A072213 -id:A072213 - cp's OEIS frontend

A128626 A triangular array distributing the values of sequence A072213 (cf. A115994).

1, 4, 1, 9, 20, 1, 16, 140, 74, 1, 25, 572, 1136, 224, 1, 36, 1785, 8866, 6685, 604, 1, 49, 4600, 47152, 88380, 31851, 1492, 1, 64, 10416, 194282, 737059, 665542, 130808, 3458, 1, 81, 21320, 665769, 4512584, 8211274, 4105870, 479826, 7602, 1, 100, 40425

Offset: 0

Alford Arnold, Mar 15 2007

			A115994 distributes the numeric partition sequence A000041 and A072213 records partition values for sequence A000290 (the squares).
Therefore the table begins:
   1;
   4,    1;
   9,   20,     1;
  16,  140,    74,     1;
  25,  572,  1136,   224,     1;
  36, 1785,  8866,  6685,   604,    1;
  49, 4600, 47152, 88380, 31851, 1492, 1;
  ...

Nathaniel Johnston, Table of n, a(n) for n = 0..209

Cf. A000041, A000290, A115994.

nn:=8: g:=sum(t^k*q^(k^2)/product((1-q^j)^2, j=1..k), k=1..nn): gser:=series(g, q=0, nn^2+1): for n from 1 to nn do P[n]:=coeff(gser, q^(n^2)) od: for n from 1 to nn do seq(coeff(P[n], t^j), j=1..n); od; # Nathaniel Johnston, Apr 30 2011

A072243 Number of distinct partitions of n^2.

Original entry on oeis.org

1, 1, 2, 8, 32, 142, 668, 3264, 16444, 84756, 444793, 2368800, 12769602, 69545358, 382075868, 2114965120, 11784471548, 66043042088, 372022512608, 2105220502772, 11962163400706, 68223286792200, 390406746862530, 2240962117491470, 12899456450932840

Offset: 0

Robert G. Wilson v, Jul 06 2002

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1260 (terms 0..200 from Alois P. Heinz)

Cf. A000009, A000041, A000290, A072213, A281489.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= n-> b(n^2):
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 22 2017

Mathematica
```
Table[ PartitionsQ[n^2], {n, 1, 24}]
```

a(n) ~ exp(Pi*n/sqrt(3)) / (4*3^(1/4)*n^(3/2)). - Vaclav Kotesovec, Dec 01 2015

a(n) = A000009(A000290(n)). - Alois P. Heinz, Jan 22 2017

a(0)=1 prepended by Alois P. Heinz, Jan 22 2017

A238640 Position of [n, n, ..., n] (n n's) in Mathematica-ordered list of partitions of n^2.

Original entry on oeis.org

1, 1, 3, 19, 168, 1582, 15546, 157051, 1625368, 17159223, 184277224, 2008388660, 22172275440, 247558926150, 2791793968821, 31764451979736, 364283594455091, 4207485803818522, 48908343969469479, 571811846280602486, 6720473048598172508, 79363083519870386700

Offset: 0

Clark Kimberling, Mar 04 2014

nonn

			The partitions of 4 in Mathematica order are 4, 31, 22, 211, 1111.  The position of 22 is a(2) = 3.

Alois P. Heinz, Table of n, a(n) for n = 0..140

Cf. A000290, A072213, A080577 (Mathematica ordering), A238638, A238639, A330661, A332706.

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1) +`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> 1 +add(b(n^2-j, j), j=n+1..n^2):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 03 2014

r[n_] := Table[n, {k, 1, n}]; Flatten[Table[Position[IntegerPartitions[n^2], r[n]], {n, 0, 8}]]
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1]+If[i>n, 0, b[n-i, i]]]]; a[n_] := 1+Sum[b[n^2-j, j], {j, n+1, n^2}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Oct 28 2015, after Alois P. Heinz *)

a(9)-a(21) from Alois P. Heinz, Sep 03 2014

A284594 Numbers whose square has a prime number of partitions.

Original entry on oeis.org

2, 6, 29, 36, 2480, 14881

Offset: 1

Serge Batalov, Mar 29 2017

Because asymptotically A072213(n) = A000041(n^2) ~ exp(Pi*sqrt(2/3)*n) / (4*sqrt(3)*n^2), the sum of the prime probabilities ~ 1/log(A072213(n)) is diverging and there are no obvious restrictions on primality; therefore, this sequence may be conjectured to be infinite.
Curiously, both A000041(6^2) and A000041(6^4) are prime; in addition, A000041(6^3) and A000041(6^1) are prime, but for no other powers A000041(6^k) is known (or can be expected) to be prime.
a(7) > 649350.

			a(2) = 6 is in the sequence because A000041(6^2) = 17977 is a prime.

