A340606 Numbers whose prime indices (A112798) are all divisors of the number of prime factors (A001222).

1, 2, 4, 6, 8, 9, 16, 20, 24, 32, 36, 50, 54, 56, 64, 81, 84, 96, 125, 126, 128, 144, 160, 176, 189, 196, 216, 240, 256, 294, 324, 360, 384, 400, 416, 441, 486, 512, 540, 576, 600, 624, 686, 729, 810, 864, 896, 900, 936, 968, 1000, 1024, 1029, 1040, 1088, 1215

Offset: 1

Author

Gus Wiseman, Jan 24 2021

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 sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
   9: {2,2}
  16: {1,1,1,1}
  20: {1,1,3}
  24: {1,1,1,2}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  50: {1,3,3}
  54: {1,2,2,2}
  56: {1,1,1,4}
  64: {1,1,1,1,1,1}
  81: {2,2,2,2}
  84: {1,1,2,4}
  96: {1,1,1,1,1,2}
		

Crossrefs

Note: Heinz numbers are given in parentheses below.

The reciprocal version is A143773 (A316428).

These partitions are counted by A340693.

A120383 lists numbers divisible by all of their prime indices.

A324850 lists numbers divisible by the product of their prime indices.

A003963 multiplies together the prime indices of n.

A018818 counts partitions of n into divisors of n (A326841).

A047993 counts balanced partitions (A106529).

A067538 counts partitions of n whose length divides n (A316413).

A056239 adds up the prime indices of n.

A061395 selects the maximum prime index.

A067538 counts partitions of n whose maximum divides n (A326836).

A072233 counts partitions by sum and length.

A112798 lists the prime indices of each positive integer.

A168659 = partitions whose length is divisible by their maximum (A340609).

A168659 = partitions whose maximum is divisible by their length (A340610).

A289509 lists numbers with relatively prime prime indices.

A326842 = partitions of n whose length and parts all divide n (A326847).

A326843 = partitions of n whose length and maximum both divide n (A326837).

A340852 have a factorization with factors dividing length.

Cf. A000720, A001222, A006141, A067539, A200750, A298423, A324925, A326149/A326155, A340608, A340827.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And@@IntegerQ/@(PrimeOmega[#]/primeMS[#])&]