cp's OEIS Frontend

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

%I A381632 #6 Mar 24 2025 22:35:11
%S A381632 2,9,24,54,72,80,108,125,216,224,400,704,960,1215,1250,1568,1664,2000,
%T A381632 2401,2500,2688,2880,4352,4800,5000,5103,6075,7290,7744,8064,8448,
%U A381632 8640,8960,9375,9728,10000,10976,14400,14580,18816,19968,21632,23552,24000,24057
%N A381632 Numbers such that (greatest prime exponent) = (sum of distinct prime indices).
%C A381632 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.
%F A381632 A051903(a(n)) = A066328(a(n)).
%e A381632 The terms together with their prime indices begin:
%e A381632       2: {1}
%e A381632       9: {2,2}
%e A381632      24: {1,1,1,2}
%e A381632      54: {1,2,2,2}
%e A381632      72: {1,1,1,2,2}
%e A381632      80: {1,1,1,1,3}
%e A381632     108: {1,1,2,2,2}
%e A381632     125: {3,3,3}
%e A381632     216: {1,1,1,2,2,2}
%e A381632     224: {1,1,1,1,1,4}
%e A381632     400: {1,1,1,1,3,3}
%e A381632     704: {1,1,1,1,1,1,5}
%e A381632     960: {1,1,1,1,1,1,2,3}
%t A381632 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A381632 Select[Range[100],Max@@Last/@FactorInteger[#]==Total[Union[prix[#]]]&]
%Y A381632 For (length) instead of (sum of distinct) we have A000961.
%Y A381632 Including number of parts gives A062457 (degenerate).
%Y A381632 Counting partitions by the LHS gives A091602, rank statistic A051903.
%Y A381632 Counting partitions by the RHS gives A116861, rank statistic A066328.
%Y A381632 Partitions of this type are counted by A381079.
%Y A381632 A001222 counts prime factors, distinct A001221.
%Y A381632 A047993 counts partitions with max part = length, ranks A106529.
%Y A381632 A051903 gives greatest prime exponent, least A051904.
%Y A381632 A055396 gives least prime index, greatest A061395.
%Y A381632 A056239 adds up prime indices, row sums of A112798.
%Y A381632 A239455 counts Look-and-Say partitions, complement A351293.
%Y A381632 A239964 counts partitions with max multiplicity = length, ranks A212166.
%Y A381632 A240312 counts partitions with max = max multiplicity, ranks A381542.
%Y A381632 A382302 counts partitions with max = max multiplicity = distinct length, ranks A381543.
%Y A381632 Cf. A000720, A048767, A130091, A246655, A317090, A380955, A381437, A381439.
%K A381632 nonn
%O A381632 1,1
%A A381632 _Gus Wiseman_, Mar 24 2025