A380216 Numbers whose prime indices have (product)/(sum) equal to an integer > 1.

49, 63, 65, 81, 125, 150, 154, 165, 169, 190, 198, 259, 273, 333, 351, 361, 364, 385, 390, 435, 442, 468, 481, 490, 495, 506, 525, 561, 580, 595, 609, 630, 658, 675, 700, 714, 741, 765, 781, 783, 810, 840, 841, 846, 874, 900, 918, 925, 931, 935, 952, 988

Offset

1,1

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.

Links

Table of n, a(n) for n=1..52.

Example

The terms together with their prime indices begin:
   49: {4,4}
   63: {2,2,4}
   65: {3,6}
   81: {2,2,2,2}
  125: {3,3,3}
  150: {1,2,3,3}
  154: {1,4,5}
  165: {2,3,5}
  169: {6,6}
  190: {1,3,8}
  198: {1,2,2,5}
  259: {4,12}
  273: {2,4,6}
  333: {2,2,12}
  351: {2,2,2,6}
  361: {8,8}
  364: {1,1,4,6}
For example, 198 has prime indices {1,2,2,5}, and 20/10 is an integer > 1, so 198 is in the sequence.

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[2, 1000], Divisible[Times@@prix[#], Total[prix[#]]]&&!SameQ[Times@@prix[#], Total[prix[#]]]&]

Crossrefs

The fraction A003963(n)/A056239(n) reduces to A326153(n)/A326154(n).

The non-proper version is A326149, superset of A326150.

Also a superset of A326151.

The squarefree case is A326158 without first term.

Partitions of this type are counted by A380219.

A379666 counts partitions by sum and product.

Counting and ranking multisets by comparing sum and product:

- same: A001055, ranks A301987

- multiple: A057567, ranks A326155

- divisor: A057568 (strict A379733), ranks A326149, see A379735, A380217.

- 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. A000720, A001222, A028422, A036844, A112798, A301988, A319000, A324850, A324851, A326156, A379319, A379844.

Sequence in context: A178951 A202001 A038640 * A326151 A216165 A374937

Adjacent sequences: A380213 A380214 A380215 * A380217 A380218 A380219

Keyword

nonn

Author

Gus Wiseman, Jan 23 2025

Status

approved