cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 117 results. Next

A324588 Heinz numbers of integer partitions of n into perfect squares (A001156).

Original entry on oeis.org

1, 2, 4, 7, 8, 14, 16, 23, 28, 32, 46, 49, 53, 56, 64, 92, 97, 98, 106, 112, 128, 151, 161, 184, 194, 196, 212, 224, 227, 256, 302, 311, 322, 343, 368, 371, 388, 392, 419, 424, 448, 454, 512, 529, 541, 604, 622, 644, 661, 679, 686, 736, 742, 776, 784, 827, 838
Offset: 1

Views

Author

Gus Wiseman, Mar 08 2019

Keywords

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Also products of elements of A011757.

Examples

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   4: {1,1}
   7: {4}
   8: {1,1,1}
  14: {1,4}
  16: {1,1,1,1}
  23: {9}
  28: {1,1,4}
  32: {1,1,1,1,1}
  46: {1,9}
  49: {4,4}
  53: {16}
  56: {1,1,1,4}
  64: {1,1,1,1,1,1}
  92: {1,1,9}
  97: {25}
  98: {1,4,4}

Crossrefs

Cf. A001156, A011757, A033461, A052335, A056239, A062457, A078135, A112798, A117144, A118914, A276078.
Cf. A109298, A324524, A324525, A324571, A324572, A324587.

Programs

  • Mathematica
    Select[Range[100],And@@Cases[FactorInteger[#],{p_,_}:>IntegerQ[Sqrt[PrimePi[p]]]]&]

A294529 Binomial transform of A001156.

Original entry on oeis.org

1, 2, 4, 8, 17, 38, 86, 192, 420, 905, 1939, 4163, 8987, 19494, 42368, 91990, 199127, 429345, 921982, 1972553, 4206909, 8949412, 19001874, 40293048, 85373962, 180826115, 382957231, 811027414, 1717497958, 3636335170, 7695599294, 16275268520, 34389570596
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 02 2017

Keywords

Links

Crossrefs

Cf. A001156, A218481, A266232, A294500, A294530.

Programs

  • Mathematica
    nmax = 40; s = CoefficientList[Series[Product[1/(1 - x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x]; Table[Sum[Binomial[n, k] * s[[k+1]], {k, 0, n}], {n, 0, nmax}]

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * A001156(k).
a(n) ~ exp(3 * 2^(-5/3) * Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3)) * Zeta(3/2)^(2/3) * 2^(n - 7/6) / (sqrt(3) * Pi^(7/6) * n^(7/6)).
G.f.: (1/(1 - x))*Product_{k>=1} 1/(1 - x^(k^2)/(1 - x)^(k^2)). - Ilya Gutkovskiy, Aug 20 2018

A045842 Expansion of Product_{k>=0} 1/(1 - x^(k+1))^A001156(k).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 19, 29, 43, 65, 94, 138, 197, 284, 403, 571, 801, 1124, 1562, 2170, 2992, 4118, 5636, 7700, 10467, 14201, 19189, 25873, 34763, 46614, 62305, 83113, 110565, 146791, 194408, 256985, 338934, 446211, 586231, 768855, 1006450, 1315304, 1715882, 2234957, 2906250
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			1/((1-x)^1 * (1-x^2)^1 * (1-x^3)^1 * (1-x^4)^1 * (1-x^5)^2 * (1-x^6)^2 * (1-x^7)^2 * (1-x^8)^2 * (1-x^9)^3 * ...) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 + 13*x^6 + 19*x^7 + 29*x^8 + 43*x^9 + ... .

Links

Crossrefs

Cf. A001156.

Formula

G.f.: Product_{k>=0} 1/(1 - x^(k+1))^A001156(k).

Extensions

More terms from Seiichi Manyama, Nov 10 2018

A285218 Indices of primes in A001156.

Original entry on oeis.org

4, 5, 6, 7, 8, 12, 21, 25, 28, 32, 34, 36, 44, 51, 58, 68, 71, 73, 76, 84, 85, 105, 117, 131, 132, 148, 150, 160, 162, 163, 170, 172, 188, 190, 216, 226, 233, 249, 252, 253, 264, 273, 284, 307, 338, 356, 358, 372, 378, 383, 390, 424, 435, 449, 456, 468, 479
Offset: 1

Views

Author

Vaclav Kotesovec, Apr 14 2017

Keywords

Examples

			51 is in the sequence because A001156(51) = 107 is prime.

Links

Crossrefs

Cf. A001156, A046063.

A053216 Number of integers that can be partitioned into squares in n different ways, or the number of times n occurs in A001156.

Original entry on oeis.org

4, 4, 1, 3, 1, 3, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0
Offset: 1

Views

Author

Asher Auel, Dec 27 1999

Keywords

Crossrefs

Cf. A001156, A053217, A000290, A002828.

