A240088 -id:A240088 - cp's OEIS frontend

A275999 Smallest nonnegative number k such that A240088(k) = n.

0, 3, 1, 5, 10, 19, 15, 22, 31, 51, 61, 37, 82, 71, 126, 96, 92, 136, 162, 187, 206, 276, 191, 261, 236, 247, 317, 302, 401, 292, 422, 547, 456, 544, 551, 612, 591, 577, 521, 666, 742, 726, 682, 877, 796, 1052, 961, 1046, 1171, 1027, 954, 1017, 1006, 1207, 1396, 1262, 1311, 1496, 1482, 1571, 1717

Offset: 1

Robert G. Wilson v, Aug 17 2016

nonn

			a(1) = 0 since A240088(0) = 1, namely 0+0+0;
a(2) = 3 since A240088(3) = 2, namely 1+1+1 or 3+0+0;
a(3) = 1 since A240088(1) = 3, namely 1+0+0 or 0+1+0 or 0+0+1;
with triangular number followed by square number followed by pentagonal number.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1007 terms from Robert G. Wilson v)

Cf. A240088.

f[n_] := Block[{c = pi = 0, pn, plmt = Floor[(Sqrt[1 + 24 n] + 7)/6], ti, tlmt}, While[pi < plmt, ti = 0; pn = pi (3pi -1)/2; tlmt = Floor[(Sqrt[1 +8(n - pn)] + 1)/2] +1; While[ti < tlmt, If[IntegerQ[Sqrt[n - pn - ti (ti + 1)/2]], c++]; ti++]; pi++]; c]; t = 0*Range@ 1000; k = 0; While[k < 100001, a = f@ k; If[ t[[a]] == 0, t[[a]] = k]; k++]; t

A160324 Number of ways to express n as the sum of a square, a pentagonal number and a hexagonal number.

Original entry on oeis.org

1, 3, 3, 1, 1, 3, 4, 3, 1, 2, 4, 3, 2, 2, 2, 4, 5, 4, 2, 2, 3, 3, 5, 3, 3, 2, 3, 5, 4, 5, 2, 5, 5, 2, 2, 1, 6, 8, 5, 2, 3, 5, 4, 3, 4, 5, 3, 3, 2, 5, 7, 7, 5, 4, 7, 4, 4, 3, 4, 4, 3, 6, 3, 2, 5, 5, 9, 7, 3, 3, 6, 9, 5, 3, 1, 8, 7, 6, 2, 5, 6, 3, 10, 4, 3, 3, 8, 7, 5, 4, 1, 4, 10, 7, 5, 4, 8, 6, 2, 8, 6, 10, 7, 5

Offset: 0

Zhi-Wei Sun, May 08 2009

In April 2009, Zhi-Wei Sun conjectured that a(n)>0 for every n=0,1,2,3,.... Note that pentagonal numbers and hexagonal numbers are more sparse than squares and that there are infinitely many positive integers which cannot be written as the sum of three squares.
On Aug 12 2009, Zhi-Wei Sun made the following general conjecture on diagonal representations by polygonal numbers: For each integer m>2, any natural number n can be written in the form p_{m+1}(x_1)+...+p_{2m}(x_m) with x_1,...,x_m nonnegative integers, where p_k(x)=(k-2)x(x-1)/2+x (x=0,1,2,...) are k-gonal numbers. Sun has verified this with m=3 for n up to 10^6, and with m=4,5,6,7,8,9,10 for n up to 5*10^5. - Zhi-Wei Sun, Aug 15 2009
On Aug 21 2009, Zhi-Wei Sun formulated the following strong version for his conjecture on diagonal representations by polygonal numbers: For any integer m>2, each natural number n can be expressed as p_{m+1}(x_1)+p_{m+2}(x_2)+p_{m+3}(x_3)+r with x_1,x_2,x_3 nonnegative integers and r an integer among 0,...,m-3. For m=3 and m=4,5,6,7,8,9,10, Sun has verified this conjecture for n up to 10^6 and 5*10^5 respectively. Sun also guessed that for each m=3,4,... all sufficiently large integers have the form p_{m+1}(x_1)+p_{m+2}(x_2)+p_{m+3}(x_3) with x_1,x_2,x_3 nonnegative integers. For example, it seems that 387904 is the largest integer not in the form p_{20}(x_1)+p_{21}(x_2)+p_{22}(x_3). - Zhi-Wei Sun, Aug 21 2009
On Sep 04 2009, Zhi-Wei Sun conjectured that the sequence contains every positive integer. For n=1,2,3,... let s(n) denote the least nonnegative integer m such that a(m)=n. Here is the list of s(1),...,s(30): 0, 9, 1, 6, 16, 36, 50, 37, 66, 82, 167, 121, 162, 236, 226, 276, 302, 446, 478, 532, 457, 586, 677, 521, 666, 852, 976, 877, 1006, 1046. - Zhi-Wei Sun, Sep 04 2009
Let r be the rank (a noninteger r-polygonal number) which is the average of the number of squares, the number of pentagonal numbers and the number of hexagonal numbers less than x for sufficiently large values of x. r ~= 4.826378432581159594... a(n) ~= sqrt(n/r). - Robert G. Wilson v, Sep 03 2025

