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

49, 63, 65, 81, 150, 154, 190, 198, 364, 468, 580, 840, 952, 1080, 1224, 1480, 2128, 2288, 2736, 3440, 5152, 5280, 6624, 8480, 9408, 10816, 12096, 12992, 15552, 16704, 19520, 24960, 26752, 27776, 35712, 44800, 45440, 56576, 57600, 66304, 85248, 101120, 118272

Offset: 1

Author

Gus Wiseman, Jun 09 2019

Keywords

• nonn

Comments

The only squarefree terms are 65, 154, and 190. See A326157 for a proof.

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 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.

Examples

			The sequence of terms together with their prime indices begins:
     49: {4,4}
     63: {2,2,4}
     65: {3,6}
     81: {2,2,2,2}
    150: {1,2,3,3}
    154: {1,4,5}
    190: {1,3,8}
    198: {1,2,2,5}
    364: {1,1,4,6}
    468: {1,1,2,2,6}
    580: {1,1,3,10}
    840: {1,1,1,2,3,4}
    952: {1,1,1,4,7}
   1080: {1,1,1,2,2,2,3}
   1224: {1,1,1,2,2,7}
   1480: {1,1,1,3,12}
   2128: {1,1,1,1,4,8}
   2288: {1,1,1,1,5,6}
   2736: {1,1,1,1,2,2,8}
   3440: {1,1,1,1,3,14}
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000 (first 80 terms from Jinyuan Wang)

Crossrefs

Satisfies A003963(a(n)) = 2 * A056239(a(n)).

Cf. A112798, A301987, A325037, A325038, A325044, A326149, A326152, A326153/A326154, A326156, A326157.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Times@@primeMS[#]==2*Plus@@primeMS[#]&]

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