A318573 Numerator of the reciprocal sum of the integer partition with Heinz number n.

0, 1, 1, 2, 1, 3, 1, 3, 1, 4, 1, 5, 1, 5, 5, 4, 1, 2, 1, 7, 3, 6, 1, 7, 2, 7, 3, 9, 1, 11, 1, 5, 7, 8, 7, 3, 1, 9, 2, 10, 1, 7, 1, 11, 4, 10, 1, 9, 1, 5, 9, 13, 1, 5, 8, 13, 5, 11, 1, 17, 1, 12, 5, 6, 1, 17, 1, 15, 11, 19, 1, 4, 1, 13, 7, 17, 9, 5, 1, 13, 2, 14, 1, 11, 10, 15, 3, 16, 1, 7, 5, 19, 13, 16, 11, 11, 1, 3

Offset: 1

Author

Gus Wiseman, Aug 29 2018

Keywords

• nonn
• frac

Comments

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
• Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to Heinz numbers

Crossrefs

Positions of 1's are A316857.

Cf. A051908, A056239, A058360, A112798, A289506, A289507, A296150, A316854, A316855, A316856, A318574, A325704.

Programs

Mathematica

Table[Sum[pr[[2]]/PrimePi[pr[[1]]],{pr,If[n==1,{},FactorInteger[n]]}],{n,100}]//Numerator

PARI

A318573(n) = { my(f=factor(n)); numerator(sum(i=1,#f~,f[i, 2]/primepi(f[i, 1]))); }; \\ Antti Karttunen, Nov 17 2019

Formula

If n = Product prime(x_i)^y_i is the prime factorization of n, then a(n) is the numerator of Sum y_i/x_i.

Extensions

More terms from Antti Karttunen, Nov 17 2019