A085543 -id:A085543 - cp's OEIS frontend

A154789 Where records occurs in A085543 for positive values of n.

Original entry on oeis.org

1, 2, 7, 9, 15, 19, 21, 39, 59, 66, 75, 96, 124, 125, 180, 327, 358, 434, 699, 999, 1685, 1779, 3561, 4184, 4835

Offset: 1

Omar E. Pol, Jan 25 2009

Essentially the same as A085544.

Cf. A000041, A085543, A085544, A154788, A154790.

A154788 Records in A085543.

Original entry on oeis.org

1, 2, 4, 8, 10, 12, 24, 40, 48, 56, 128, 144, 192, 256, 384, 480, 576, 768, 1536, 4608, 6144, 6912, 15360, 36864, 294912

Offset: 1

Omar E. Pol, Jan 25 2009

Cf. A000005, A000041, A085543, A085544, A154789, A154790.

DeleteDuplicates[DivisorSigma[0,PartitionsP[Range[0,2000]]],GreaterEqual] (* The program generates the first 22 terms of the sequence. *) (* Harvey P. Dale, Dec 09 2024 *)

a(n) = A085543(A085544(n)) = A085543(A154789(n)). - Max Alekseyev, Apr 26 2010

More terms from Max Alekseyev, Apr 26 2010

A236103 Number of distinct partition numbers dividing n.

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 1, 3, 4, 2, 1, 3, 1, 3, 3, 4, 1, 3, 2, 2, 2, 3, 1, 6, 1, 2, 3, 2, 3, 3, 1, 2, 2, 3, 1, 5, 1, 4, 4, 2, 1, 3, 2, 3, 2, 2, 1, 3, 3, 4, 2, 2, 1, 6, 1, 2, 3, 2, 2, 5, 1, 2, 2, 4, 1, 3, 1, 2, 4, 2, 4, 3, 1, 3, 2, 2, 1, 5, 2, 2, 2, 4, 1, 6

Offset: 1

Omar E. Pol, Jan 21 2014

nonn

			For n = 20 the divisors of 20 are 1, 2, 4, 5, 10, 20 and three of them are also partition numbers: 1, 2, 5, so a(20) = 3.
For n = 42 the divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42 and five of them are also partition numbers: 1, 2, 3, 7, 42, so a(42) = 5.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000041, A001221, A049575, A078506, A085543, A167392, A236102, A236105, A236107, A236108.

p = {1}; Table[If[n >= Last@p, AppendTo[p, PartitionsP[1 + Length@p]]]; Length@Select[p, Mod[n, #] == 0 &], {n, 90}] (* Giovanni Resta, Jan 22 2014 *)

From Amiram Eldar, Jan 01 2024: (Start)
a(n) = Sum_{d|n} A167392(d).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A078506 = 2.510597... . (End)

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)).

A085544 n-th partition number (A000041) sets a new record for number of divisors.

Original entry on oeis.org

0, 2, 7, 9, 15, 19, 21, 39, 59, 66, 75, 96, 124, 125, 180, 327, 358, 434, 699, 999, 1685, 1779, 3561, 4184, 4835

Offset: 1

Jason Earls, Jul 03 2003

No more terms < 6000. - David Wasserman, Feb 22 2005

Where records occur in A085543. - Omar E. Pol, Jan 25 2009

See A154789 for another version.

Cf. A085543. - Omar E. Pol, Jan 25 2009

More terms from Don Reble, Jul 13 2003
More terms from David Wasserman, Feb 22 2005
Offset changed to 1 by Jinyuan Wang, Mar 15 2020

A154790 Highly composite partition numbers.

Original entry on oeis.org

1, 2, 15, 30, 176, 490, 792, 31185, 831820, 2323520, 8118264, 118114304, 2841940500, 3163127352, 684957390936, 60105349839666544, 471314064268398780, 52527070729108240605, 57601805366500810491219000, 23127843459154899464880444632250

Offset: 1

Omar E. Pol, Jan 25 2009

nonn

Partition numbers with record values for the number of divisors.

