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

1,2

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.

Links

Table of n, a(n) for n=1..69.

Formula

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

Example

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.

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}]

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.

Sequence in context: A327498 A111615 A358668 * A387717 A348401 A324932

Adjacent sequences: A385213 A385214 A385215 * A385217 A385218 A385219

Keyword

nonn

Author

Gus Wiseman, Jul 05 2025

Status

approved