cp's OEIS Frontend

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

%I A379319 #7 Jan 19 2025 09:35:49
%S A379319 2,30,84,108,150,154,190,198,200,264,364,390,442,468,490,506,580,624,
%T A379319 630,658,700,714,810,840,846,874,900,918,952,988,1020,1080,1110,1118,
%U A379319 1120,1224,1254,1330,1430,1440,1480,1596,1632,1666,1708,1710,1716,1786,1794
%N A379319 Even numbers whose product of prime indices is a multiple of their sum of prime indices.
%C A379319 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 A379319 The prime indices of 150 are {1,2,3,3}, with sum 9 and product 18, so 150 is in the sequence.
%e A379319 The terms together with their prime indices begin:
%e A379319      2: {1}
%e A379319     30: {1,2,3}
%e A379319     84: {1,1,2,4}
%e A379319    108: {1,1,2,2,2}
%e A379319    150: {1,2,3,3}
%e A379319    154: {1,4,5}
%e A379319    190: {1,3,8}
%e A379319    198: {1,2,2,5}
%e A379319    200: {1,1,1,3,3}
%e A379319    264: {1,1,1,2,5}
%e A379319    364: {1,1,4,6}
%e A379319    390: {1,2,3,6}
%e A379319    442: {1,6,7}
%e A379319    468: {1,1,2,2,6}
%e A379319    490: {1,3,4,4}
%t A379319 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A379319 Select[Range[1000],MemberQ[prix[#],1]&&Divisible[Times@@prix[#],Total[prix[#]]]&]
%Y A379319 Even terms of A326149, which is counted by A057568 (strict A379733).
%Y A379319 For nonprime instead of even we have A326150.
%Y A379319 For odd instead of even we have A379318, counted by A379734 (strict A379735).
%Y A379319 Partitions of this type are counted by A379320.
%Y A379319 For squarefree instead of even we have A379844.
%Y A379319 The squarefree case is A379845.
%Y A379319 Divide all terms by 2 to get A380217.
%Y A379319 A000040 lists the prime numbers, differences A001223.
%Y A379319 A003963 multiplies together prime indices.
%Y A379319 A055396 gives least prime index, greatest A061395.
%Y A379319 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A379319 Counting and ranking multisets by comparing sum and product:
%Y A379319 - same: A001055, ranks A301987
%Y A379319 - divisible: A057567, ranks A326155
%Y A379319 - greater than: A096276 shifted right, ranks A325038
%Y A379319 - greater or equal: A096276, ranks A325044
%Y A379319 - less than: A114324, ranks A325037, see A318029, A379720
%Y A379319 - less or equal: A319005, ranks A379721, see A025147
%Y A379319 - different: A379736, ranks A379722, see A111133
%Y A379319 Cf. A069016, A301988, A318950, A319000, A324851, A325041, A379671, A379678.
%K A379319 nonn
%O A379319 1,1
%A A379319 _Gus Wiseman_, Jan 17 2025