A331384 Numbers whose sum of prime factors is equal to their product of prime indices.

35, 65, 95, 98, 154, 324, 364, 476, 623, 763, 791, 812, 826, 938, 994, 1036, 1064, 1106, 1144, 1148, 1162, 1288, 1484, 1708, 1736, 2044, 2408, 2632, 4320, 5408, 6688, 6974, 8000, 10208, 12623, 12701, 12779, 14144, 19624, 23144, 25784, 26048, 44176, 47696

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.

Numbers k such that A001414(k) = A003963(k). - Jason Yuen, Dec 19 2024

Links

Table of n, a(n) for n=1..44.

Example

The sequence of terms together with their prime indices begins:
     35: {3,4}
     65: {3,6}
     95: {3,8}
     98: {1,4,4}
    154: {1,4,5}
    324: {1,1,2,2,2,2}
    364: {1,1,4,6}
    476: {1,1,4,7}
    623: {4,24}
    763: {4,29}
    791: {4,30}
    812: {1,1,4,10}
    826: {1,4,17}
    938: {1,4,19}
    994: {1,4,20}
   1036: {1,1,4,12}
   1064: {1,1,1,4,8}
   1106: {1,4,22}
   1144: {1,1,1,5,6}
   1148: {1,1,4,13}
For example, 476 has prime factors {2,2,7,17} and prime indices {1,1,4,7}, and 2+2+7+17 = 28 = 1*1*4*7, so 476 is in the sequence.

Mathematica

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

Crossrefs

These are the Heinz numbers of the partitions counted by A331383.

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.

Partitions whose Heinz number is divisible by their sum of primes: A330953.

Sum of prime factors is divisible by sum of prime indices: A331380

Partitions whose product divides their sum of primes are A331381.

Cf. A000040, A001414, A003963, A324850, A330954, A331378, A331379, A331382, A331415, A331416.

Sequence in context: A292005 A338008 A331378 * A297150 A292081 A162832

Adjacent sequences: A331381 A331382 A331383 * A331385 A331386 A331387

Keyword

nonn,changed

Author

Gus Wiseman, Jan 16 2020

Status

approved