A385216 Greatest Heinz number of a sparse submultiset of the prime indices of n, where a multiset is sparse iff 1 is not a first difference.

1, 2, 3, 4, 5, 3, 7, 8, 9, 10, 11, 4, 13, 14, 5, 16, 17, 9, 19, 20, 21, 22, 23, 8, 25, 26, 27, 28, 29, 10, 31, 32, 33, 34, 7, 9, 37, 38, 39, 40, 41, 21, 43, 44, 9, 46, 47, 16, 49, 50, 51, 52, 53, 27, 55, 56, 57, 58, 59, 20, 61, 62, 63, 64, 65, 33, 67, 68, 69

Offset: 1

Author

Gus Wiseman, Jul 05 2025

Keywords

• nonn

Comments

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.

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.

Examples

			The prime indices of 12 are {1,1,2}, with sparse submultisets {{},{1},{2},{1,1}}, with Heinz numbers {1,2,3,4}, so a(12) = 4.
The prime indices of 36 are {1,1,2,2}, with sparse submultisets {{},{1},{2},{1,1},{2,2}}, with Heinz numbers {1,2,3,4,9}, so a(36) = 9.
The prime indices of 462 are {1,2,4,5}, with sparse submultisets {{},{1},{2},{4},{5},{1,4},{2,4},{1,5},{2,5}}, with Heinz numbers {1,2,3,7,11,14,21,22,33}, so a(462) = 33.
		

Crossrefs

Sparse submultisets are counted by A166469, maximal A385215.

The union is A319630 (Heinz numbers of sparse multisets), complement A104210.

For binary instead of prime indices we have A374356, see A245564, A384883.

A000005 counts divisors (or submultisets of prime indices).

A001222 counts prime factors, distinct A001221.

A051903 gives greatest prime exponent, least A051904, counted by A091602.

A055396 gives least prime index, greatest A061395, counted by A008284.

A056239 adds up prime indices, row sums of A112798.

A212166 ranks partitions with max multiplicity = length, counted by A239964.

A381542 ranks partitions with max part = max multiplicity, counted by A240312.

Cf. A000720, A005117, A122111, A130091, A268193, A316476.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Max@@Select[Divisors[n],FreeQ[Differences[prix[#]],1]&],{n,100}]

Formula

a(n) = n iff n belongs to A319630.