			For n=10 the a(10)=4 solutions are 4+0+6, 4+5+1, 9+0+1, 9+1+0.

Zhi-Wei Sun, Table of n, a(n) for n = 0..50000
M. B. Nathanson, A short proof of Cauchy's polygonal number theorem, Proc. Amer. Math. Soc. 99(1987), 22-24.
G. Pall, Large positive integers are sums of four or five values of a quadratic function, Amer. J. Math. 54(1932), 66-78.
Zhi-Wei Sun, Various new conjectures involving polygonal numbers and primes (a message to Number Theory List), May 2009.
Zhi-Wei Sun, Mixed Sums of Primes and Other Terms (a webpage).
Zhi-Wei Sun, On universal sums of polygonal numbers, arXiv:0905.0635 [math.NT], 2009-2015.

Cf. A000290, A000326, A000384, A008443, A240088, A165141.

a = Compile[{{n, Integer}}, Block[{c = 0}, Do[ c += Boole[ Mod[ Sqrt[n - i(2i -1) -j(3j -1)/2], 1] == 0], {i, 0, (1 + Sqrt[1 +8n])/4}, {j, 0, (1 + Sqrt[1 +24(n - i(2i -1))])/6}]; c]]; Array[a, 111, 0] (* _Robert G. Wilson v, Sep 03 2025 *)

a(n) = |{: x,y,z=0,1,2,... & x^2+(3y^2-y)/2+(2z^2-z)=n}|.

A085263 Number of ways to write n as the sum of a squarefree number (A005117) and a positive square (A000290).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 2, 0, 3, 3, 2, 2, 3, 3, 2, 2, 3, 4, 2, 1, 4, 4, 2, 1, 5, 4, 3, 2, 2, 5, 2, 3, 6, 6, 3, 2, 6, 4, 3, 2, 5, 6, 3, 2, 5, 6, 3, 2, 4, 6, 4, 3, 4, 6, 4, 1, 7, 5, 3, 3, 7, 6, 4, 4, 6, 8, 3, 3, 6, 7, 2, 4, 8, 5, 4, 3, 7, 9, 4, 2, 8, 9, 4, 3, 6, 6, 5, 4, 7, 9, 5, 3, 8, 4, 3, 5, 9

Offset: 1

Reinhard Zumkeller, Jun 23 2003

a(A085265(n))>0; a(A085266(n))=1; a(A085267(n))>1.
a(A085264(n))=n and a(i)<>n for i < A085264(n).
First occurrence of k: 2, 6, 11, 23, 30, 38, 62, 71, 83, 110, 138, 155, 182, 203, 227, 263, 302, 327, 383, 435, 447, 503, 542, 602, 635, ..., . Conjecture: For each k above, there is a finite number of terms; for example, only the two numbers 1 and 13 cannot be represented as the sum of a squarefree number and a square. The number of k terms beginning with 0: 2, 9, 19, 27, 38, 36, 57, 63, 62, 74, 94, ..., . - Robert G. Wilson v, May 16 2014

			a(11)=3:
11 = 1 + 10 = A000290(1) + A005117(7)
   = 4 + 7  = A000290(2) + A005117(6)
   = 9 + 2  = A000290(3) + A005117(2).

