A167392 -id:A167392 - cp's OEIS frontend

A186114 Triangle of regions and partitions of integers (see Comments lines for definition).

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

Offset: 1

Omar E. Pol, Aug 08 2011

Let r = T(n,k) be a record in the sequence. The consecutive records "r" are the natural numbers A000027. Consider the first n rows; the triangle T(n,k) has the property that the columns, without the zeros, from k..1, are also the partitions of r in juxtaposed reverse-lexicographical order, so k is also A000041(r), the number of partitions of r. Note that a record r is always the final term of a row if such row contains 1’s. The number of positive integer a(1)..r is A006128(r). The sums a(1)..r is A066186(r). Here the set of positive integers in every row (from 1 to n) is called a “region” of r. The number of regions of r equals the number of partitions of r. If T(n,1) = 1 then the row n is formed by the smallest parts, in nondecreasing order, of all partitions of T(n,n).

			Triangle begins:
1,
1, 2,
1, 1, 3,
0, 0, 0, 2,
1, 1, 1, 2, 4,
0, 0, 0, 0, 0, 3,
1, 1, 1, 1, 1, 2, 5,
0, 0, 0, 0, 0, 0, 0, 2,
0, 0, 0, 0, 0, 0, 0, 2, 4,
0, 0, 0, 0, 0, 0, 0, 0, 0, 3,
1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 6
...
The row n = 11 contains the 6th record in the sequence:  a(66) = T(11,11) = 6, then consider the first 11 rows of triangle. Note that the columns, from k = 11..1, without the zeros, are also the 11 partitions of 6 in juxtaposed reverse-lexicographical order: [6], [3, 3], [4, 2], [2, 2, 2], [5, 1], [3, 2, 1], [4, 1, 1], [2, 2, 1, 1], [3, 1, 1, 1], [2, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1]. See A026792.

Robert Price, Table of n, a(n) for n = 1..196878, rows 1-627.
Omar E. Pol, Illustration of the seven regions of 5

Mirror of triangle A193870. Column 1 gives A167392. Right diagonal gives A141285.

Cf. A000041, A135010, A138121, A183152, A186412, A187219, A194436-A194439, A194446-A194448, A206437.

A206437 = Cases[Import["https://oeis.org/A206437/b206437.txt",
     "Table"], {, }][[All, 2]];
A194446 = Cases[Import["https://oeis.org/A194446/b194446.txt",
    "Table"], {, }][[All, 2]];
f[n_] := Module[{v},
   v = Take[A206437, A194446[[n]]];
   A206437 = Drop[A206437, A194446[[n]]];
   Reverse[PadRight[v, n]]];
Table[f[n], {n, PartitionsP[20]}]  // Flatten (* Robert Price, Apr 26 2020 *)

T(n,1) = A167392(n).

T(n,k) = A141285(n), if k = n.

A193870 Triangle of regions and partitions of integers (see Comments lines for definition).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 07 2011

Triangle T(n,k) read by rows in which, from rows 1..n, if r = T(n,k) is a record in the sequence then the set of positive integers in every row (from 1 to n) is called a “region” of r. Note that n, the number of regions of r is also the number of partitions of r. The consecutive records "r" are the natural numbers A000027. The triangle has the property that, for rows n..1, the diagonals (without the zeros) are also the partitions of r, in juxtaposed reverse-lexicographical order. Note that a record "r" is the initial term of a row if such row contains 1’s. If T(n,k) is a record in the sequence then A000041(T(n,k)) = n. Note that if T(n,k) < 2 is not the last term of the row n then T(n,k+1) = T(n,k). The union of the rows that contain 1's gives A182715.

			Triangle begins:
1,
2, 1,
3, 1, 1,
2, 0, 0, 0,
4, 2, 1, 1, 1,
3, 0, 0, 0, 0, 0,
5, 2, 1, 1, 1, 1, 1,
2, 0, 0, 0, 0, 0, 0, 0,
4, 2, 0, 0, 0, 0, 0, 0, 0,
3, 0, 0, 0, 0, 0, 0, 0, 0, 0,
6, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1,
…
For n = 11 note that the row n contains the 6th record in the sequence: T(11,1) = a(56) = 6, then consider the first 11 rows of triangle. Note that the diagonals d, from d = n..1, without the zeros, are also the partitions of 6 in juxtaposed reverse-lexicographical order: [6], [3, 3], [4, 2], [2, 2, 2], [5, 1], [3, 2, 1], [4, 1, 1], [2, 2, 1, 1], [3, 1, 1, 1], [2, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1]. See A026792.

