A085561 -id:A085561 - cp's OEIS frontend

A087175 Number of distinct primes dividing the n-th partition number.

0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 2, 2, 1, 2, 2, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 1, 3, 3, 4, 4, 4, 3, 3, 3, 3, 4, 4, 3, 3, 3, 3, 5, 3, 5, 4, 3, 3, 4, 5, 3, 5, 4, 3, 5, 2, 4, 2, 4, 3, 4, 3, 3, 3, 4, 6, 2, 1, 4, 4, 4, 2, 4, 3, 5, 2, 5, 2, 4, 3, 2, 3, 2, 2, 6, 2, 4, 7, 3, 2, 5, 3, 3

Offset: 1

Reinhard Zumkeller, Aug 23 2003

nonn

			A000041(14) = 135 = 3^3 * 5, so a(14) = 2.
A000041(97) = 133230930 = 2*3*5*7*29*131*167, so a(97)=7.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Distinct Prime Factors
Eric Weisstein's World of Mathematics, Partition Function
Wikipedia, Partition function

Cf. A000041, A001221, A085561. See also A071963, A192885.

Table[If[n==1,0,Length[FactorInteger[PartitionsP[n]]]],{n,1,100}] (* Jonathan Sondow, Aug 19 2011 *)

a(n)={omega(numbpart(n))} \\ Andrew Howroyd, Dec 28 2017

a(n) = A001221(A000041(n)).

A278241 Least number with the same prime signature as the n-th partition number: a(n) = A046523(A000041(n)).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 6, 6, 30, 30, 24, 6, 2, 24, 48, 30, 24, 30, 60, 30, 360, 30, 6, 180, 30, 420, 210, 60, 30, 60, 30, 60, 180, 30, 60, 2, 30, 60, 1680, 420, 210, 30, 240, 60, 30, 210, 420, 30, 60, 30, 60, 2310, 60, 2310, 420, 30, 30, 420, 4620, 30, 2310, 420, 30, 2310, 6, 6720, 6, 420, 30, 3360, 30, 30, 30, 2520, 120120, 6, 2, 420, 420, 1260, 6, 840, 30, 4620, 12

Offset: 0

Antti Karttunen, Nov 16 2016

nonn

This sequence works as a "sentinel" for partition numbers by matching to any sequence that is obtained as f(A000041(n)), where f(n) is any function that depends only on the prime signature of n (see the index entry for "sequences computed from exponents in ..."). The last line in Crossrefs section lists such sequences that were present in the database as of Nov 11 2016.

Amiram Eldar, Table of n, a(n) for n = 0..10000 (terms 0..2527 from Antti Karttunen)
Index entries for sequences computed from exponents in factorization of n

Cf. A000041, A046523, A278245, A278248.

Sequences that partition N into same or coarser equivalence classes: A085543, A085561, A087175.

A046523(n) = my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]) \\ From Charles R Greathouse IV, Aug 17 2011
A278241(n) = A046523(numbpart(n));
for(n=0, 2310, write("b278241.txt", n, " ", A278241(n)));

(define (A278241 n) (A046523 (A000041 n)))

a(n) = A046523(A000041(n)).

cp's OEIS Frontend

A087175 Number of distinct primes dividing the n-th partition number.

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