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Number of divisors of n with the same sum of prime indices as their quotient. Central column of A321144, taking gaps as 0's.
13

%I #12 Jan 20 2025 22:52:24

%S 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 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 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 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 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 Antti Karttunen, <a href="/A357879/b357879.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.

%F a(n) = Sum_{d|n} [A056239(d) = A056239(n/d)], where [ ] is the Iverson bracket. - _Antti Karttunen_, Jan 20 2025

%e The a(3600) = 5 divisors, their prime indices, and the prime indices of their quotients:

%e 45: {2,2,3} * {1,1,1,1,3}

%e 50: {1,3,3} * {1,1,1,2,2}

%e 60: {1,1,2,3} * {1,1,2,3}

%e 72: {1,1,1,2,2} * {1,3,3}

%e 80: {1,1,1,1,3} * {2,2,3}

%t sumprix[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]];

%t Table[Length[Select[Divisors[n],sumprix[#]==sumprix[n]/2&]],{n,100}]

%o (PARI)

%o A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1])));

%o A357879(n) = sumdiv(n,d, A056239(d)==A056239(n/d)); \\ _Antti Karttunen_, Jan 20 2025

%Y Positions of nonzero terms are A357976, counted by A002219.

%Y A001222 counts prime factors, distinct A001221.

%Y A056239 adds up prime indices, row sums of A112798.

%Y Cf. A033879, A033880, A064914, A181819, A213074, A235130, A237258, A276107, A300061, A321144, A357975.

%K nonn

%O 1,12

%A _Gus Wiseman_, Oct 27 2022

%E Data section extended to a(108) by _Antti Karttunen_, Jan 20 2025