Robert Price, Table of n, a(n) for n = 1..196878, rows 1-627.
Omar E. Pol, Illustration of the seven regions of 5

Mirror of triangle A186114. Column 1 gives A141285. Right diagonal gives A167392.

Cf. A046746, A135010, A138121, A182699, A182709, A183152, A186412, A187219, A194436-A194439, A194446-A194448, A206437.

A206437 = Cases[Import["https://oeis.org/A206437/b206437.txt",
     "Table"], {, }][[All, 2]];
A194446 = Cases[Import["https://oeis.org/A194446/b194446.txt",
     "Table"], {, }][[All, 2]];f[n_] := Module[{v},
   v = Take[A206437, A194446[[n]]];
   A206437 = Drop[A206437, A194446[[n]]];
   PadRight[v, n]];
Table[f[n], {n, PartitionsP[20]}] // Flatten (* Robert Price, Apr 26 2020 *)

T(n,1) = A141285(n).

T(n,k) = A167392(n), if k = n.

A049575 Prime partition numbers.

Original entry on oeis.org

2, 3, 5, 7, 11, 101, 17977, 10619863, 6620830889, 80630964769, 228204732751, 1171432692373, 1398341745571, 10963707205259, 15285151248481, 10657331232548839, 790738119649411319, 18987964267331664557, 74878248419470886233, 1394313503224447816939

Offset: 1

G. L. Honaker, Jr.

Alois P. Heinz, Table of n, a(n) for n = 1..100
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
G. K. Patil, Ramanujan's Life And His Contributions In The Field Of Mathematics, International Journal of Scientific Research and Engineering Studies (IJSRES), 1(6) (2014), ISSN: 2349-8862.
Eric Weisstein's World of Mathematics, Partition Function P Congruences.
Eric Weisstein's World of Mathematics, Partition Function P.

Intersection of A000040 and A000041.
Cf. A046063, A051143.
Cf. A038753, A065728. - Reinhard Zumkeller, Nov 03 2009

lst={};Do[a=PartitionsP[n];If[PrimeQ[a],AppendTo[lst,a]],{n,2*6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 14 2009 *)
Select[PartitionsP[Range[1000]],PrimeQ] (* Harvey P. Dale, Mar 11 2013 *)

for(n=0,10^3,my(p=numbpart(n));if(isprime(p),print1(p,", "))); \\ Joerg Arndt, May 09 2013

a(n) = A000041(A046063(n)) = A000040(A051143(n)). - M. F. Hasler, Oct 19 2008

A010051(a(n))*A167392(a(n)) = 1. - Reinhard Zumkeller, Nov 03 2009

More terms from James Sellers and Christian G. Bower, Oct 15 1999.

A052001 Even partition numbers.

Original entry on oeis.org

2, 22, 30, 42, 56, 176, 490, 792, 1002, 1958, 2436, 3010, 3718, 5604, 6842, 12310, 37338, 53174, 89134, 105558, 124754, 204226, 451276, 614154, 715220, 831820, 1300156, 1741630, 2012558, 2323520, 4087968, 7089500, 8118264, 12132164

Offset: 1

Patrick De Geest, Nov 15 1999

nonn

Intersection of A005843 and A000041; A059841(a(n)) * A167392(a(n)) = 1. [Reinhard Zumkeller, Nov 03 2009]

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A000041, A001560, A005843, A052002, A052003, A059841, A167392, A213179.

a052001 n = a052001_list !! (n-1)
a052001_list = filter even a000041_list
-- Reinhard Zumkeller, Nov 03 2015

Select[PartitionsP[Range[100]], EvenQ] (* Jean-François Alcover, Mar 01 2019 *)

for(n=1, 100, if((k=numbpart(n))%2==0, print1(k", "))) \\ Altug Alkan, Nov 02 2015

a(n) = 2*A213179(n). - Omar E. Pol, May 08 2013

Offset corrected by Reinhard Zumkeller, Nov 03 2015

A087183 Partition numbers of the form 3*k.