Chris K. Caldwell, Top twenty prime partition numbers, The Prime Pages.
Eric Weisstein's World of Mathematics, Partition Function P
Eric Weisstein's World of Mathematics, Integer Sequence Primes

Cf. A000041, A046063, A072213, A285086, A285087, A285088.

for(n=1,2500,if(ispseudoprime(numbpart(n^2)),print1(n,", ")))

A285086 Numbers n such that the number of partitions of n^2+1 (=A000041(n^2+1)) is prime.

Original entry on oeis.org

1, 2, 3914

Offset: 1

Serge Batalov, Apr 09 2017

Because asymptotically A000041(n^2+1) ~ exp(Pi*sqrt(2/3*(n^2+1))) / (4*sqrt(3)*(n^2+1)), the sum of the prime probabilities ~ 1/log(A000041(n^2+1)) is diverging and there are no obvious restrictions on primality; therefore, this sequence may be conjectured to be infinite.

a(4) > 90000.

			a(2) = 2 is in the sequence because A000041(2^2+1) = 7 is a prime.

Chris K. Caldwell, Top twenty prime partition numbers, The Prime Pages.
Eric Weisstein's World of Mathematics, Partition Function P
Eric Weisstein's World of Mathematics, Integer Sequence Primes

Cf. A000041, A046063, A072213, A284594, A285087, A285088.

for(n=1,3920,if(ispseudoprime(numbpart(n^2+1)),print1(n,", ")))

A285087 Numbers n such that the number of partitions of n^2-1 is prime.

Original entry on oeis.org

2, 13, 21, 46909

Offset: 1

Serge Batalov, Apr 09 2017

Because asymptotically A000041(n^2-1) ~ exp(Pi*sqrt(2/3*(n^2-1))) / (4*sqrt(3)*(n^2-1)), the sum of the prime probabilities ~1/log(A000041(n^2-1)) is diverging and there are no obvious restrictions on primality; therefore, this sequence may be conjectured to be infinite.

a(5) > 50000.

			13 is in the sequence because A000041(13^2-1) = 228204732751 is a prime.

Chris K. Caldwell, Top twenty prime partition numbers, The Prime Pages.
Eric Weisstein's World of Mathematics, Partition Function P
Eric Weisstein's World of Mathematics, Integer Sequence Primes

Cf. A000041, A046063, A072213, A284594, A285086, A285088.

for(n=1,2000,if(ispseudoprime(numbpart(n^2-1)),print1(n,", ")))

from itertools import count, islice
from sympy import isprime, npartitions
def A285087_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: isprime(npartitions(n**2-1)), count(max(startvalue,1)))
A285087_list = list(islice(A285087_gen(),3)) # Chai Wah Wu, Nov 20 2023

{n: A000041(n^2-1) in A000040}.

A285088 Numbers n such that the number of partitions of n(n+1)/2 (=A000041(A000217(n))) is prime.

Original entry on oeis.org

2, 3, 8, 3947, 43968, 61681

Offset: 1

Serge Batalov, Apr 09 2017

Because asymptotically A000041(n*(n+1)/2) ~ exp(Pi*sqrt(2/3*(n*(n+1)/2))) / (4*sqrt(3)*(n*(n+1)/2)), the sum of the prime probabilities ~1/log(A000041(n*(n+1)/2)) is diverging and there are no obvious restrictions on primality; therefore, this sequence may be conjectured to be infinite.

			a(3) = 8 is in the sequence because A000041(8*9/2) = 17977 is a prime.

Chris K. Caldwell, Top twenty prime partition numbers, The Prime Pages.
Eric Weisstein's World of Mathematics, Partition Function P
Eric Weisstein's World of Mathematics, Integer Sequence Primes

Cf. A000041, A046063, A072213, A284594, A285086, A285087.

for(n=1,2000,if(ispseudoprime(numbpart(n*(n+1)/2)),print1(n,", ")))

A347466 Number of factorizations of n^2.

Original entry on oeis.org

1, 2, 2, 5, 2, 9, 2, 11, 5, 9, 2, 29, 2, 9, 9, 22, 2, 29, 2, 29, 9, 9, 2, 77, 5, 9, 11, 29, 2, 66, 2, 42, 9, 9, 9, 109, 2, 9, 9, 77, 2, 66, 2, 29, 29, 9, 2, 181, 5, 29, 9, 29, 2, 77, 9, 77, 9, 9, 2, 269, 2, 9, 29, 77, 9, 66, 2, 29, 9, 66, 2, 323, 2, 9, 29, 29

Offset: 1

Gus Wiseman, Sep 23 2021

nonn

A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

			The a(1) = 1 through a(8) = 11 factorizations:
  ()  (4)    (9)    (16)       (25)   (36)       (49)   (64)
      (2*2)  (3*3)  (2*8)      (5*5)  (4*9)      (7*7)  (8*8)
                    (4*4)             (6*6)             (2*32)
                    (2*2*4)           (2*18)            (4*16)
                    (2*2*2*2)         (3*12)            (2*4*8)
                                      (2*2*9)           (4*4*4)
                                      (2*3*6)           (2*2*16)
                                      (3*3*4)           (2*2*2*8)
                                      (2*2*3*3)         (2*2*4*4)
                                                        (2*2*2*2*4)
                                                        (2*2*2*2*2*2)

