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A331378 Numbers whose product of prime indices is divisible by their sum of prime factors. 14
35, 65, 95, 98, 154, 189, 297, 324, 363, 364, 375, 450, 476, 585, 623, 702, 763, 765, 791, 812, 826, 918, 938, 994, 1036, 1064, 1106, 1144, 1148, 1162, 1197, 1225, 1287, 1288, 1300, 1305, 1309, 1449, 1470, 1484, 1517, 1566, 1593, 1665, 1708, 1710, 1736, 1769 (list; graph; refs; listen; history; text; internal format)
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..48.

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}

   189: {2,2,2,4}

   297: {2,2,2,5}

   324: {1,1,2,2,2,2}

   363: {2,5,5}

   364: {1,1,4,6}

   375: {2,3,3,3}

   450: {1,2,2,3,3}

   476: {1,1,4,7}

   585: {2,2,3,6}

   623: {4,24}

   702: {1,2,2,2,6}

   763: {4,29}

   765: {2,2,3,7}

   791: {4,30}

   812: {1,1,4,10}

For example, 450 = prime(1)*prime(2)*prime(2)*prime(3)*prime(3) has prime indices {1,2,2,3,3} and prime factors {2,3,3,5,5}, and since 36 is divisible by 18, 450 is in the sequence.

MATHEMATICA

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Select[Range[2, 1000], Divisible[Times@@primeMS[#], Total[Prime/@primeMS[#]]]&]

CROSSREFS

These are the Heinz numbers of the partitions counted by A330954.

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

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

Sum of prime indices divides product of prime indices: A326149.

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

Partitions whose product divides their sum of primes are A331381.

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

Cf. A000040, A001414, A056239, A057568, A324850, A330953, A331379, A331381, A331382, A331383.

Sequence in context: A108172 A176875 A292005 * A331384 A297150 A292081

Adjacent sequences:  A331375 A331376 A331377 * A331379 A331380 A331381

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 15 2020

STATUS

approved

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Last modified February 19 00:35 EST 2020. Contains 332028 sequences. (Running on oeis4.)