Original entry on oeis.org

3, 15, 30, 42, 135, 231, 297, 627, 792, 1002, 1575, 2436, 5604, 8349, 10143, 14883, 31185, 37338, 44583, 63261, 105558, 147273, 239943, 281589, 329931, 614154, 1121505, 1505499, 3087735, 4087968, 4697205, 8118264, 15796476, 44108109

Offset: 1

Reinhard Zumkeller, Aug 23 2003

nonn

The numbers m such that 3 divides A000041(m) are given in A083214. Klarreich writes: no one has proved whether there are infinitely many partition numbers divisible by 3, although it is known that there are infinitely many partition numbers divisible by 2. - Jonathan Vos Post, Jul 31 2008

Intersection of A008585 and A000041. - Reinhard Zumkeller, Nov 03 2009

Erica Klarreich, Pieces of numbers: a proof brings closure to a dramatic tale of partitions and primes, Science News, Jun 18 2005.

Paul Tek, Table of n, a(n) for n = 1..10000
Erica Klarreich, Pieces of numbers: a proof brings closure to a dramatic tale of partitions and primes, Science News, Jun 18 2005.
Eric Weisstein's World of Mathematics, Partition Function.
Eric Weisstein's World of Mathematics, Partition Function P Congruences.

Cf. A000041, A008585, A068907, A052001, A052003, A079978, A083214, A087180, A087184, A087185, A167392, A213365.

Select[PartitionsP@Range[120], Divisible[#, 3] &] (* Vladimir Reshetnikov, Nov 05 2015 *)

for(n=9, 1e3, t=numbpart(n); if(t%3, , print1(t", "))) \\ Charles R Greathouse IV, May 08 2013

A079978(a(n))*A167392(a(n)) = 1. - Reinhard Zumkeller, Nov 03 2009
a(n) = 3*A213365(n). - Omar E. Pol, May 08 2013
a(n) = A000041(A083214(n)). - Amiram Eldar, May 22 2025

A052003 Odd partition numbers.

Original entry on oeis.org

1, 3, 5, 7, 11, 15, 77, 101, 135, 231, 297, 385, 627, 1255, 1575, 4565, 8349, 10143, 14883, 17977, 21637, 26015, 31185, 44583, 63261, 75175, 147273, 173525, 239943, 281589, 329931, 386155, 526823, 966467, 1121505, 1505499, 2679689, 3087735

Offset: 1

Patrick De Geest, Nov 15 1999

nonn

Intersection of A005408 and A000041; A000035(a(n))*A167392(a(n)) = 1; a(n) = A000041(A052002(n+1)). - Reinhard Zumkeller, Nov 03 2015

Sean A. Irvine, Table of n, a(n) for n = 1..1000 (terms 1..529 from G. C. Gruebel).
K. Ono, Odd values of the partition function
K. Ono, Odd values of the partition function, Discrete Mathematics 169, 1997, pp. 263-268.

Cf. A052002, A000041.

a052003 n = a052003_list !! n
a052003_list = filter odd a000041_list
-- Reinhard Zumkeller, Nov 03 2015

select(type, [seq(combinat:-numbpart(n),n=1..200)],odd); # Robert Israel, Nov 03 2015

Select[PartitionsP[Range[50]], OddQ] (* Vladimir Reshetnikov, Nov 02 2015 *)

for(n=1, 100, if((k=numbpart(n))%2==1, print1(k", "))) \\ Altug Alkan, Nov 02 2015

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)

A236108 Nonprimes whose proper divisors are partition numbers.

