A371733 Maximal length of a factorization of n into factors > 1 all having the same sum of prime indices.

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 5, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4

Offset: 1

Author

Gus Wiseman, Apr 13 2024

Keywords

• nonn

Comments

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. Sum of prime indices is given by A056239.

Factorizations into factors all having the same sum of prime indices are counted by A321455.

Examples

			The factorizations of 588 of this type are (7*7*12), (21*28), (588), so a(588) = 3.
The factorizations of 900 of this type are (5*5*6*6), (9*10*10), (25*36), (30*30), (900), so a(900) = 4.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.
• Index entries for sequences related to prime indices in the factorization of n.

Crossrefs

Positions of 1's are A321453, counted by A321451.

Positions of terms > 1 are A321454, counted by A321452.

Factorizations of this type are counted by A321455, different sums A321469.

For different sums instead of same sums we have A371734.

For set partitions of binary indices we have A371735.

A001055 counts factorizations.

A002219 (aerated) counts biquanimous partitions, ranks A357976.

A112798 lists prime indices, reverse A296150, length A001222, sum A056239.

A321142 and A371794 count non-biquanimous strict partitions.

A371789 counts non-quanimous sets, differences A371790.

A371796 counts quanimous sets, differences A371797.

Cf. A035470, A279787, A305551, A322794, A326515, A326518, A326534, A336137, A371783, A371791.

Programs

Mathematica

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
Table[Max[Length/@Select[facs[n],SameQ@@hwt/@#&]],{n,100}]

PARI

A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1])));
all_have_same_sum_of_pis(facs) = if(!#facs, 1, (#Set(apply(A056239,facs)) == 1));
A371733(n, m=n, facs=List([])) = if(1==n, if(all_have_same_sum_of_pis(facs),#facs,0), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s = max(s, A371733(n/d, d, newfacs)))); (s)); \\ Antti Karttunen, Jan 20 2025

Extensions

Data section extended to a(108) by Antti Karttunen, Jan 20 2025