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A357879 Number of divisors of n with the same sum of prime indices as their quotient. Central column of A321144, taking gaps as 0's.

%I A357879 #12 Jan 20 2025 22:52:24
%S A357879 1,0,0,1,0,0,0,0,1,0,0,2,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,2,0,0,0,0,
%T A357879 0,1,0,0,0,2,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,1,0,0,0,0,
%U A357879 0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2
%N A357879 Number of divisors of n with the same sum of prime indices as their quotient. Central column of A321144, taking gaps as 0's.
%C A357879 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%H A357879 Antti Karttunen, <a href="/A357879/b357879.txt">Table of n, a(n) for n = 1..65537</a>
%H A357879 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%F A357879 a(n) = Sum_{d|n} [A056239(d) = A056239(n/d)], where [ ] is the Iverson bracket. - _Antti Karttunen_, Jan 20 2025
%e A357879 The a(3600) = 5 divisors, their prime indices, and the prime indices of their quotients:
%e A357879   45: {2,2,3} * {1,1,1,1,3}
%e A357879   50: {1,3,3} * {1,1,1,2,2}
%e A357879   60: {1,1,2,3} * {1,1,2,3}
%e A357879   72: {1,1,1,2,2} * {1,3,3}
%e A357879   80: {1,1,1,1,3} * {2,2,3}
%t A357879 sumprix[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]];
%t A357879 Table[Length[Select[Divisors[n],sumprix[#]==sumprix[n]/2&]],{n,100}]
%o A357879 (PARI)
%o A357879 A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1])));
%o A357879 A357879(n) = sumdiv(n,d, A056239(d)==A056239(n/d)); \\ _Antti Karttunen_, Jan 20 2025
%Y A357879 Positions of nonzero terms are A357976, counted by A002219.
%Y A357879 A001222 counts prime factors, distinct A001221.
%Y A357879 A056239 adds up prime indices, row sums of A112798.
%Y A357879 Cf. A033879, A033880, A064914, A181819, A213074, A235130, A237258, A276107, A300061, A321144, A357975.
%K A357879 nonn
%O A357879 1,12
%A A357879 _Gus Wiseman_, Oct 27 2022
%E A357879 Data section extended to a(108) by _Antti Karttunen_, Jan 20 2025