Original entry on oeis.org

4, 6, 9, 10, 14, 15, 21, 22, 25, 33, 35, 49, 55, 77, 121, 202, 303, 505, 707, 1111, 10201, 35954, 53931, 89885, 125839, 197747, 1815677, 21239726, 31859589, 53099315, 74339041, 116818493, 323172529, 1072606163, 13241661778, 19862492667, 33104154445, 46345816223, 72829139779

Offset: 1

Omar E. Pol, Jan 22 2014

nonn

Known terms are squares of A049575 or products of 2 distinct terms of A049575. - Michel Marcus, Jan 25 2023

This conjecture holds for terms <= 10^16. - David A. Corneth, Jan 25 2023

			10 is in the sequence because 10 is a nonprime number and the proper divisors of 10 are 1, 2, 5, which are also partition numbers.

David A. Corneth, Table of n, a(n) for n = 1..81

Cf. A000041, A018252, A049575, A167392.

Cf. A236102, A236103, A236105, A236107, A236110.

isA000041 := proc(n)
    local k,P;
    for k from 1 do
        P := combinat[numbpart](k) ;
        if P > n then
            return false;
        elif P = n then
            return true ;
        end if;
    end do:
end proc:
isA236108 := proc(n)
    local pdvs,d ;
    if n =1 or isprime(n) then
        return false;
    end if;
    pdvs := numtheory[divisors](n) minus {n} ;
    for d in pdvs do
        if not isA000041(d) then
            return false;
        end if;
    end do:
    return true;
end proc:
for n from 1 to 300000 do
    if isA236108(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jan 29 2014

partitionNumbers = Table[PartitionsP[n], {n, 1, 1000}];
Select[Range[2, 10000],
 If[! PrimeQ[#],
ContainsOnly[Divisors[#][[2 ;; -2]], partitionNumbers]] &] (* Julien Kluge, Dec 03 2016 *)

a(17)-a(26) from R. J. Mathar, Jan 29 2014
a(27)-a(32) from Jon E. Schoenfield, Feb 05 2014
a(33)-a(34) from Michel Marcus, Jan 24 2023
More terms from David A. Corneth, Jan 25 2023

A065728 Partition numbers (A000041) that are semiprimes (A001358).

Original entry on oeis.org

15, 22, 77, 1255, 2012558, 2679689, 9289091, 18004327, 38887673, 56634173, 72533807, 82010177, 104651419, 2056148051, 2552338241, 20390982757, 27517052599, 118159068427, 749474411781, 5134205287973, 18028182516671

Offset: 1

Patrick De Geest, Nov 18 2001

nonn

Enoch Haga asks if this is a finite sequence. The larger these numbers get, the more opportunity for more factors.

			E.g., the 808th partition number 8151756509675604512522473567 = 5963320232189 * 1366982853893003.

Harry J. Smith, Table of n, a(n) for n=1,...,100
Hisanori Mishima, Factorization results
Eric Weisstein's World of Mathematics, Semiprime.

Intersection of A001358 and A000041.

Cf. A049575, A065729.

Select[PartitionsP[Range[0,450]],PrimeOmega[#]==2&] (* Harvey P. Dale, Sep 19 2016 *)

{ n=0; for (m=1, 10^9, p=numbpart(m); if (bigomega(p) == 2, write("b065728.txt", n++, " ", p); if (n==100, return)) ) } \\ Harry J. Smith, Oct 28 2009

A064911(a(n))*A167392(a(n)) = 1. [From Reinhard Zumkeller, Nov 03 2009]

OFFSET changed from 0,1 to 1,1 by Harry J. Smith, Oct 28 2009

A167376 Complement of the partition numbers.

Original entry on oeis.org

0, 4, 6, 8, 9, 10, 12, 13, 14, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82

Offset: 1

Reinhard Zumkeller, Nov 03 2009

nonn

Numbers not occurring in A000041;

A167392(a(n)) = 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000041, A167377, A167392.

import Data.List.Ordered (minus)
a167376 n = a167376_list !! n
a167376_list = minus [0..] a000041_list
-- Reinhard Zumkeller, Nov 03 2015

cp's OEIS Frontend

A186114 Triangle of regions and partitions of integers (see Comments lines for definition).

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A193870 Triangle of regions and partitions of integers (see Comments lines for definition).

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A049575 Prime partition numbers.

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A052001 Even partition numbers.

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A087183 Partition numbers of the form 3*k.

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A052003 Odd partition numbers.

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A236103 Number of distinct partition numbers dividing n.

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