cp's OEIS Frontend

A326151 Numbers whose product of prime indices is twice their sum of prime indices.

%I A326151 #17 Jun 28 2020 10:20:11
%S A326151 49,63,65,81,150,154,190,198,364,468,580,840,952,1080,1224,1480,2128,
%T A326151 2288,2736,3440,5152,5280,6624,8480,9408,10816,12096,12992,15552,
%U A326151 16704,19520,24960,26752,27776,35712,44800,45440,56576,57600,66304,85248,101120,118272
%N A326151 Numbers whose product of prime indices is twice their sum of prime indices.
%C A326151 The only squarefree terms are 65, 154, and 190. See A326157 for a proof.
%C A326151 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.
%C A326151 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose product of parts is twice their sum of parts. The enumeration of these partitions by sum is given by A326152.
%H A326151 David A. Corneth, <a href="/A326151/b326151.txt">Table of n, a(n) for n = 1..10000</a> (first 80 terms from Jinyuan Wang)
%e A326151 The sequence of terms together with their prime indices begins:
%e A326151      49: {4,4}
%e A326151      63: {2,2,4}
%e A326151      65: {3,6}
%e A326151      81: {2,2,2,2}
%e A326151     150: {1,2,3,3}
%e A326151     154: {1,4,5}
%e A326151     190: {1,3,8}
%e A326151     198: {1,2,2,5}
%e A326151     364: {1,1,4,6}
%e A326151     468: {1,1,2,2,6}
%e A326151     580: {1,1,3,10}
%e A326151     840: {1,1,1,2,3,4}
%e A326151     952: {1,1,1,4,7}
%e A326151    1080: {1,1,1,2,2,2,3}
%e A326151    1224: {1,1,1,2,2,7}
%e A326151    1480: {1,1,1,3,12}
%e A326151    2128: {1,1,1,1,4,8}
%e A326151    2288: {1,1,1,1,5,6}
%e A326151    2736: {1,1,1,1,2,2,8}
%e A326151    3440: {1,1,1,1,3,14}
%t A326151 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A326151 Select[Range[1000],Times@@primeMS[#]==2*Plus@@primeMS[#]&]
%o A326151 (PARI) is(k) = {my(f=factor(k)); for(i=1, #f~, f[i, 1]=primepi(f[i, 1])); factorback(f)==2*sum(i=1, #f~, f[i, 2]*f[i, 1]); } \\ _Jinyuan Wang_, Jun 27 2020
%Y A326151 Satisfies A003963(a(n)) = 2 * A056239(a(n)).
%Y A326151 Cf. A112798, A301987, A325037, A325038, A325044, A326149, A326152, A326153/A326154, A326156, A326157.
%K A326151 nonn
%O A326151 1,1
%A A326151 _Gus Wiseman_, Jun 09 2019