A355746 - OEIS

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A355746

Number of different multisets that can be obtained by choosing a prime index (or a prime factor) of each integer from 2 to n.

1, 1, 1, 1, 1, 2, 2, 2, 2, 4, 4, 6, 6, 12, 20, 20, 20, 26, 26, 36, 58, 116, 116, 140, 140, 280, 280, 384, 384, 536, 536, 536, 844, 1688, 2380, 2716, 2716, 5432, 8484, 10152, 10152, 13308, 13308, 18064, 21616, 43232, 43232, 47648, 47648, 54656, 84480, 114304, 114304

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

LINKS

Michael S. Branicky, Table of n, a(n) for n = 1..75

FORMULA

a(n) = A355744(A070826(n)).

a(p) = a(p-1) for p prime. - Michael S. Branicky, Aug 03 2022

EXAMPLE

The a(n) multisets for n = 2, 6, 10, 12:

{1} {1,1,1,2,3} {1,1,1,1,1,2,2,3,4} {1,1,1,1,1,1,2,2,3,4,5}

{1,1,2,2,3} {1,1,1,1,2,2,2,3,4} {1,1,1,1,1,2,2,2,3,4,5}

{1,1,1,1,2,2,3,3,4} {1,1,1,1,1,2,2,3,3,4,5}

{1,1,1,2,2,2,3,3,4} {1,1,1,1,2,2,2,2,3,4,5}

{1,1,1,1,2,2,2,3,3,4,5}

{1,1,1,2,2,2,2,3,3,4,5}

MATHEMATICA

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

Table[Length[Union[Sort/@Tuples[primeMS/@Range[2, n]]]], {n, 15}]

PROG

(Python)

from sympy import factorint

from itertools import count, islice

def agen():

s = {(1, )}

for n in count(2):

yield len(s)

s = set(tuple(sorted(t+(d, ))) for t in s for d in factorint(n))

print(list(islice(agen(), 53))) # Michael S. Branicky, Aug 03 2022

CROSSREFS

The sum of the same integers is A000096.

The product of the same integers is A000142, Heinz number A070826.

The integers themselves are the rows of A131818 (shifted).

Counting sequences instead of multisets: A355537, with multiplicity A327486.

Using prime indices instead of 2..n gives A355744, for sequences A355741.

The version for divisors instead of prime factors is A355747.

A000040 lists the prime numbers.

A001221 counts distinct prime factors, with sum A001414.

A001222 counts prime factors with multiplicity.

A003963 multiplies together the prime indices of n.

A056239 adds up prime indices, row sums of A112798.

Cf. A000720, A002110, A076610, A355538, A355731, A355733, A355740, A355742, A355745.

Sequence in context: A325710 A214927 A326115 * A239485 A237756 A010334

Adjacent sequences: A355743 A355744 A355745 * A355747 A355748 A355749

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 20 2022

EXTENSIONS

a(28) and beyond from Michael S. Branicky, Aug 03 2022

STATUS

approved