A071963 -id:A071963 - cp's OEIS frontend

A192885 A071963(n) - n, where A071963(n) is the largest prime factor of p(n), the n-th partition number A000041(n).

1, 0, 0, 0, 1, 2, 5, -2, 3, -4, -3, -4, -1, 88, -9, -4, -5, -6, -7, -12, -1, -10, 145, 228, -17, 64, 3, 16, -15, 54, 437, 280, -9, -10, 1197, 6, 17941, 244, 5, -28, 87, 152, 2375, 28, 53, 1042, 195, 20, 6965, 582, 9233, 610, 1, 5184, 5, 172, 963, 102302

Offset: 0

Jonathan Sondow, Aug 16 2011

sign

It appears that if n > 39, then a(n) is positive, i.e., A071963(n) > n. This has been checked up to n = 2500.

Cilleruelo and Luca proved that A071963(n) > log log n for almost all n, a much weaker statement. Earlier Schinzel and Wirsing proved that for all large N the product p(1)*p(2)*...*p(N) has at least C*log N distinct prime factors, for any positive constant C < 1/log 2.

			There are 77 partitions of 12, and 77 = 7*11, so a(12) = 11 - 12 = -1.

T. D. Noe, Table of n, a(n) for n = 0..1000
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, Greatest Prime Factor
Eric Weisstein's World of Mathematics, Partition function
Wikipedia, Partition function

Cf. A000041, A006530, A071963.

Table[First[Last[FactorInteger[PartitionsP[n]]]] - n, {n, 0, 100}]

a(n)=if(n<2,!n,my(f=factor(numbpart(n))[,1]);f[#f]-n) \\ Charles R Greathouse IV, Feb 04 2013

a(n) = A006530(A000041(n)) - n

A087174 Duplicate of A071963.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 5, 11, 5, 7, 7, 11, 101, 5, 11, 11, 11, 11, 7, 19, 11, 167, 251, 7, 89, 29

Offset: 1

dead

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

Original entry on oeis.org

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)).

A194262 Largest 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, 97, 1087, 241, 67, 7013, 631, 9283, 661, 53, 5237, 59, 227, 1019, 102359, 3251, 199, 409, 971

Offset: 1

Jonathan Sondow, Aug 21 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, A071963, A194259, A194260, A194261.

with(combinat): with(numtheory):
b:= proc(n) option remember;
      `if`(n=1, {}, b(n-1) union factorset(numbpart(n)))
    end:
a:= n-> `if`(n=1, 1, max(1, (b(n) minus b(n-1))[])):
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} // Last; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 28 2014 *)

A087173 Smallest prime factor of n-th partition number.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 3, 2, 2, 2, 2, 7, 101, 3, 2, 3, 3, 5, 2, 3, 2, 2, 5, 3, 2, 2, 2, 2, 5, 2, 2, 3, 3, 2, 3, 17977, 7, 5, 3, 2, 3, 2, 3, 5, 2, 2, 2, 3, 5, 2, 3, 3, 3, 5, 2, 11, 2, 2, 2, 17, 3, 2, 3, 2, 2, 2, 1181, 3, 5, 2, 3, 11, 23, 2, 2, 7, 10619863, 2, 2, 2, 11, 5, 7, 2, 11, 2, 11, 3, 5, 2473

Offset: 1

Reinhard Zumkeller, Aug 23 2003

nonn

			A000041(100) = 190569292 = 2*2*43*59*89*211, therefore a(100)=2.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Least Prime Factor
Eric Weisstein's World of Mathematics, Partition Function
Eric Weisstein's World of Mathematics, Partition Function P Congruences

Cf. A071963.

FactorInteger[#][[1,1]]&/@PartitionsP[Range[90]] (* Harvey P. Dale, May 20 2023 *)

spf(n) = if (n==1, 1, vecmin(factor(n)[,1]));
a(n) = spf(numbpart(n)); \\ Michel Marcus, Feb 24 2023

a(n) = A020639(A000041(n)).

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.

A254269 Largest prime factor of the strict partition numbers Q(n) (partitions into distinct parts, A000009).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 2, 5, 3, 2, 5, 3, 5, 3, 11, 3, 2, 19, 23, 3, 2, 19, 89, 13, 61, 71, 11, 3, 37, 2, 37, 17, 13, 7, 2, 13, 167, 19, 3, 491, 53, 7, 31, 23, 227, 2, 3, 37, 97, 17, 59, 241, 79, 5, 953, 1063, 1777, 29, 367, 17, 17, 3019, 181, 29, 4111

Offset: 0

Jean-François Alcover, Jan 27 2015

nonn

A035359 is the sequence of indices n such that a(n) = A000009(n).

Jean-François Alcover and Giovanni Resta, Table of n, a(n) for n = 0..10000 (first 1001 terms from Jean-François Alcover)
Eric Weisstein's MathWorld, Partition Function Q

Cf. A000009, A035359, A071963.

Table[FactorInteger[PartitionsQ[n]][[-1, 1]], {n, 0, 100}]

A360169 Numbers k such that the partition number p(k) = A000041(k) is a term of A057109, i.e., it is not a divisor of the factorial of its greatest prime factor.

Original entry on oeis.org

14, 19, 24, 28, 118

Offset: 1

Pontus von Brömssen, Jan 28 2023

There are no more terms below 3000.

			p(14) = 3^3 * 5, but 5! has only one factor 3.
p(19) = 2 * 5 * 7^2, but 7! has only one factor 7.
p(24) = 3^2 * 5^2 * 7, but 7! has only one factor 5.
p(28) = 2 * 11 * 13^2, but 13! has only one factor 13.
p(118) = 11 * 197 * 827^2, but 827! has only one factor 827.

Cf. A000041, A057109, A071963.

cp's OEIS Frontend

A192885 A071963(n) - n, where A071963(n) is the largest prime factor of p(n), the n-th partition number A000041(n).

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A087174 Duplicate of A071963.

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

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A194262 Largest 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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A087173 Smallest prime factor of n-th partition number.

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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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A254269 Largest prime factor of the strict partition numbers Q(n) (partitions into distinct parts, A000009).

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Mathematica

A360169 Numbers k such that the partition number p(k) = A000041(k) is a term of A057109, i.e., it is not a divisor of the factorial of its greatest prime factor.

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A194262 Largest 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.