A381632 Numbers such that (greatest prime exponent) = (sum of distinct prime indices).

2, 9, 24, 54, 72, 80, 108, 125, 216, 224, 400, 704, 960, 1215, 1250, 1568, 1664, 2000, 2401, 2500, 2688, 2880, 4352, 4800, 5000, 5103, 6075, 7290, 7744, 8064, 8448, 8640, 8960, 9375, 9728, 10000, 10976, 14400, 14580, 18816, 19968, 21632, 23552, 24000, 24057

Offset: 1

Author

Gus Wiseman, Mar 24 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, sum A056239.

Examples

			The terms together with their prime indices begin:
      2: {1}
      9: {2,2}
     24: {1,1,1,2}
     54: {1,2,2,2}
     72: {1,1,1,2,2}
     80: {1,1,1,1,3}
    108: {1,1,2,2,2}
    125: {3,3,3}
    216: {1,1,1,2,2,2}
    224: {1,1,1,1,1,4}
    400: {1,1,1,1,3,3}
    704: {1,1,1,1,1,1,5}
    960: {1,1,1,1,1,1,2,3}
		

Crossrefs

For (length) instead of (sum of distinct) we have A000961.

Including number of parts gives A062457 (degenerate).

Counting partitions by the LHS gives A091602, rank statistic A051903.

Counting partitions by the RHS gives A116861, rank statistic A066328.

Partitions of this type are counted by A381079.

A001222 counts prime factors, distinct A001221.

A047993 counts partitions with max part = length, ranks A106529.

A051903 gives greatest prime exponent, least A051904.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A239455 counts Look-and-Say partitions, complement A351293.

A239964 counts partitions with max multiplicity = length, ranks A212166.

A240312 counts partitions with max = max multiplicity, ranks A381542.

A382302 counts partitions with max = max multiplicity = distinct length, ranks A381543.

Cf. A000720, A048767, A130091, A246655, A317090, A380955, A381437, A381439.

Programs

Mathematica

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

Formula

A051903(a(n)) = A066328(a(n)).