A379722 Numbers whose prime indices do not have the same sum as product.

1, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 85, 86, 87, 88, 90, 91, 92, 93

Offset: 1

Author

Gus Wiseman, Jan 08 2025

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.

Partitions of this type are counted by A379736.

The complement is A301987, counted by A001055.

Examples

			The terms together with their prime indices begin:
    1: {}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   25: {3,3}
   26: {1,6}
   27: {2,2,2}
   28: {1,1,4}
   32: {1,1,1,1,1}
		

Crossrefs

Nonzeros of A325036.

A000040 lists the primes, differences A001223.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798, counted by A001222.

A324851 finds numbers > 1 divisible by the sum of their prime indices.

A379666 counts partitions by sum and product, without 1's A379668.

A379681 gives sum plus product of prime indices, firsts A379682.

Counting and ranking multisets by comparing sum and product:

- same: A001055 (strict A045778), ranks A301987

- divisible: A057567, ranks A326155

- divisor: A057568, ranks A326149, see A379733

- greater: A096276 shifted right, ranks A325038

- greater or equal: A096276, ranks A325044

- less: A114324, ranks A325037, see A318029

- less or equal: A319005, ranks A379721

- different: A379736, ranks A379722 (this)

Cf. A000720, A003963, A025147, A075254, A111133, A178503, A319000, A325041.

Programs

Mathematica

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