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A337479 Primitive elements of A337386: numbers k for which sigma(A003961(k)) >= 2*A003961(k), but none of the proper divisors of k satisfy the same condition.

%I A337479 #24 Oct 06 2020 02:37:18
%S A337479 120,180,300,420,504,630,660,780,924,990,1020,1050,1092,1140,1170,
%T A337479 1380,1470,1650,1740,1860,2220,2310,2460,2580,2730,2820,2856,3168,
%U A337479 3180,3192,3432,3540,3570,3660,3864,3990,4020,4260,4284,4290,4380,4488,4590,4740,4752,4788,4830
%N A337479 Primitive elements of A337386: numbers k for which sigma(A003961(k)) >= 2*A003961(k), but none of the proper divisors of k satisfy the same condition.
%C A337479 Equivalently, numbers k such that A003961(k) is in A006039, i.e., numbers that become an (odd) primitive nondeficient number when prime-shifted once.
%C A337479 Conjecture: every positive integer is either a (possibly trivial) multiple of a sequence term or divides infinitely many terms of this sequence. - _Peter Munn_, Sep 24 2020
%H A337479 Antti Karttunen, <a href="/A337479/b337479.txt">Table of n, a(n) for n = 1..16350; terms less than 2^25</a>
%H A337479 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A337479 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A337479 For all n >= 1, A337690(a(n)) = A337539(n).
%t A337479 Block[{f}, f[1] = 1; f[n_] := Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}]; Select[Range[5000], And[DivisorSigma[1, Last[#]] >= 2 Last[#], NoneTrue[Most[#], DivisorSigma[1, #] >= 2 # &]] &@ Map[f, Divisors@ #] &] ] (* _Michael De Vlieger_, Oct 05 2020 *)
%o A337479 (PARI)
%o A337479 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A337479 isA337386(n) = { my(x=A003961(n)); (sigma(x)>=2*x); };
%o A337479 isA337479(n) = (1==sumdiv(n,d,isA337386(d)));
%Y A337479 Cf. A000203, A003961, A003973, A006039, A337386, A337539, A337690.
%Y A337479 Cf. also A071395, A091191, A337372, A337543.
%K A337479 nonn
%O A337479 1,1
%A A337479 _Antti Karttunen_, Sep 01 2020