A355524 Minimal difference between adjacent prime indices of n > 1, or 0 if n is prime.

0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 2, 4, 0, 0, 0, 5, 0, 0, 0, 1, 0, 0, 3, 6, 1, 0, 0, 7, 4, 0, 0, 1, 0, 0, 0, 8, 0, 0, 0, 0, 5, 0, 0, 0, 2, 0, 6, 9, 0, 0, 0, 10, 0, 0, 3, 1, 0, 0, 7, 1, 0, 0, 0, 11, 0, 0, 1, 1, 0, 0, 0, 12, 0, 0, 4, 13, 8

Offset: 2

Author

Gus Wiseman, Jul 10 2022

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.

Examples

			The prime indices of 9842 are {1,4,8,12}, with differences (3,4,4), so a(9842) = 3.
		

Links

• Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs

Crossrefs found in the link are not repeated here.

Positions of first appearances are A077017 w/o the first term.

Positions of terms > 0 are A120944.

Positions of zeros are A130091.

Triangle A238353 counts m such that A056239(m) = n and a(m) = k.

For maximal difference we have A286470 or A355526.

Positions of terms > 1 are A325161.

If singletons (k) have minimal difference k we get A355525.

Positions of 1's are A355527.

Prepending 0 to the prime indices gives A355528.

A115720 and A115994 count partitions by their Durfee square.

A287352, A355533, A355534, A355536 list the differences of prime indices.

Cf. A000005, A066312, A238354, A238710, A286469, A351294, A352822, A355530, A355531, A355532.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[PrimeQ[n],0,Min@@Differences[primeMS[n]]],{n,2,100}]