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

%I A341610 #17 Mar 06 2021 01:34:03
%S A341610 8,12,16,18,20,24,27,28,30,32,36,40,42,44,45,48,50,52,54,56,60,63,64,
%T A341610 66,68,70,72,75,76,78,80,81,84,88,90,92,96,98,99,100,102,104,105,108,
%U A341610 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
%N A341610 Nonprimitive terms of A246282: numbers k that have more than one divisor d|k such that A003961(d) > 2*d.
%C A341610 Numbers k for which A337345(k) > 1, or equally, for which A337346(k) > 0.
%C A341610 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.
%C A341610 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.
%H A341610 Antti Karttunen, <a href="/A341610/b341610.txt">Table of n, a(n) for n = 1..10000</a>
%H A341610 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A341610 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%t A341610 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 *)
%o A341610 (PARI)
%o A341610 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A341610 isA341610(n) = (1<sumdiv(n,d,A003961(d)>(d+d)));
%Y A341610 Cf. A337345.
%Y A341610 Positions of nonzero terms in A337346.
%Y A341610 Setwise difference A246282 \ A337372.
%Y A341610 Conjectured subsequence: A005101. (Clearly abundant numbers are all in A246282).
%Y A341610 Cf. A341611, A341614.
%Y A341610 Differs from its subsequence A033942 for the first time at n=52, with a(52) = 126, while A033942(52) = 125.
%K A341610 nonn
%O A341610 1,1
%A A341610 _Antti Karttunen_, Feb 22 2021