A331380 Numbers whose sum of prime factors is divisible by their sum of prime indices.

2, 4, 8, 16, 32, 33, 39, 55, 64, 65, 66, 74, 77, 78, 86, 91, 110, 128, 130, 132, 154, 156, 164, 182, 188, 220, 256, 260, 264, 308, 312, 364, 371, 411, 440, 459, 512, 513, 520, 528, 530, 616, 624, 636, 689, 728, 746, 755, 765, 766, 855, 880, 906, 915, 918, 1007

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.

Links

Table of n, a(n) for n=1..56.

Example

The sequence of terms together with their prime indices begins:
     2: {1}
     4: {1,1}
     8: {1,1,1}
    16: {1,1,1,1}
    32: {1,1,1,1,1}
    33: {2,5}
    39: {2,6}
    55: {3,5}
    64: {1,1,1,1,1,1}
    65: {3,6}
    66: {1,2,5}
    74: {1,12}
    77: {4,5}
    78: {1,2,6}
    86: {1,14}
    91: {4,6}
   110: {1,3,5}
   128: {1,1,1,1,1,1,1}
   130: {1,3,6}
   132: {1,1,2,5}
For example, 132 has prime factors {2,2,3,11} and prime indices {1,1,2,5}, and 18 is divisible by 9, so 132 is in the sequence.

Mathematica

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

Crossrefs

These are the Heinz numbers of the partitions counted by A331380.

Numbers divisible by the sum of their prime factors are A036844.

Partitions whose product is divisible by their sum are A057568.

Numbers divisible by the sum of their prime indices are A324851.

Product of prime indices is divisible by sum of prime indices: A326149.

Partitions whose Heinz number is divisible by their sum are A330950.

Heinz number is divisible by sum of primes: A330953.

Partitions whose product divides their sum of primes are A331381.

Partitions whose product is equal to their sum of primes are A331383.

Product of prime indices equals sum of prime factors: A331384.

Cf. A000040, A001414, A056239, A330954, A331378, A331379, A331382, A331415, A331416.

Sequence in context: A036124 A070339 A070338 * A115423 A221467 A337696

Adjacent sequences: A331377 A331378 A331379 * A331381 A331382 A331383

Keyword

nonn

Author

Gus Wiseman, Jan 16 2020

Status

approved