A337487 De Polignac numbers k > 1 such that k - 2^m is a de Polignac number for every 1 < 2^m < k.

1117175145, 2544265305, 3147056235, 3366991695, 3472109835, 3621922845, 3861518805, 4447794915, 4848148485, 5415281745, 5693877405, 7525056375, 7602256605, 9055691835, 9217432215, 13431856995, 16819230075, 19373391165, 21468020835, 24358769685, 27002844795, 30252463305, 33359739795

Offset: 1

Author

Thomas Ordowski, Aug 29 2020

Keywords

• nonn

Comments

Odd integers k > 3 that are not of the form p + 2^m + 2^n with m,n >= 0, where p is a prime.

These are de Polignac numbers k > 1 in A156695. Numbers k in A156695 such that k - 2 is composite.

Problem: are there infinitely many such numbers?

Links

• Amiram Eldar, Table of n, a(n) for n = 1..204 (terms below 10^12, calculated using the b-file at A156695)

Crossrefs

An intersection of A006285 > 1 and A156695 (with m,n >= 1).

Programs

Mathematica

A156695 = Cases[Import["https://oeis.org/A156695/b156695.txt", "Table"], {, }][[;; , 2]]; dePolQ[n_] := n > 3 && AllTrue[n - 2^Range[Floor[Log[2, n]]], !PrimeQ[#] &]; Select[A156695, dePolQ] (* Amiram Eldar, Aug 29 2020 *)