A194262 -id:A194262 - cp's OEIS frontend

A194259 Number of distinct prime factors of p(1)p(2)...*p(n), where p(n) is the n-th partition number.

0, 1, 2, 3, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 8, 9, 9, 10, 11, 12, 13, 14, 15, 16, 17, 17, 18, 19, 20, 21, 21, 21, 22, 23, 24, 25, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 39, 40, 42, 43, 44, 45, 47, 48, 49, 51, 52, 53, 54, 56, 57, 59, 60, 61

Offset: 1

Jonathan Sondow, Aug 20 2011

nonn

Schinzel and Wirsing proved that a(n) > C*log n, for any positive constant C < 1/log 2 and all large n. In fact, it appears that a(n) > n for all n > 115 (see A194260).

It also appears that a(n) > a(n-1), for all n > 97, so that some prime factor of p(n) does not divide p(1)*p(2)*...*p(n-1). See A194261, A194262.

			p(1)*p(2)*...*p(8) = 1*2*3*5*7*11*15*22 = 2^2 * 3^2 * 5^2 * 7 * 11^2, so a(8) = 5.

Alois P. Heinz and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 2000 terms from Alois P. Heinz)
J. Cilleruelo and F. Luca, On the largest prime factor of the partition function of n
A. Schinzel and E. Wirsing, Multiplicative properties of the partition function, Proc. Indian Acad. Sci., Math. Sci. (Ramanujan Birth Centenary Volume), 97 (1987), 297-303; alternative link.
Eric Weisstein's World of Mathematics, Partition Function
Wikipedia, Partition function

Cf. A000041, A001221, A087175, A194260, A194261, A194262.

with(combinat): with(numtheory):
b:= proc(n) option remember;
      `if`(n=1, {}, b(n-1) union factorset(numbpart(n)))
    end:
a:= n-> nops(b(n)):
seq(a(n), n=1..100); # Alois P. Heinz, Aug 20 2011

a[n_] := Product[PartitionsP[k], {k, 1, n}] // PrimeNu; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 28 2014 *)
PrimeNu[FoldList[Times,PartitionsP[Range[80]]]] (* Harvey P. Dale, May 29 2025 *)

a(n)=my(v=[]);for(k=2,n,v=concat(v,factor(numbpart(k))[,1])); #vecsort(v,,8) \\ Charles R Greathouse IV, Feb 01 2013

a(n) = A001221(product(k=1..n, A000041(k))).

A194261 Smallest prime that divides the n-th partition number p(n) but does not divide p(1)p(2)...*p(n-1), or 1 if none.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 1, 1, 1, 1, 1, 1, 101, 1, 1, 1, 1, 1, 1, 19, 1, 167, 251, 1, 89, 29, 43, 13, 83, 467, 311, 23, 1, 1231, 41, 17977, 281, 1, 1, 127, 193, 2417, 71, 31, 1087, 73, 67, 7013, 631, 9283, 661, 53, 5237, 17, 227, 47, 102359, 3251, 199, 139, 971, 2273

Offset: 1

Jonathan Sondow, Aug 20 2011

It appears that a(n) is prime for all n > 97. See A194259 and A194260 for additional comments and links.

Alois P. Heinz and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 2000 terms from Alois P. Heinz)

Cf. A000041, A194259, A194260, A194262.

with(combinat): with(numtheory):
b:= proc(n) option remember;
      `if`(n=1, {}, b(n-1) union factorset(numbpart(n)))
    end:
m:= proc(n) option remember; min((b(n) minus b(n-1))[]) end:
a:= n-> `if`(n=1, 1, `if`(m(n)=infinity, 1, m(n))):
seq(a(n), n=1..120);  # Alois P. Heinz, Aug 21 2011

a[n_] := Complement[FactorInteger[PartitionsP[n]][[All, 1]], FactorInteger[Product[PartitionsP[k], {k, 1, n-1}]][[All, 1]]] /. {} -> {1} // First; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 28 2014 *)

A194345 Numbers k for which the largest prime factor of p(k) divides p(1)p(2)...*p(k-1), where p(k) is the number of partitions of k.

Original entry on oeis.org

1, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 24, 33, 38, 39, 82, 97, 158, 166, 180

Offset: 1

Jonathan Sondow, Aug 21 2011

It appears that for all k > 180, the largest prime factor of p(k) does not divide p(1)*p(2)*...*p(k-1). This has been checked up to k = 2000. [Checked up to k = 10000, using A071963 b-file. - Pontus von Brömssen, Jun 05 2023]

See A071963 and A194259 for links and additional comments.

			1 is in the sequence because p(1) = 1 and 1 has no prime factor, so the condition is vacuously true.
For k = 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 24, 33, 38, 39, 82, 97, every prime factor of p(k) divides p(1)*p(2)*...*p(k-1).
For k = 158, 166, 180, not every prime factor of p(k) divides p(1)*p(2)*...*p(k-1), but the largest one does.

Cf. A000041, A071963, A087173, A192885, A194259, A194260, A194261, A194262.

cp's OEIS Frontend

A194259 Number of distinct prime factors of p(1)p(2)...*p(n), where p(n) is the n-th partition number.

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A194261 Smallest prime that divides the n-th partition number p(n) but does not divide p(1)p(2)...*p(n-1), or 1 if none.

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A194345 Numbers k for which the largest prime factor of p(k) divides p(1)p(2)...*p(k-1), where p(k) is the number of partitions of k.

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cp's OEIS Frontend

A194259 Number of distinct prime factors of p(1)*p(2)*...*p(n), where p(n) is the n-th partition number.

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Maple

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PARI

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A194261 Smallest prime that divides the n-th partition number p(n) but does not divide p(1)*p(2)*...*p(n-1), or 1 if none.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A194345 Numbers k for which the largest prime factor of p(k) divides p(1)*p(2)*...*p(k-1), where p(k) is the number of partitions of k.

Views

Author

Keywords

Comments

Examples

Crossrefs

A194259 Number of distinct prime factors of p(1)p(2)...*p(n), where p(n) is the n-th partition number.

A194261 Smallest prime that divides the n-th partition number p(n) but does not divide p(1)p(2)...*p(n-1), or 1 if none.

A194345 Numbers k for which the largest prime factor of p(k) divides p(1)p(2)...*p(k-1), where p(k) is the number of partitions of k.