A378658 - OEIS

%I #26 Dec 12 2024 15:13:22

%S 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 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 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 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 For all n, a(n) > 1. This follows from a proof given in A337372. See also A378662.

%C Among the initial 10 million terms, there are 7835064 2's.

%H Antti Karttunen, <a href="/A378658/b378658.txt">Table of n, a(n) for n = 1..100000</a>

%F {A337345(k) for k such that A080224(k) = 1}.

%F a(n) = 1+A378662(A091191(n)).

%t s = Select[Range[2^11], DivisorSigma[1, #] > 2 # && Times @@ Boole@ Map[DivisorSigma[1, #] <= 2 # &, Most@ Divisors@ #] == 1 &];

%t 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 (PARI)

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

%o A337345(n) = sumdiv(n,d,A003961(d)>(2*d));

%o is_A091191(n) = if(sigma(n)<=2*n, 0, fordiv(n,d,if(d<n && sigma(d)>2*d, return(0))); (1));

%o k=0; n=0; while(k<100000, n++; if(is_A091191(n), k++; print1(A337345(n), ", "); write("b378658.txt", k, " ", A337345(n))));

%Y Cf. A000203, A003961, A080224, A091191, A337345, A337372, A378662.

%K nonn,look

%O 1,1

%A _Antti Karttunen_, Dec 05 2024