A087175 -id:A087175 - cp's OEIS frontend

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

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)).

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

Original entry on oeis.org

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))).

A194260 A194259(n) - n, where A194259(n) is the number of distinct prime factors of p(1)p(2)...*p(n) and p(n) is the n-th partition number.

Original entry on oeis.org

-1, -1, -1, -1, -1, -1, -2, -3, -4, -5, -6, -7, -7, -8, -9, -10, -11, -12, -13, -13, -14, -14, -14, -15, -15, -15, -15, -15, -15, -15, -15, -15, -16, -16, -16, -16, -16, -17, -18, -18, -18, -18, -18, -17, -17, -16, -16, -16, -16, -16, -16, -16, -16, -15, -15, -14, -14, -14, -14, -13, -13, -13, -12, -12, -12, -12, -11, -11, -10, -10, -10, -10, -9, -9, -9, -9, -9, -8, -7, -7, -7, -8, -8, -8, -8, -7, -7, -7, -7, -6, -5, -4, -4, -4, -3, -3, -4, -4, -4, -4, -4, -3, -3, -3, -3, -3, -3, -3, -3, -2, -2, -2, -2, -2, 0, 1

Offset: 1

Jonathan Sondow, Aug 20 2011

sign

Schinzel and Wirsing proved that a(n) > C*log n - n, for any positive constant C < 1/log 2 and all large n. In fact, it appears that a(n) > 0 for all n > 115.

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).

			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 - 8 = -3.

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

Cf. A000041, A001221, A087175, A194259.

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)) -n:
seq(a(n), n=1..116); # Alois P. Heinz, Aug 20 2011

a[n_] := PrimeNu[Product[PartitionsP[k], {k, 1, n}]] - n; Table[a[n], {n, 1, 116}] (* Jean-François Alcover, Jan 28 2014 *)

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

A111353 Number of distinct prime factors of P(6*n+1) where P(m) is the partition number.

Original entry on oeis.org

2, 1, 3, 3, 3, 3, 3, 3, 4, 5, 2, 3, 4, 2, 3, 7, 4, 5, 5, 2, 4, 4, 3, 5, 3, 1, 4, 3, 3, 2, 4, 5, 3, 3, 5, 3, 6, 4, 2, 2, 3, 4, 5, 4, 4, 3, 4, 4, 4, 3, 5, 3, 5, 5, 6, 3, 4, 4, 3, 3, 3, 5, 6, 3, 4, 6, 4, 4, 4, 4, 4, 3, 6, 7, 3, 4, 4, 7, 2, 8

Offset: 1

Parthasarathy Nambi, Nov 05 2005

nonn

			If n=1 then the number of distinct prime factors of P(6*n+1) = P(7) is 2, which is the first term in the sequence.

Table[Length[FactorInteger[PartitionsP[6n + 1]]], {n, 1, 80}] (* Stefan Steinerberger, Feb 17 2006 *)
PrimeNu[PartitionsP[6*Range[80]+1]] (* Harvey P. Dale, Mar 31 2019 *)

a(n) = A087175(6n+1). - R. J. Mathar, Aug 25 2011

More terms from Stefan Steinerberger, Feb 17 2006

A366581 a(n) = phi(p(n)), where phi is Euler's totient function (A000010) and p(n) is the number of partitions of n (A000041).

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 10, 8, 10, 8, 12, 24, 60, 100, 72, 80, 120, 180, 240, 168, 360, 240, 332, 1000, 720, 880, 672, 1008, 1560, 3280, 1864, 3100, 4840, 5544, 4920, 8800, 17976, 16800, 18480, 12960, 10584, 23040, 24160, 37800, 57600, 43440, 34560, 49896, 84144

Offset: 0

Sean A. Irvine, Oct 13 2023

nonn

Cf. A000041, A000010, A087175, A139041.

Table[EulerPhi[PartitionsP[n]], {n, 0, 48}] (* Paul F. Marrero Romero, Oct 14 2023 *)

a(n) = eulerphi(numbpart(n)); \\ Michel Marcus, Oct 14 2023

a(n) = A000010(A000041(n)).

cp's OEIS Frontend

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

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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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A194260 A194259(n) - n, where A194259(n) is the number of distinct prime factors of p(1)*p(2)*...*p(n) and p(n) is the n-th partition number.

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Author

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Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A111353 Number of distinct prime factors of P(6*n+1) where P(m) is the partition number.

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Author

Keywords

Examples

Programs

Mathematica

Formula

Extensions

A366581 a(n) = phi(p(n)), where phi is Euler's totient function (A000010) and p(n) is the number of partitions of n (A000041).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

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

A194260 A194259(n) - n, where A194259(n) is the number of distinct prime factors of p(1)p(2)...*p(n) and p(n) is the n-th partition number.