A340387 Numbers whose sum of prime indices is twice their number, counted with multiplicity in both cases.

1, 3, 9, 10, 27, 28, 30, 81, 84, 88, 90, 100, 208, 243, 252, 264, 270, 280, 300, 544, 624, 729, 756, 784, 792, 810, 840, 880, 900, 1000, 1216, 1632, 1872, 2080, 2187, 2268, 2352, 2376, 2430, 2464, 2520, 2640, 2700, 2800, 2944, 3000, 3648, 4896, 5440, 5616

Offset

1,2

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.

Also Heinz numbers of integer partitions whose sum is twice their length, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). Like partitions in general (A000041), these are also counted by A000041.

Links

Table of n, a(n) for n=1..50.

Formula

All terms satisfy A056239(a(n)) = 2*A001222(a(n)).

Example

The sequence of terms together with their prime indices begins:
      1: {}
      3: {2}
      9: {2,2}
     10: {1,3}
     27: {2,2,2}
     28: {1,1,4}
     30: {1,2,3}
     81: {2,2,2,2}
     84: {1,1,2,4}
     88: {1,1,1,5}
     90: {1,2,2,3}
    100: {1,1,3,3}
    208: {1,1,1,1,6}
    243: {2,2,2,2,2}
    252: {1,1,2,2,4}

Mathematica

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

Crossrefs

Partitions of 2n into n parts are counted by A000041.

The number of prime indices alone is A001222.

The sum of prime indices alone is A056239.

Allowing sum to be any multiple of length gives A067538, ranked by A316413.

A000569 counts graphical partitions, ranked by A320922.

A027187 counts partitions of even length, ranked by A028260.

A058696 counts partitions of even numbers, ranked by A300061.

A301987 lists numbers whose sum of prime indices equals their product, with nonprime case A301988.

Cf. A000720, A001221, A001414, A006125, A006129, A112798, A316428, A320911, A325037, A325044, A330950, A331385, A331416.

Sequence in context: A077560 A326003 A060140 * A025616 A305091 A282635

Adjacent sequences: A340384 A340385 A340386 * A340388 A340389 A340390

Keyword

nonn

Author

Gus Wiseman, Jan 09 2021

Status

approved