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Square Numbers.
Eric Weisstein's World of Mathematics, Squarefree
Robert G. Wilson v, Plot of first 100000 terms

Cf. A000290, A002636, A005117, A115288, A160324, A160325, A160326, A240088, A242442, A242443.

f[n_] := Count[ SquareFreeQ@# & /@ (n - Range[1, Floor[ Sqrt[ n]]]^2), True]; Array[f, 105] (* Robert G. Wilson v, May 16 2014 *)

a(n) = sum(k=1, n-1, issquare(k) * issquarefree(n-k)); \\ Michel Marcus, Oct 30 2020

a(n+1) = Sum_{k=1..n} A008966(k)*A010052(n-k+1). - Reinhard Zumkeller, Nov 04 2009
a(n) < sqrt(n). - Robert G. Wilson v, May 17 2014
G.f.: (Sum_{i>=1} x^(i^2))*(Sum_{j>=1} mu(j)^2*x^j). - Ilya Gutkovskiy, Feb 06 2017

A242442 Number of ways of writing n, a positive integer, as an unordered sum of a triangular number (A000217), an odd square (A016754) and a pentagonal number (A000326).

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 2, 1, 2, 2, 2, 3, 2, 2, 2, 4, 2, 0, 2, 2, 2, 2, 3, 4, 2, 3, 3, 2, 4, 3, 5, 2, 2, 3, 2, 4, 5, 4, 1, 3, 3, 4, 1, 2, 3, 5, 5, 1, 3, 5, 5, 4, 4, 4, 4, 2, 5, 4, 5, 4, 5, 4, 2, 5, 4, 4, 4, 4, 2, 4, 5, 5, 2, 2, 6, 5, 4, 2, 4, 6, 7, 7, 2, 3, 5, 6, 5, 5, 5, 2, 5, 9, 3, 5, 2, 8, 6, 1, 8, 3

Offset: 1

Robert G. Wilson v, May 14 2014

It is conjectured that only 18 cannot be so represented. See Sun, p. 4, Remark 1.2 (b).

Zhi-Wei Sun, On Universal Sums Of Polygonal Numbers, arXiv:0905.0635v18 [math.NT] 26 Oct 2011

Cf. A002636, A115288, A160324, A160325, A160326, A240088, A242443.

planeFigurative[n_, r_] := (n - 2) Binomial[r, 2] + r; s = Sort@ Flatten@ Table[ planeFigurative[3, i] + planeFigurative[4, j] + planeFigurative[5, k], {i, 0, 20}, {j, 1, 11, 2}, {k, 0, 8}]; Table[ Count[s, n], {n, 0, 104}]

A242443 Number of ways of writing n, a positive integer, as an unordered sum of a triangular number (A000217), an even square (A016742) and a generalized pentagonal number (A001318).

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 3, 4, 1, 4, 3, 4, 2, 2, 5, 3, 5, 3, 5, 4, 5, 7, 3, 4, 4, 6, 6, 4, 6, 3, 5, 7, 6, 4, 1, 7, 7, 6, 5, 6, 9, 5, 7, 7, 8, 6, 8, 4, 6, 6, 7, 9, 4, 10, 3, 6, 9, 7, 8, 5, 9, 7, 6, 7, 5, 11, 9, 7, 3, 7, 12, 13, 7, 7, 6, 9, 11, 6, 11, 8, 7, 10, 10, 8, 8, 8, 11, 5, 8, 5, 8, 11, 10, 10, 6, 14, 10, 6, 7, 7

Offset: 1

Robert G. Wilson v, May 14 2014

It is conjectured (1.1) and then proved by theorem 1.2 that all positive integers can be so represented [Sun, pp. 4-5].

Zhi-Wei Sun, On Universal Sums Of Polygonal Numbers, arXiv:0905.0635v18 [math.NT] 26 Oct 2011

Cf. A002636, A115288, A160324, A160325, A160326, A240088, A242442.

planeFigurative[n_, r_] := pf[n, r] = (n - 2) Binomial[r, 2] + r; s = Sort@ Table[ planeFigurative[3, i] + planeFigurative[3, j] + planeFigurative[3, k], {i, 0, 14}, {j, 0, 10, 2}, {k, -8, 8}]; Table[ Count[s, n], {n, 0, 50}]

A327792 a(n) is the greatest nonnegative number which has a partition into a triangular number (A000217), a square number (A000290), and a pentagonal number (A000326) in n different ways.