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Positions of 2's are the primes (A000040), which have squares A001248.
The restriction to powers of 2 is A058696.
The additive version (partitions) is A072213.
The case of integer alternating product is A347459, nonsquared A347439.
A000290 lists squares, complement A000037.
A001055 counts factorizations.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A347050 = factorizations with alternating permutation, complement A347706.
Cf. A000041, A062312, A120452, A144338, A273013, A330972, A345957, A346635, A347437, A347438, A347457, A347460, A347464.

b:= proc(n, k) option remember; `if`(n>k, 0, 1)+`if`(isprime(n), 0,
      add(`if`(d>k, 0, b(n/d, d)), d=numtheory[divisors](n) minus {1, n}))
    end:
a:= proc(n) option remember; b((l-> mul(ithprime(i)^l[i], i=1..nops(l)))(
      sort(map(i-> i[2], ifactors(n^2)[2]), `>`))$2)
    end:
seq(a(n), n=1..76);  # Alois P. Heinz, Oct 14 2021

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[facs[n^2]],{n,25}]

A001055(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A001055(n/d, d))); (s));
A347466(n) = A001055(n^2); \\ Antti Karttunen, Oct 13 2021

a(n) = A001055(A000290(n)).

A128854 Number of partitions of n^3.

Original entry on oeis.org

1, 1, 22, 3010, 1741630, 3163127352, 15285151248481, 175943559810422753, 4453575699570940947378, 233202632378520643600875145, 24061467864032622473692149727991, 4700541557913558825461268913956492487

Offset: 0

Zerinvary Lajos, Apr 16 2007

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..93

Cf. A000041, A000578, A072213.

Table[PartitionsP[n^3],{n,0,11}] (* James C. McMahon, Jan 12 2025 *)

combinat::partitions::count(n^3) $n=0..15

a(n) = numbpart(n^3); \\ Michel Marcus, Jul 12 2023

a(n) = A000041(A000578(n)). - Michel Marcus, Aug 15 2013

a(n) ~ exp(Pi*sqrt(2/3)*n^(3/2))/(4*sqrt(3)*n^3). - Ilya Gutkovskiy, Jan 13 2017

A093115 Number of partitions of n^2 into squares not greater than n.

Original entry on oeis.org

1, 1, 1, 1, 5, 7, 10, 13, 17, 108, 159, 228, 317, 430, 572, 748, 5753, 8125, 11266, 15376, 20672, 27430, 35942, 46575, 59717, 523905, 708028, 946875, 1253880, 1645224, 2140099, 2761318, 3535658, 4494602, 5674753, 7118724, 69766770, 90940578, 117756370

Offset: 0

Reinhard Zumkeller, Mar 21 2004

nonn

			n=6: 6^2 = 9*2^2 = 8*2^2+4*1^2 = 7*2^2+8*1^2 = 6*2^2+12*1^2 = 5*2^2+16*1^2 = 4*2^2+20*1^2 = 3*2^2+24*1^2 = 2*2^2+28*1^2 = 1*2^2+32*1^2 = 36*1^2, therefore a(6)=10.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A093116, A092362, A001156, A037444, A078134.
Cf. A072925.
Cf. A072213, A161407. [Reinhard Zumkeller, Jun 10 2009]

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<1, 0, b(n, i-1) +`if`(i^2>n, 0, b(n-i^2, i))))
    end:
a:= proc(n) local r; r:= isqrt(n);
      b(n^2, r-`if`(r^2>n, 1, 0))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 15 2013

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i^2 > n, 0, b[n-i^2, i]]]]; a[n_] := (r = Sqrt[n] // Floor; b[n^2, r - If[r^2 > n, 1, 0]]); Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 29 2015, after Alois P. Heinz *)

Coefficient of x^(n^2) in the series expansion of Product_{k=1..floor(sqrt(n))} 1/(1 - x^(k^2)). - Vladeta Jovovic, Mar 24 2004

More terms from Vladeta Jovovic, Mar 24 2004

Corrected a(0) by Alois P. Heinz, Apr 15 2013

cp's OEIS Frontend

A128626 A triangular array distributing the values of sequence A072213 (cf. A115994).

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A072243 Number of distinct partitions of n^2.

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A238640 Position of [n, n, ..., n] (n n's) in Mathematica-ordered list of partitions of n^2.

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A284594 Numbers whose square has a prime number of partitions.

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A285086 Numbers n such that the number of partitions of n^2+1 (=A000041(n^2+1)) is prime.

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A285087 Numbers n such that the number of partitions of n^2-1 is prime.

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A285088 Numbers n such that the number of partitions of n(n+1)/2 (=A000041(A000217(n))) is prime.

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A347466 Number of factorizations of n^2.

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