A072213 -id:A072213 - cp's OEIS frontend

A093116 Number of partitions of n^2 into squares not less than n.

1, 1, 1, 1, 2, 2, 2, 1, 2, 5, 4, 4, 5, 9, 15, 23, 24, 13, 20, 32, 55, 84, 113, 185, 303, 545, 167, 298, 435, 716, 1055, 1701, 2584, 4213, 6471, 10218, 15884, 4856, 7376, 11231, 17221, 26054, 39583, 60109, 91622, 138569, 209951, 318368, 483098, 730183

Offset: 0

Reinhard Zumkeller, Mar 21 2004

nonn

			n=10: 10^2 = 100 = 64+36 = 36+16+16+16+16 = 25+25+25+25, all other partitions of 100 into squares contain parts < 10, therefore a(10) = 4.

Alois P. Heinz, Table of n, a(n) for n = 0..250

Cf. A093115, A092362, A001156, A037444, A078134.

Cf. A072213, A161408. [Reinhard Zumkeller, Jun 10 2009]

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

b[n_, i_] := b[n, i] = If[n==0, 1, If[i^2>n, 0, b[n, i+1] + b[n-i^2, i]]]; a[n_] := With[{r = Sqrt[n]//Floor}, b[n^2, r + If[r^2Jean-François Alcover, Oct 26 2015, after Alois P. Heinz *)

A161407 Number of partitions of n^2 into parts smaller than n.

Original entry on oeis.org

1, 0, 1, 5, 30, 185, 1226, 8442, 60289, 442089, 3314203, 25295011, 195990980, 1538069121, 12203218743, 97746332667, 789480879664, 6423539487002, 52607252796831, 433368610079872, 3588859890833443, 29862449600982149, 249560820679038935, 2093852201126089073

Offset: 0

Reinhard Zumkeller, Jun 10 2009

nonn

			a(3) = #{2+2+2+2+1, 2+2+2+1+1+1, 2+2+5x1, 2+7x1, 9x1} = 5.

Alois P. Heinz, Table of n, a(n) for n = 0..320

Cf. A072213, A093115, A109655, A161408.

a := proc (n) local G, Gser: G := 1/(product(1-x^j, j = 1 .. n-1)): Gser := series(G, x = 0, n^2+5): coeff(Gser, x, n^2) end proc: 1, seq(a(n), n = 1 .. 23); # Emeric Deutsch, Jun 20 2009
# second Maple program:
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-> b(n^2, n-1):
seq(a(n), n=0..30);  # Alois P. Heinz, Dec 21 2014

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_] := b[n^2, n-1]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 30 2015, after Alois P. Heinz *)

a(n) ~ c * d^n / n^2, where d = A258268 = 9.153370192454122461948530292401354... and c = 0.0881548837986971165169272782933415... - Vaclav Kotesovec, Sep 08 2021

More terms from Emeric Deutsch, Jun 20 2009

a(0)=1 from Alois P. Heinz, Dec 21 2014

A161408 Number of partitions of n^2 into parts greater than n.

Original entry on oeis.org

1, 0, 1, 2, 6, 17, 56, 179, 607, 2076, 7269, 25736, 92360, 334506, 1222463, 4499892, 16673446, 62130710, 232700539, 875483029, 3307244256, 12539455600, 47702381244, 182021195608, 696487788847, 2671877845634, 10274126646175, 39593295985708, 152889766657797

Offset: 0

Reinhard Zumkeller, Jun 10 2009

nonn

			a(4) = #{16, 11+5, 10+6, 9+7, 8+8, 6+5+5} = 6.

Seiichi Manyama, Table of n, a(n) for n = 0..100

Cf. A072213, A093116, A161407.

a := proc (n) local G, Gser: G := 1/(product(1-x^j, j = n+1 .. n^2)): Gser := series(G, x = 0, n^2+5): coeff(Gser, x, n^2) end proc: 1, seq(a(n), n = 1 .. 27); # Emeric Deutsch, Jun 22 2009

a[n_] := a[n] = 1/Product[1 - x^j, {j, n + 1, n^2}] + O[x]^(n^2 + 1) // CoefficientList[#, x]& // Last;
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 28}] (* Jean-François Alcover, May 18 2017, adapted from Maple *)

a(n) ~ c * d^n / n^(3/2), where d = 4.076293... and c = 0.187307... - Vaclav Kotesovec, Sep 08 2021