Original entry on oeis.org

0, 18, 168, 78, 243, 130, 553, 455, 515, 658, 865, 945, 633, 1918, 2258, 1385, 1583, 2828, 2135, 2335, 2785, 4533, 3168, 3478, 2790, 3868, 4193, 7328, 4953, 5278, 6390, 8148, 8015, 4585, 9160, 10485, 7613, 12333, 12025, 10178, 9923, 9720, 12558, 11340, 17420, 11753, 14893, 16155, 16415, 14343, 18053, 19803, 16608, 27283

Offset: 1

Robert G. Wilson v, Sep 25 2019

nonn

The largest nonnegative number k such that A240088(k) = n.

			a(0) does not exist since all numbers can be represented as the sum of a triangular, square & pentagonal number.

Robert G. Wilson v, Table of n, a(n) for n = 1..200

Cf. A000217, A000290, A000326, A240088, A275999, A327793.

f[n_] := Block[{j, k = 1, lenq, lenr, v = {}, t = PolygonalNumber[3, Range[0, 1 + Sqrt[2 n]]], s = PolygonalNumber[4, Range[0, 1 + Sqrt[n]]], p = PolygonalNumber[5, Range[0, 2 + Sqrt[2 n/3]]]}, u = Select[Union[Join[t, s, p]], # < n + 1 &]; q = IntegerPartitions[n, {3}, u]; lenq = 1 + Length@q; While[k < lenq, j = 1; r = q[[k]]; rr = Permutations@r; lenr = 1 + Length@rr; While[j < lenr, If[ MemberQ[t, rr[[j, 1]]] && MemberQ[s, rr[[j, 2]]] && MemberQ[p, rr[[j, 3]]], AppendTo[v, rr[[j]]]]; j++]; k++]; Length@v];

A374409 Number of ways to write n as an ordered sum of a triangular number, a pentagonal number and a hexagonal number.

Original entry on oeis.org

1, 3, 3, 2, 2, 2, 4, 5, 3, 2, 2, 4, 5, 3, 2, 4, 7, 4, 3, 3, 2, 7, 6, 5, 2, 2, 5, 4, 8, 6, 5, 3, 3, 6, 6, 4, 5, 7, 6, 5, 3, 6, 5, 8, 4, 3, 7, 5, 5, 4, 8, 11, 6, 4, 3, 5, 12, 7, 6, 1, 8, 7, 3, 6, 4, 6, 7, 12, 6, 5, 4, 9, 11, 11, 4, 1, 5, 8, 11, 7, 6, 9, 8, 4, 6, 7, 10, 3, 8, 4, 4, 10, 8, 9, 9, 11, 7, 7, 8, 10, 4

Offset: 0

Seiichi Manyama, Jul 08 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000217, A000326, A000384, A240088.

G.f.: (Sum_{k>=0} x^(k*(k+1)/2)) * (Sum_{k>=0} x^(k*(3*k-1)/2)) * (Sum_{k>=0} x^(k*(2*k-1))).

A327793 The number of nonnegative numbers that can be partitioned into a triangular number (A000217), a square number (A000290), and a pentagonal number (A000326) in n different ways.

Original entry on oeis.org

1, 2, 10, 12, 13, 10, 23, 25, 16, 36, 31, 34, 27, 45, 36, 50, 68, 61, 53, 68, 57, 72, 60, 59, 61, 87, 85, 88, 82, 97, 91, 106, 95, 98, 127, 93, 111, 125, 127, 124, 109, 127, 152, 122, 114, 146, 147, 132, 157, 169, 118, 180, 156, 158, 163, 168, 180, 178, 190, 184, 187, 196, 207, 191, 210, 204, 207, 206, 190, 227, 231, 203, 195, 219, 264

Offset: 1

Robert G. Wilson v, Sep 25 2019

nonn

The number of nonnegative numbers k such that A240088(k) = n.

