A387411 Numbers k such that the odd part of (1+k) divides (1+A003961(k)), where A003961 is fully multiplicative with a(p) = nextprime(p).

1, 3, 4, 7, 10, 15, 18, 23, 27, 31, 47, 57, 63, 95, 119, 127, 255, 348, 383, 415, 447, 511, 575, 695, 767, 959, 1023, 1054, 1071, 1535, 1919, 2047, 2626, 3471, 3839, 4095, 4415, 6815, 8191, 8703, 13823, 16383, 31743, 32767, 39895, 42367, 48127, 64607, 65535, 68727, 74495, 81919, 92159, 98303, 113535, 124671, 131071

Offset: 1

Author

Antti Karttunen, Sep 01 2025

Keywords

• nonn
• new

Comments

Like in many sequences of this type, the criterion seems to select for numbers with a long tail of trailing 1-bits. Terms that are not in A004767 are: 1, 4, 10, 18, 57, 348, 1054, 2626, 675348, 1869741, 12371554, 14070141, 1158654378, 1673018314, etc.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..159
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000265, A003961, A004767.

Subsequences: A000225, A348514 (which is also a subsequence of A387414).

For similar sequences, see A336700, A387410, A387415, A387410, A387418, A387419.

Programs

Mathematica

a3961[x_] := Apply[Times, Prime[PrimePi[#1] + 1]^#2 & @@@ FactorInteger[x]] - Boole[x == 1];
a265[x_] := x/2^IntegerExponent[x, 2];
Select[Range[2^17], Divisible[1 + a3961[#], a265[# + 1] ] &] (* Michael De Vlieger, Sep 01 2025 *)

PARI

A000265(n) = (n>>valuation(n,2));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA387411(n) = !((1+A003961(n))%A000265(1+n));