Extended by Emeric Deutsch, Jun 22 2009

a(0)=1 from Alois P. Heinz, Dec 21 2014

A126683 Number of partitions of the n-th triangular number n(n+1)/2 into distinct odd parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 8, 16, 33, 68, 144, 312, 686, 1523, 3405, 7652, 17284, 39246, 89552, 205253, 472297, 1090544, 2525904, 5867037, 13663248, 31896309, 74628130, 174972341, 411032475, 967307190, 2280248312, 5383723722, 12729879673, 30141755384, 71462883813

Offset: 0

Moshe Shmuel Newman, Feb 15 2007

nonn

Also the number of self-conjugate partitions of the n-th triangular number.

			The 5th triangular number is 15. Writing this as a sum of distinct odd numbers: 15 = 11 + 3 + 1 = 9 + 5 + 1 = 7 + 5 + 3 are all the possibilities. So a(5) = 4.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Sequences A066655 and A104383 do the same thing for triangular numbers, with partitions or distinct partitions. Sequences A072213 and A072243 are analogs for squares rather than triangular numbers.

Cf. A000217.

g:= mul(1+x^(2*j+1),j=0..900): seq(coeff(g,x,n*(n+1)/2),n=0..40); # Emeric Deutsch, Feb 27 2007
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i^2n, 0, b(n-2*i+1, i-1))))
    end:
a:= n-> b(n*(n+1)/2, ceil(n*(n+1)/4)*2-1):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 31 2018

a[n_] := SeriesCoefficient[QPochhammer[-x, x^2], {x, 0, n*(n+1)/2}];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, May 25 2018 *)

More terms from Emeric Deutsch, Feb 27 2007

a(0)=1 prepended by Alois P. Heinz, Jan 31 2018

A347585 Number of partitions of n^2 into n or more parts.

Original entry on oeis.org

1, 1, 4, 25, 201, 1773, 16751, 165083, 1681341, 17562238, 187255089, 2030853040, 22344663465, 248900855994, 2802367768848, 31848644363490, 364960085991118, 4212964989100093, 48953036382441044, 572178690287957687, 6723501191850208483, 79388206896842420091

Offset: 0

Seiichi Manyama, Sep 08 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..100

Cf. A058399, A072213, A161408, A206226, A206240.

a(n) = [x^(n^2)] Sum_{k>=n} x^k / Product_{j=1..k} (1 - x^j).
a(n) = A072213(n) + A206240(n) - A206226(n).
a(n) ~ exp(Pi*sqrt(2/3)*n) / (4*sqrt(3)*n^2). - Vaclav Kotesovec, Sep 14 2021

A347607 Number of partitions of n^n.

Original entry on oeis.org

1, 1, 5, 3010, 365749566870782, 8630901377559029573671524821295260243701883575513498104067

Offset: 0

Seiichi Manyama, Sep 08 2021

nonn

The next term a(6) = 1.30449952...*10^235 is too large to include.

a(7) = 1.5782589391...*10^1004. - Chai Wah Wu, Sep 09 2021

Seiichi Manyama, Table of n, a(n) for n = 0..6

Main diagonal of A347615.

Cf. A000041, A000312, A064682, A072213, A128854.

a:= n-> combinat[numbpart](n^n):
seq(a(n), n=0..6);  # Alois P. Heinz, Sep 09 2021

PARI
```
a(n) = numbpart(n^n);
```

from sympy.functions import partition
def A347607(n): return partition(n**n) # Chai Wah Wu, Sep 09 2021

a(n) = A000041(n^n).

A347615 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) is the number of partitions of n^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 1, 1, 1, 22, 30, 5, 1, 1, 1, 231, 3010, 231, 7, 1, 1, 1, 8349, 18004327, 1741630, 1958, 11, 1, 1, 1, 1741630, 133978259344888, 365749566870782, 3163127352, 17977, 15, 1, 1, 1, 4351078600, 233202632378520643600875145, 61847822068260244309086870983975, 1606903190858354689128371, 15285151248481, 173525, 22, 1

Offset: 0

Seiichi Manyama, Sep 08 2021

			Square array begins:
  1, 1,   1,       1,               1, ...
  1, 1,   1,       1,               1, ...
  1, 2,   5,      22,             231, ...
  1, 3,  30,    3010,        18004327, ...
  1, 5, 231, 1741630, 365749566870782, ...