A053217 Numbers that do not occur in A001156.

Original entry on oeis.org

7, 11, 15, 17, 18, 22, 24, 25, 29, 30, 32, 33, 35, 36, 39, 40, 41, 42, 44, 45, 47, 48, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 64, 65, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 79, 80, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 99, 100, 101, 102, 103, 105, 106
Offset: 1

Views

Author

Asher Auel, Dec 27 1999

Keywords

Crossrefs

Cf. A001156, A053216.

A304046 Numbers k such that A001156(k) is divisible by k.

Original entry on oeis.org

1, 49, 987, 1044, 7152, 7695, 9477, 20464, 21375, 56887, 115354, 163871
Offset: 1

Views

Author

Vaclav Kotesovec, May 05 2018

Keywords

Comments

No other terms below 250000.

Examples

			987 is in the sequence because A001156(987) = 3514989282 = 3561286 * 987.

Crossrefs

Cf. A001156.

A007864 Number of matrix bundles of codimension n (Euler transform of A001156).

Original entry on oeis.org

1, 2, 4, 7, 11, 19, 30, 49, 76, 118, 180, 276, 411, 614, 908, 1336, 1944, 2824, 4067, 5839, 8326, 11829, 16719, 23557, 33019, 46142, 64226, 89117, 123198, 169841, 233373, 319817, 436982, 595554, 809503, 1097714, 1484805, 2003938, 2698410
Offset: 1

Views

Author

Peter J. Cameron

Keywords

References

  • V. I. Arnold, Singularity Theory, Cambridge Univ. Press, 1981, p. 57.

Links

Extensions

Corrected and extended by Vladeta Jovovic, Sep 05 2002

A033461 Number of partitions of n into distinct squares.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 0, 2, 2, 0, 0, 2, 3, 1, 1, 2, 2, 1, 1, 1, 1, 1, 0, 2, 3, 1, 1, 4, 3, 0, 1, 2, 2, 1, 0, 1, 4, 3, 0, 2, 4, 2, 1, 3, 2, 1, 2, 3, 3, 2, 1, 3, 6, 3, 0, 2, 5, 3, 0, 1, 3, 3, 3, 4
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

"WEIGH" transform of squares A000290.
a(n) = 0 for n in {A001422}, a(n) > 0 for n in {A003995}. - Alois P. Heinz, May 14 2014
Number of partitions of n in which each part i has multiplicity i. Example: a(50)=3 because we have [1,2,2,3,3,3,6,6,6,6,6,6], [1,7,7,7,7,7,7,7], and [3,3,3,4,4,4,4,5,5,5,5,5]. - Emeric Deutsch, Jan 26 2016
The Heinz numbers of integer partitions into distinct pairs are given by A324587. - Gus Wiseman, Mar 09 2019
From Gus Wiseman, Mar 09 2019: (Start)
Equivalent to Emeric Deutsch's comment, a(n) is the number of integer partitions of n where the multiplicities (where if x < y the multiplicity of x is counted prior to the multiplicity of y) are equal to the distinct parts in increasing order. The Heinz numbers of these partitions are given by A109298. For example, the first 30 terms count the following integer partitions:
1: (1)
4: (22)
5: (221)
9: (333)
10: (3331)
13: (33322)
14: (333221)
16: (4444)
17: (44441)
20: (444422)
21: (4444221)
25: (55555)
25: (4444333)
26: (555551)
26: (44443331)
29: (5555522)
29: (444433322)
30: (55555221)
30: (4444333221)
The case where the distinct parts are taken in decreasing order is A324572, with Heinz numbers given by A324571.
(End)

Examples

			a(50)=3 because we have [1,4,9,36], [1,49], and [9,16,25]. - _Emeric Deutsch_, Jan 26 2016
From _Gus Wiseman_, Mar 09 2019: (Start)
The first 30 terms count the following integer partitions:
   1: (1)
   4: (4)
   5: (4,1)
   9: (9)
  10: (9,1)
  13: (9,4)
  14: (9,4,1)
  16: (16)
  17: (16,1)
  20: (16,4)
  21: (16,4,1)
  25: (25)
  25: (16,9)
  26: (25,1)
  26: (16,9,1)
  29: (25,4)
  29: (16,9,4)
  30: (25,4,1)
  30: (16,9,4,1)
(End)

References

  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 288-289.

Links

Crossrefs

Cf. A001422, A003995, A078434, A242434 (the same for compositions), A279329.
Cf. A001156 (non-strict case), A001462, A005117, A052335, A078135, A109298, A114638, A117144, A324571, A324572, A324587, A324588.
Row sums of A341040.

