cp's OEIS Frontend

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

%I A380216 #8 Jan 25 2025 19:27:52
%S A380216 49,63,65,81,125,150,154,165,169,190,198,259,273,333,351,361,364,385,
%T A380216 390,435,442,468,481,490,495,506,525,561,580,595,609,630,658,675,700,
%U A380216 714,741,765,781,783,810,840,841,846,874,900,918,925,931,935,952,988
%N A380216 Numbers whose prime indices have (product)/(sum) equal to an integer > 1.
%C A380216 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.
%e A380216 The terms together with their prime indices begin:
%e A380216    49: {4,4}
%e A380216    63: {2,2,4}
%e A380216    65: {3,6}
%e A380216    81: {2,2,2,2}
%e A380216   125: {3,3,3}
%e A380216   150: {1,2,3,3}
%e A380216   154: {1,4,5}
%e A380216   165: {2,3,5}
%e A380216   169: {6,6}
%e A380216   190: {1,3,8}
%e A380216   198: {1,2,2,5}
%e A380216   259: {4,12}
%e A380216   273: {2,4,6}
%e A380216   333: {2,2,12}
%e A380216   351: {2,2,2,6}
%e A380216   361: {8,8}
%e A380216   364: {1,1,4,6}
%e A380216 For example, 198 has prime indices {1,2,2,5}, and 20/10 is an integer > 1, so 198 is in the sequence.
%t A380216 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A380216 Select[Range[2,1000],Divisible[Times@@prix[#],Total[prix[#]]]&&!SameQ[Times@@prix[#],Total[prix[#]]]&]
%Y A380216 The fraction A003963(n)/A056239(n) reduces to A326153(n)/A326154(n).
%Y A380216 The non-proper version is A326149, superset of A326150.
%Y A380216 Also a superset of A326151.
%Y A380216 The squarefree case is A326158 without first term.
%Y A380216 Partitions of this type are counted by A380219.
%Y A380216 A379666 counts partitions by sum and product.
%Y A380216 Counting and ranking multisets by comparing sum and product:
%Y A380216 - same: A001055, ranks A301987
%Y A380216 - multiple: A057567, ranks A326155
%Y A380216 - divisor: A057568 (strict A379733), ranks A326149, see A379735, A380217.
%Y A380216 - greater than: A096276 shifted right, ranks A325038
%Y A380216 - greater or equal: A096276, ranks A325044
%Y A380216 - less than: A114324, ranks A325037, see A318029, A379720
%Y A380216 - less or equal: A319005, ranks A379721, see A025147
%Y A380216 - different: A379736, ranks A379722, see A111133
%Y A380216 Cf. A000720, A001222, A028422, A036844, A112798, A301988, A319000, A324850, A324851, A326156, A379319, A379844.
%K A380216 nonn
%O A380216 1,1
%A A380216 _Gus Wiseman_, Jan 23 2025