Columns k=0..3 give A000012, A000041, A072213, A128854.
Rows n=0+1, 2-10 give A000012, A068413, A248728, A068413(2*n), A248730, A248732, A248734, A068413(3*n), A248728(2*n), A070177.
Main diagonal gives A347607.
Cf. A238016, A347617, A347618, A347621.

PARI
```
T(n, k) = numbpart(n^k);
```

T(n,k) = A000041(n^k).

A355704 Indices k of partition function p where p(k) and p(k) + 2 are twin primes.

Original entry on oeis.org

3, 4, 6, 13, 2335166

Offset: 1

Serge Batalov, Jul 14 2022

Because asymptotically size of partitions number function p(n) ~ O(exp(sqrt(n))), and probability of primality of p(n) ~ O(1/sqrt(n)) and combined probability of primality of p(n) and p(n)+-2 is ~ O(1/n), the sum of the prime probabilities is diverging and there are no obvious restrictions on primality; therefore, this sequence may be conjectured to be infinite.

a(6) > 10^7.

			13 is a term because A000041(13) = 101 is prime and 103 is prime.

Cf. A000041, A046063, A072213, A285086, A285087, A285088, A284594, A355705, A355706.

for(n=1, 2500, if(ispseudoprime(p=numbpart(n))&&ispseudoprime(p+2), print1(n,", ")))

A355706 Indices k of partition function p where p(k) is a twin prime.

Original entry on oeis.org

3, 4, 5, 6, 13, 186, 3542, 2335166

Offset: 1

Serge Batalov, Jul 15 2022

Because asymptotically the size of the partition number function p(n) is ~ O(exp(sqrt(n))), and the probability of primality of p(n) is ~ O(1/sqrt(n)) and the combined probability of primality of p(n) and p(n)+-2 is ~ O(1/n), the sum of the prime probabilities is diverging and there are no obvious restrictions on primality; therefore, this sequence may be conjectured to be infinite.
p(2335166), a 1696-digit number, was known to be prime and proven prime by F. Morain using his software (ca. April 2001), but the primality of p(2335166)+2 was found by targeted search (for this sequence) in July 2022.
a(9) > 10^7.

			13 is a term because A000041(13) = 101 is prime and 101 and 103 are twin primes.

Subsequence of A046063.
Union of A355704 and A355705.
Cf. A000041, A072213, A285086, A285087, A285088, A284594.

for(n=1, 3600, if(ispseudoprime(p=numbpart(n))&&(ispseudoprime(p-2)||ispseudoprime(p+2)), print1(n, ", ")))

A355705 Indices k of partition function p where p(k) and p(k) - 2 are twin primes.

Original entry on oeis.org

4, 5, 186, 3542

Offset: 1

Serge Batalov, Jul 15 2022

Because asymptotically size of partitions number function p(n) ~ O(exp(sqrt(n))), and probability of primality of p(n) ~ O(1/sqrt(n)) and combined probability of primality of p(n) and p(n)+-2 is ~ O(1/n), the sum of the prime probabilities is diverging and there are no obvious restrictions on primality; therefore, this sequence may be conjectured to be infinite.

a(5) > 10^7.

			4 is a term because A000041(4) = 5, and 3 and 5 are twin primes.
5 is a term because A000041(5) = 7, and 5 and 7 are twin primes.

Cf. A000041, A046063, A072213, A285086, A285087, A285088, A284594, A355704, A355706.

for(n=1, 3600, if(ispseudoprime(p=numbpart(n))&&ispseudoprime(p-2), print1(n, ", ")))

cp's OEIS Frontend

A093116 Number of partitions of n^2 into squares not less than n.

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Mathematica

A161407 Number of partitions of n^2 into parts smaller than n.

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Maple

Mathematica

Formula

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A161408 Number of partitions of n^2 into parts greater than n.

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Maple

Mathematica

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A126683 Number of partitions of the n-th triangular number n(n+1)/2 into distinct odd parts.

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Maple

Mathematica

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A347585 Number of partitions of n^2 into n or more parts.

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A347607 Number of partitions of n^n.

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Python

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A347615 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) is the number of partitions of n^k.

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PARI

Formula

A355704 Indices k of partition function p where p(k) and p(k) + 2 are twin primes.

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