A363222 Numbers whose multiset of prime indices satisfies (maximum) - (minimum) = (length).

10, 21, 28, 42, 55, 70, 88, 91, 98, 99, 132, 165, 187, 198, 208, 220, 231, 247, 308, 312, 325, 330, 351, 363, 391, 455, 462, 468, 484, 520, 544, 550, 551, 585, 702, 713, 715, 726, 728, 770, 780, 816, 819, 833, 845, 975, 1073, 1078, 1092, 1144, 1170, 1210, 1216

Offset: 1

Author

Gus Wiseman, May 29 2023

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 terms together with their prime indices begin:
    10: {1,3}
    21: {2,4}
    28: {1,1,4}
    42: {1,2,4}
    55: {3,5}
    70: {1,3,4}
    88: {1,1,1,5}
    91: {4,6}
    98: {1,4,4}
    99: {2,2,5}
   132: {1,1,2,5}
   165: {2,3,5}
   187: {5,7}
   198: {1,2,2,5}
		

Crossrefs

The RHS is A001222.

Partitions of this type are counted by A237832.

The LHS (maximum minus minimum) is A243055.

A001221 (omega) counts distinct prime factors.

A112798 lists prime indices, sum A056239.

A360005 gives median of prime indices, distinct A360457.

Cf. A067801, A111907, A118096, A237821, A361205, A361908, A361909.

Programs

Mathematica

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

Formula

A061395(a(n)) - A055396(a(n)) = A001222(a(n)).