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

1,1

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.

Links

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

Example

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}

Mathematica

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

Prog

(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

Crossrefs

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

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

Sequence in context: A178951 A202001 A038640 * A216165 A065807 A038678

Adjacent sequences: A326148 A326149 A326150 * A326152 A326153 A326154

Keyword

nonn

Author

Gus Wiseman, Jun 09 2019

Status

approved