Programs

  • Maple
    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-1))))
    end:
a:= n-> b(n, isqrt(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, May 14 2014
  • Mathematica
    nn=10; CoefficientList[Series[Product[(1+x^(k*k)), {k,nn}], {x,0,nn*nn}], x] (* T. D. Noe, Jul 24 2006 *)
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-1]]]]; a[n_] := b[n, Floor[Sqrt[n]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Sep 21 2015, after Alois P. Heinz *)
nmax = 20; poly = ConstantArray[0, nmax^2 + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j + 1]] += poly[[j - k^2 + 1]], {j, nmax^2, k^2, -1}];, {k, 2, nmax}]; poly (* Vaclav Kotesovec, Dec 09 2016 *)
Table[Length[Select[IntegerPartitions[n],Reverse[Union[#]]==Length/@Split[#]&]],{n,30}] (* Gus Wiseman, Mar 09 2019 *)
  • PARI
    a(n)=polcoeff(prod(k=1,sqrt(n),1+x^k^2), n)
  • PARI
    first(n)=Vec(prod(k=1,sqrtint(n),1+'x^k^2,O('x^(n+1))+1)) \\ Charles R Greathouse IV, Sep 03 2015
  • Python
    from functools import cache
from sympy.core.power import isqrt
@cache
def b(n,i):
  # Code after Alois P. Heinz
  if n == 0: return 1
  if i == 0: return 0
  i2 = i*i
  return b(n, i-1) + (0 if i2 > n else b(n - i2, i-1))
a = lambda n: b(n, isqrt(n))
print([a(n) for n in range(1, 101)]) # Darío Clavijo, Nov 30 2023

Formula

G.f.: Product_{n>=1} ( 1+x^(n^2) ).
a(n) ~ exp(3 * 2^(-5/3) * Pi^(1/3) * ((sqrt(2)-1)*zeta(3/2))^(2/3) * n^(1/3)) * ((sqrt(2)-1)*zeta(3/2))^(1/3) / (2^(4/3) * sqrt(3) * Pi^(1/3) * n^(5/6)), where zeta(3/2) = A078434. - Vaclav Kotesovec, Dec 09 2016
See Murthy, Brack, Bhaduri, Bartel (2018) for a more complete asymptotic expansion. - N. J. A. Sloane, Aug 17 2018

Extensions

More terms from Michael Somos

A006456 Number of compositions (ordered partitions) of n into squares.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 5, 7, 11, 16, 22, 30, 43, 62, 88, 124, 175, 249, 354, 502, 710, 1006, 1427, 2024, 2870, 4068, 5767, 8176, 11593, 16436, 23301, 33033, 46832, 66398, 94137, 133462, 189211, 268252, 380315, 539192, 764433, 1083764, 1536498, 2178364
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A280542.
Row sums of A337165.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<0, 0,
      `if`(n=0, 1, add(a(n-j^2), j=1..isqrt(n))))
    end:
seq(a(n), n=0..44);  # Alois P. Heinz, Jul 26 2025
  • Mathematica
    a[n_]:=a[n]=If[n==0, 1, Sum[a[n - k], {k, Select[Range[n], IntegerQ[Sqrt[#]] &]}]]; Table[a[n], {n,0,  100}] (* Indranil Ghosh, Jul 28 2017, after David W. Wilson's formula *)
  • PARI
    N=66;  x='x+O('x^N);
Vec( 1/( 1 - sum(k=1,1+sqrtint(N), x^(k^2) ) ) )
/* Joerg Arndt, Sep 30 2012 */
  • Python
    from gmpy2 import is_square
class Memoize:
    def _init_(self, func):
        self.func=func
        self.cache={}
    def _call_(self, arg):
        if arg not in self.cache:
            self.cache[arg] = self.func(arg)
        return self.cache[arg]
@Memoize
def a(n): return 1 if n==0 else sum([a(n - k) for k in range(1, n + 1) if is_square(k)])
print([a(n) for n in range(101)]) # Indranil Ghosh, Jul 28 2017, after David W. Wilson's formula

Formula

a(0) = 1; a(n) = Sum_{1 <= k^2 <= n} a(n-k^2), if n > 0. - David W. Wilson
G.f.: 1/(1-x-x^4-x^9-....) - Jon Perry, Jul 04 2004
a(n) ~ c * d^n, where d is the root of the equation EllipticTheta(3, 0, 1/d) = 3, d = 1.41774254618138831428829091099000662953179532057717725688..., c = 0.46542113389379672452973940263069782869244877335179331541... - Vaclav Kotesovec, May 01 2014, updated Jan 05 2017
G.f.: 2/(3 - theta_3(q)), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 08 2018

Extensions

Name corrected by Bob Selcoe, Feb 12 2014
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