Amiram Eldar, Table of n, a(n) for n = 1..25

Cf. A000005, A000041, A085543, A085544, A154788, A154789.

a(n) = A000041(A085544(n)). - Amiram Eldar, Apr 09 2024

More terms from D. S. McNeil, May 10 2010

a(20) from Amiram Eldar, Apr 09 2024

A272209 Number of partitions of the number of divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 2, 5, 3, 5, 2, 11, 2, 5, 5, 7, 2, 11, 2, 11, 5, 5, 2, 22, 3, 5, 5, 11, 2, 22, 2, 11, 5, 5, 5, 30, 2, 5, 5, 22, 2, 22, 2, 11, 11, 5, 2, 42, 3, 11, 5, 11, 2, 22, 5, 22, 5, 5, 2, 77, 2, 5, 11, 15, 5, 22, 2, 11, 5, 22, 2, 77, 2, 5, 11, 11, 5, 22, 2, 42, 7, 5, 2, 77

Offset: 1

Omar E. Pol, Apr 25 2016

nonn

			For n = 12 the divisors of 12 are 1, 2, 3, 4, 6, 12. There are 6 divisors of 12 and the number of partitions of 6 is A000041(6) = 11, so a(12) = 11.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A000041, A035116, A058699, A085543, A272024.

Table[PartitionsP@ DivisorSigma[0, n], {n, 120}] (* Michael De Vlieger, Apr 25 2016 *)

a(n) = numbpart(numdiv(n)); \\ Michel Marcus, Apr 26 2016

a(n) = p(d(n)) = A000041(A000005(n)).

A258596 Number of divisors of the number of partitions of n into distinct parts, a(n) = A000005(A000009(n)).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 2, 4, 4, 4, 6, 4, 6, 4, 4, 6, 4, 4, 8, 7, 6, 2, 8, 4, 4, 8, 14, 8, 9, 8, 12, 16, 14, 10, 12, 6, 16, 24, 4, 8, 36, 8, 16, 8, 12, 27, 16, 16, 28, 8, 4, 8, 22, 8, 8, 6, 16, 16, 42, 32, 6, 8, 20, 6, 48, 12, 16, 28, 8, 2, 12, 20, 16, 12, 4, 32

Offset: 0

Giovanni Resta, Jun 04 2015

nonn

a(n) is odd for n = 0, 1, 2, 6, 20, 29, 46, ... ; see A035142. - Michel Marcus, Jun 19 2015

Giovanni Resta, Table of n, a(n) for n = 0..10000

Cf. A000009, A085543.

DivisorSigma[0, PartitionsQ@ Range[0, 100]]

a(n) = numdiv(polcoeff(prod(k=1, n, 1 + x^k, 1 + x * O(x^n)), n)); \\ Michel Marcus, Jun 19 2015

a(n) = A000005(A000009(n)).

A316887 Numbers k whose partition numbers have more than k nontrivial divisors.

Original entry on oeis.org

21, 75, 94, 96, 97, 109, 124, 125, 128, 129, 131, 136, 149, 180, 213, 327, 334, 347, 358, 374, 424, 434, 449, 481, 581, 644, 699, 765, 776, 789, 859, 896, 974, 999, 1216, 1240, 1243, 1249, 1267, 1269, 1324, 1398, 1442, 1499, 1524, 1587, 1685, 1699, 1752, 1779

Offset: 1

Pierandrea Formusa, Jul 15 2018

nonn

			21 belongs to this sequence as the partition number of 21 is 792, which has 22 nontrivial divisors (divisors different from 1 and itself), and 22 > 21.

Cf. A000005, A000041, A085543.

Select[Range[1000], DivisorSigma[0, PartitionsP[#]] > # + 2 &] (* or *) First/@ Select[ Import[ "https://oeis.org/A085543/b085543.txt", "Table"], #[[2]] > #[[1]] + 2&] (* Giovanni Resta, Jul 16 2018 *)

select(n->numdiv(numbpart(n))-2 > n, [1..1000]) \\ Andrew Howroyd, Jul 15 2018

r=""
for i in range(1000):
   l=Partitions(i+1).cardinality().divisors()
   if ((len(l)-2)>i+1): r=r+str(i+1)+","
print(r)

a(35)-a(50) from Giovanni Resta, Jul 16 2018

cp's OEIS Frontend

A154789 Where records occurs in A085543 for positive values of n.

Views

Author

Keywords

Comments

Crossrefs

A154788 Records in A085543.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

Extensions

A236103 Number of distinct partition numbers dividing n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Scheme

Formula

A085544 n-th partition number (A000041) sets a new record for number of divisors.

Views

Author

Keywords

Comments

Crossrefs

Extensions

A154790 Highly composite partition numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A272209 Number of partitions of the number of divisors of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A258596 Number of divisors of the number of partitions of n into distinct parts, a(n) = A000005(A000009(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A316887 Numbers k whose partition numbers have more than k nontrivial divisors.

Views

Author

Keywords

Examples