A316437 Take the integer partition with Heinz number n, divide all parts by the GCD of the parts, then take the Heinz number of the resulting partition.

1, 2, 2, 4, 2, 6, 2, 8, 4, 10, 2, 12, 2, 14, 15, 16, 2, 18, 2, 20, 6, 22, 2, 24, 4, 26, 8, 28, 2, 30, 2, 32, 33, 34, 35, 36, 2, 38, 10, 40, 2, 42, 2, 44, 45, 46, 2, 48, 4, 50, 51, 52, 2, 54, 55, 56, 14, 58, 2, 60, 2, 62, 12, 64, 6, 66, 2, 68, 69, 70, 2, 72, 2, 74, 75, 76, 77, 78, 2, 80, 16, 82, 2, 84, 85, 86, 22, 88, 2, 90, 15

Offset: 1

Author

Gus Wiseman, Jul 03 2018

Keywords

• nonn

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

This sequence is idempotent, meaning a(a(n)) = a(n) for all n.

All terms belong to A289509.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Wikipedia, Idempotence
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000720, A056239, A289508, A289509, A290103, A296150, A316430, A316431, A316432, A316438.

Programs

Mathematica

f[n_]:=If[n==1,1,With[{pms=Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]},Times@@Prime/@(pms/GCD@@pms)]];
Table[f[n],{n,100}]

PARI

A316437(n) = if(1==n,1,my(f = factor(n), pis = apply(p -> primepi(p), f[, 1]~), es = f[, 2]~, g = gcd(pis)); factorback(vector(#f~, k, prime(pis[k]/g)^es[k]))); \\ Antti Karttunen, Aug 06 2018

Extensions

More terms from Antti Karttunen, Aug 06 2018