A379319 Even numbers whose product of prime indices is a multiple of their sum of prime indices.

2, 30, 84, 108, 150, 154, 190, 198, 200, 264, 364, 390, 442, 468, 490, 506, 580, 624, 630, 658, 700, 714, 810, 840, 846, 874, 900, 918, 952, 988, 1020, 1080, 1110, 1118, 1120, 1224, 1254, 1330, 1430, 1440, 1480, 1596, 1632, 1666, 1708, 1710, 1716, 1786, 1794

Offset: 1

Author

Gus Wiseman, Jan 17 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. The sum and product of prime indices are A056239 and A003963 respectively.

Examples

			The prime indices of 150 are {1,2,3,3}, with sum 9 and product 18, so 150 is in the sequence.
The terms together with their prime indices begin:
     2: {1}
    30: {1,2,3}
    84: {1,1,2,4}
   108: {1,1,2,2,2}
   150: {1,2,3,3}
   154: {1,4,5}
   190: {1,3,8}
   198: {1,2,2,5}
   200: {1,1,1,3,3}
   264: {1,1,1,2,5}
   364: {1,1,4,6}
   390: {1,2,3,6}
   442: {1,6,7}
   468: {1,1,2,2,6}
   490: {1,3,4,4}
		

Crossrefs

Even terms of A326149, which is counted by A057568 (strict A379733).

For nonprime instead of even we have A326150.

For odd instead of even we have A379318, counted by A379734 (strict A379735).

Partitions of this type are counted by A379320.

For squarefree instead of even we have A379844.

The squarefree case is A379845.

Divide all terms by 2 to get A380217.

A000040 lists the prime numbers, differences A001223.

A003963 multiplies together prime indices.

A055396 gives least prime index, greatest A061395.

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

Counting and ranking multisets by comparing sum and product:

- same: A001055, ranks A301987

- divisible: A057567, ranks A326155

- greater than: A096276 shifted right, ranks A325038

- greater or equal: A096276, ranks A325044

- less than: A114324, ranks A325037, see A318029, A379720

- less or equal: A319005, ranks A379721, see A025147

- different: A379736, ranks A379722, see A111133

Cf. A069016, A301988, A318950, A319000, A324851, A325041, A379671, A379678.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],MemberQ[prix[#],1]&&Divisible[Times@@prix[#],Total[prix[#]]]&]