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A378658 a(n) = A337345(A091191(n)), where A337345 is the number of divisors d of n for which A003961(d) > 2*d, and A091191 lists the primitive abundant numbers.

%I A378658 #26 Dec 12 2024 15:13:22
%S A378658 3,3,3,4,4,5,2,4,3,4,2,4,3,3,2,2,6,2,2,2,6,2,6,6,2,2,7,2,6,2,2,2,6,2,
%T A378658 6,2,6,2,5,5,2,2,2,2,6,5,2,2,4,2,2,6,2,2,5,6,2,2,2,12,2,8,2,6,2,2,2,2,
%U A378658 6,2,2,8,2,6,2,8,6,2,2,2,8,2,6,2,2,6,8,2,2,13,2,2,2,6,2,2,8,2,6,2,2,2,4,6
%N A378658 a(n) = A337345(A091191(n)), where A337345 is the number of divisors d of n for which A003961(d) > 2*d, and A091191 lists the primitive abundant numbers.
%C A378658 For all n, a(n) > 1. This follows from a proof given in A337372. See also A378662.
%C A378658 Among the initial 10 million terms, there are 7835064 2's.
%H A378658 Antti Karttunen, <a href="/A378658/b378658.txt">Table of n, a(n) for n = 1..100000</a>
%F A378658 {A337345(k) for k such that A080224(k) = 1}.
%F A378658 a(n) = 1+A378662(A091191(n)).
%t A378658 s = Select[Range[2^11], DivisorSigma[1, #] > 2 # && Times @@ Boole@ Map[DivisorSigma[1, #] <= 2 # &, Most@ Divisors@ #] == 1 &];
%t A378658 Map[Length@ Select[Divisors[#], 2 # < (Times @@ Map[Power @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi[p] + 1], e}] - Boole[# == 1]) &] &, s] (* _Michael De Vlieger_, Dec 06 2024 *)
%o A378658 (PARI)
%o A378658 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A378658 A337345(n) = sumdiv(n,d,A003961(d)>(2*d));
%o A378658 is_A091191(n) = if(sigma(n)<=2*n, 0, fordiv(n,d,if(d<n && sigma(d)>2*d, return(0))); (1));
%o A378658 k=0; n=0; while(k<100000, n++; if(is_A091191(n), k++; print1(A337345(n), ", "); write("b378658.txt", k, " ", A337345(n))));
%Y A378658 Cf. A000203, A003961, A080224, A091191, A337345, A337372, A378662.
%K A378658 nonn,look
%O A378658 1,1
%A A378658 _Antti Karttunen_, Dec 05 2024