A375400 Heinz number of the multiset of minima of maximal anti-runs in the weakly increasing prime indices of n.

1, 2, 3, 4, 5, 2, 7, 8, 9, 2, 11, 4, 13, 2, 3, 16, 17, 6, 19, 4, 3, 2, 23, 8, 25, 2, 27, 4, 29, 2, 31, 32, 3, 2, 5, 12, 37, 2, 3, 8, 41, 2, 43, 4, 9, 2, 47, 16, 49, 10, 3, 4, 53, 18, 5, 8, 3, 2, 59, 4, 61, 2, 9, 64, 5, 2, 67, 4, 3, 2, 71, 24, 73, 2, 15, 4, 7

Offset: 1

Author

Gus Wiseman, Aug 17 2024

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.

An anti-run is a sequence with no adjacent equal parts. The minima of maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the least term of each.

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 540 are (1,1,2,2,2,3), with maximal anti-runs ((1),(1,2),(2),(2,3)), with minima (1,1,2,2), with Heinz number 36, so a(540) = 36.
The prime indices of 990 are (1,2,2,3,5), with maximal anti-runs ((1,2),(2,3,5)), with minima (1,2), with Heinz number 6, so a(990) = 6.
		

Crossrefs

bigomega is A001222(a(n)) = A375136(n).

Least prime factor is A020639(a(n)) = A020639(n).

Least prime index is A055396(a(n)) = A055396(n).

Heinz weights are A056239(a(n)) = A374706(n).

The greatest prime index A061395(a(n)) is the maximum of row n of A375128.

Firsts for omega (except first term) are half A061742.

Prime indices A112798(a(n)) are row n of A375128.

Positions of prime-powers are A375396, counted by A115029.

Positions of squarefree numbers are A375398, counted by A375134.

A000041 counts integer partitions, strict A000009.

A027748 lists distinct prime factors, sum A008472.

A304038 lists distinct prime indices, sum A066328.

A number's prime factors (A027746, reverse A238689) have sum A001414, min A020639, max A006530.

A number's prime indices (A112798, reverse A296150) have sum A056239, min A055396, max A061395.

Both have length A001222, distinct A001221.

Cf. A000040, A046660, A124768, A320324, A358836, A374683, A374700, A375133.

Programs

Mathematica

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