A341610 Nonprimitive terms of A246282: numbers k that have more than one divisor d|k such that A003961(d) > 2*d.

8, 12, 16, 18, 20, 24, 27, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 124, 126, 128, 130, 132, 135, 136, 138, 140, 144, 147, 148, 150, 152, 153, 154, 156, 160, 162, 164, 165

Offset: 1

Author

Antti Karttunen, Feb 22 2021

Keywords

• nonn

Comments

Numbers k for which A337345(k) > 1, or equally, for which A337346(k) > 0.

Obviously A337346(n) = 0 for any noncomposite and for any semiprime, thus this is a subsequence of A033942. The first term of A033942 not present here is 125, as A337345(125) = 1.

Empirically checked: in range 1 .. 2^31, all abundant numbers are found in this sequence. For proving this, we should concentrate only on checking A091191, as the set A005101 \ A091191 (non-primitive abundant numbers) is certainly included, as for any divisor d for which sigma(d) > 2*d (or even sigma(d) >= 2*d), we also have A003961(d) > 2*d.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A337345.

Positions of nonzero terms in A337346.

Setwise difference A246282 \ A337372.

Conjectured subsequence: A005101. (Clearly abundant numbers are all in A246282).

Cf. A341611, A341614.

Differs from its subsequence A033942 for the first time at n=52, with a(52) = 126, while A033942(52) = 125.

Programs

Mathematica

Block[{nn = 165, s}, s = {1}~Join~Array[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] &, nn - 1, 2]; Select[Range[nn], 1 < DivisorSum[#, 1 &, s[[#]] > 2 # &] &]] (* Michael De Vlieger, Feb 22 2021 *)

PARI

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA341610(n) = (1A003961(d)>(d+d)));