A326149 Numbers whose product of prime indices is divisible by their sum of prime indices.

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 43, 47, 49, 53, 59, 61, 63, 65, 67, 71, 73, 79, 81, 83, 84, 89, 97, 101, 103, 107, 108, 109, 113, 125, 127, 131, 137, 139, 149, 150, 151, 154, 157, 163, 165, 167, 169, 173, 179, 181, 190, 191, 193, 197

Offset

1,1

Comments

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 divisible by their sum of parts. The enumeration of these partitions by sum is given by A057568.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

The sequence of terms together with their prime indices begins:
    2: {1}
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   17: {7}
   19: {8}
   23: {9}
   29: {10}
   30: {1,2,3}
   31: {11}
   37: {12}
   41: {13}
   43: {14}
   47: {15}
   49: {4,4}
   53: {16}
   59: {17}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[2, 100], Divisible[Times@@primeMS[#], Plus@@primeMS[#]]&]

Crossrefs

Satisfies A056239(a(n))|A003963(a(n)).

The nonprime case is A326150, with squarefree case A326158.

Cf. A000720, A001222, A057567, A057568, A112798, A301987.

Cf. A325037, A325042, A325044, A326151, A326153/A326154, A326155, A326156.

Sequence in context: A325623 A136327 A095415 * A301987 A353393 A316857

Adjacent sequences: A326146 A326147 A326148 * A326150 A326151 A326152

Keyword

nonn

Author

Gus Wiseman, Jun 09 2019

Status

approved