			a(0) does not exist since all numbers can be represented as the sum of a triangular, square and pentagonal number;
a(1) = 1 because A240088({0}) = 1;
a(2) = 2 because A240088({3, 18}) = 2;
a(3) = 10 because A240088({1, 2, 4, 8, 9, 13, 14, 35, 98, 168}) = 3;
a(4) = 12 because A240088({5, 6, 7, 21, 25, 30, 34, 39, 43, 48, 63, 78}) = 4;
a(5) = 13 because A240088({10, 11, 12, 17, 20, 23, 28, 33, 69, 193, 203, 230, 243}) = 5;
a(6) = 10 because A240088({19, 24, 32, 44, 53, 55, 74, 90, 111, 130}) = 6;
a(7) = 23 because A240088({15, 16, 27, 29, 40, 46, 56, 60, 62, 68, 73, 84, 85, 95, 108, 113, 123, 135, 139, 163, 165, 273, 553}) = 7; etc.

Cf. A000217, A000290, A000326, A240088, A275999, A327792.

f[n_] := Block[{j, k = 1, lenq, lenr, v = {}, t = PolygonalNumber[3, Range[0, 1 + Sqrt[2 n]]], s = PolygonalNumber[4, Range[0, 1 + Sqrt[n]]], p = PolygonalNumber[5, Range[0, 2 + Sqrt[2 n/3]]]}, u = Select[Union[Join[t, s, p]], # < n + 1 &]; q = IntegerPartitions[n, {3}, u]; lenq = 1 + Length@q; While[k < lenq, j = 1; r = q[[k]]; rr = Permutations@r; lenr = 1 + Length@rr; While[j < lenr, If[ MemberQ[t, rr[[j, 1]]] && MemberQ[s, rr[[j, 2]]] && MemberQ[p, rr[[j, 3]]], AppendTo[v, rr[[j]]]]; j++]; k++]; Length@v];

A374406 Number of ways to write n as an ordered sum of a triangular number, a square and a hexagonal number.

Original entry on oeis.org

1, 3, 3, 2, 3, 3, 3, 5, 3, 2, 6, 5, 3, 3, 1, 5, 9, 5, 3, 5, 5, 4, 7, 3, 2, 9, 5, 4, 5, 6, 6, 8, 8, 2, 6, 4, 5, 11, 8, 3, 6, 5, 4, 9, 4, 7, 11, 8, 2, 5, 8, 7, 13, 5, 6, 10, 7, 6, 4, 7, 8, 9, 4, 2, 11, 12, 5, 12, 6, 3, 15, 10, 6, 9, 7, 4, 12, 6, 5, 8, 9, 10, 14, 7, 4, 15, 4, 9, 7, 4, 5, 12, 15, 7, 10, 10, 7, 13, 10, 3

Offset: 0

Seiichi Manyama, Jul 08 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000217, A000290, A000384, A160324, A240088.

G.f.: (Sum_{k>=0} x^(k*(k+1)/2)) * (Sum_{k>=0} x^(k^2)) * (Sum_{k>=0} x^(k*(2*k-1))).

cp's OEIS Frontend

A275999 Smallest nonnegative number k such that A240088(k) = n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A160324 Number of ways to express n as the sum of a square, a pentagonal number and a hexagonal number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A085263 Number of ways to write n as the sum of a squarefree number (A005117) and a positive square (A000290).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A242442 Number of ways of writing n, a positive integer, as an unordered sum of a triangular number (A000217), an odd square (A016754) and a pentagonal number (A000326).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A242443 Number of ways of writing n, a positive integer, as an unordered sum of a triangular number (A000217), an even square (A016742) and a generalized pentagonal number (A001318).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A327792 a(n) is the greatest nonnegative number which has a partition into a triangular number (A000217), a square number (A000290), and a pentagonal number (A000326) in n different ways.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A374409 Number of ways to write n as an ordered sum of a triangular number, a pentagonal number and a hexagonal number.

Views

Author

Keywords

Links

Crossrefs

Formula

A327793 The number of nonnegative numbers that can be partitioned into a triangular number (A000217), a square number (A000290), and a pentagonal number (A000326) in n different ways.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A374406 Number of ways to write n as an ordered sum of a triangular number, a square and a hexagonal number.

Views

Author