A339216 -id:A339216 - cp's OEIS frontend

A374101 Numbers k such that k and k+2 are both self numbers (A003052).

1, 3, 5, 7, 108, 209, 310, 411, 512, 613, 714, 815, 916, 1109, 1210, 1311, 1412, 1513, 1614, 1715, 1816, 1917, 2110, 2211, 2312, 2413, 2514, 2615, 2716, 2817, 2918, 3111, 3212, 3313, 3414, 3515, 3616, 3717, 3818, 3919, 4112, 4213, 4314, 4415, 4516, 4617, 4718

Offset: 1

Amiram Eldar, Jun 28 2024

The least difference between consecutive self numbers is 2 (see Griffin N. Macris's proof at A010061 that may be adapted to other bases).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A003052.

Cf. A010061, A339216 (binary analog).

seq[max_] := Module[{c = Complement[Range[max], Table[n + DigitSum[n], {n, 1, max}]], d, ind}, d = Differences[c]; ind = Position[d, 2] // Flatten; c[[ind]]]; seq[5000]

A374103 Numbers k such that k and k+2 are both prime binary self (or Colombian) numbers (A374102).

Original entry on oeis.org

311, 599, 641, 881, 1319, 1697, 1721, 2657, 2969, 3257, 4019, 4127, 4337, 4721, 5009, 6449, 6569, 6689, 6761, 7547, 9041, 9239, 10457, 10529, 11171, 11699, 11939, 13691, 16229, 19379, 20147, 20357, 20477, 22697, 23057, 24977, 25169, 26249, 26681, 26729, 27059

Offset: 1

Amiram Eldar, Jun 28 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A001359, A339216 and A374102.

Cf. A010061, A006378.

seq[max_] := Module[{c = Complement[Prime[Range[max]], Table[n + DigitCount[n, 2, 1], {n, 0, Prime[max]}]], d, ind}, d = Differences[c]; ind = Position[d, 2] // Flatten; c[[ind]]]; seq[3000]

A339239 Binary self numbers (A010061) with a record gap to the next binary self number.

Original entry on oeis.org

1, 6, 63, 250, 131070, 1048574, 33554426, 17179869180

Offset: 1

Amiram Eldar, Nov 28 2020

The corresponding gaps are 3, 7, 8, 20, 21, 24, 37, 42, ...
a(9) <= 288230376151711738.
Apparently, the records gaps occur for pairs of consecutive binary self numbers with a power of 2 between them. If this is generally true, then the next terms are 288230376151711738, 147573952589676412923, 37778931862957161709564, 10633823966279326983230456482242756602, 5444517870735015415413993718908291383294, 43556142965880123323311949751266331066367, ..., with the corresponding gaps 70, 77, 83, 135, 136, 137, ...

			The first 4 binary self numbers are 1, 4, 6 and 13. The gaps between them are 3, 2 and 7. The record gaps are 3 and 7, and the corresponding terms are 1 and 6.

Cf. A010061, A339216.

s[n_] := n + DigitCount[n, 2, 1]; selfQ[n_] := AllTrue[Range[n, n - Floor@Log2[n], -1], s[#] != n &]; dm = 0; seq = {}; n1 = 1; Do[If[selfQ[n], d = n - n1; If[d > dm, dm = d; AppendTo[seq, n1]]; n1 = n], {n, 2, 150000}]; seq

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A374101 Numbers k such that k and k+2 are both self numbers (A003052).

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A374103 Numbers k such that k and k+2 are both prime binary self (or Colombian) numbers (A374102).

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Mathematica

A339239 Binary self numbers (A010061) with a record gap to the next binary self number.

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