11775, 11903, 12031, 12159, 12287, 12415, 12543, 12671, 12799, 12927, 13055, 13183, 13311, 13439, 13567, 13695, 13823, 13951, 14079, 14207, 14335, 14463, 14591, 14719, 14847, 14975, 15103, 15231, 15359, 15487, 15615, 15743, 15871, 15999, 16127, 16255, 16383, 16639, 16895, 17151, 17407, 17663, 17919, 18175, 18431, 18687, 18943, 19199, 19455, 19711, 19967, 20223, 20479, 20735, 20991, 21247, 21503, 21759, 22015, 22271, 22527, 22783, 23039, 23295, 23551, 23807, 24063, 24319, 24575, 24831, 25087, 25343, 25599, 25855, 26111, 26367, 26623, 26879, 27135, 27391, 27647, 27903, 28159, 28415, 28671, 28927, 29183, 29439, 29695, 29951, 30207, 30463, 30719, 30975, 31231, 31487, 31743, 31999, 32255, 32511, 32767, 33023, 33279, 33535, 33791, 34047, 34303, 34559, 34815, 35071, 35327, 35583, 35839, 36095, 36351, 36607, 36863, 37119, 37375, 37631, 37887, 38143, 38399, 38655, 38911, 39167, 39423, 39679, 39935

Total Proth numbers of the second kind between 1 and 40000: 283

Percentage of total range: 0.7075%