A353832 Heinz number of the multiset of run-sums of the prime indices of n.

1, 2, 3, 3, 5, 6, 7, 5, 7, 10, 11, 9, 13, 14, 15, 7, 17, 14, 19, 15, 21, 22, 23, 15, 13, 26, 13, 21, 29, 30, 31, 11, 33, 34, 35, 21, 37, 38, 39, 25, 41, 42, 43, 33, 35, 46, 47, 21, 19, 26, 51, 39, 53, 26, 55, 35, 57, 58, 59, 45, 61, 62, 49, 13, 65, 66, 67, 51, 69, 70, 71, 35, 73, 74, 39, 57, 77, 78, 79, 35, 19

Offset: 1

Author

Gus Wiseman, May 23 2022

Keywords

• nonn

Comments

The sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. For example, the runs of (2,2,1,1,1,3,2,2) are (2,2), (1,1,1), (3), (2,2), with sums (4,3,3,4).

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

This sequence represents the transformation f(P) described by Kimberling at A237685.

Examples

			The prime indices of 1260 are {1,1,2,2,3,4}, with run-sums (2,4,3,4), and the multiset {2,3,4,4} has Heinz number 735, so a(1260) = 735.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)
• Index entries for sequences related to prime indices in the factorization of n.

Crossrefs

The number of distinct prime factors of a(n) is A353835, weak A353861.

The version for compositions is A353847, listed A353932.

The greatest prime factor of a(n) has index A353862, least A353931.

A001222 counts prime factors, distinct A001221.

A056239 adds up prime indices, row sums of A112798 and A296150.

A300273 ranks collapsible partitions, counted by A275870.

A353833 ranks partitions with all equal run-sums, counted by A304442.

A353838 ranks partitions with all distinct run-sums, counted by A353837.

A353840-A353846 pertain to partition run-sum trajectory.

A353851 counts compositions w/ all equal run-sums, ranked by A353848.

A353864 counts rucksack partitions, ranked by A353866.

A353865 counts perfect rucksack partitions, ranked by A353867.

Cf. A005811, A047966, A071625, A073093, A181819, A182850, A182857, A304660, A323014, A353834, A353839, A353841 (1 + iterations needed to reach a squarefree number).

Programs

Mathematica

Table[Times@@Prime/@Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>PrimePi[p]*k],{n,100}]

PARI

pis_to_runs(n) = { my(runs=List([]), f=factor(n)); for(i=1,#f~,while(f[i,2], listput(runs,primepi(f[i,1])); f[i,2]--)); (runs); };
A353832(n) = if(1==n,n,my(pruns = pis_to_runs(n), m=1, runsum=pruns[1]); for(i=2,#pruns,if(pruns[i] == pruns[i-1], runsum += pruns[i], m *= prime(runsum); runsum = pruns[i])); (m*prime(runsum))); \\ Antti Karttunen, Jan 20 2025

Formula

A001222(a(n)) = A001221(n).

A001221(a(n)) = A353835(n).

A061395(a(n)) = A353862(n).

Extensions

More terms from Antti Karttunen, Jan 20 2025