A160325 -id:A160325 - cp's OEIS frontend

A165141 The least positive integer that can be written in exactly n ways as the sum of a square, a pentagonal number and a hexagonal number.

3, 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, 1277, 1381, 1857, 1556, 1507, 1657, 1832, 1732, 2336, 2299, 2007, 2677, 2326, 2117, 2591, 2502, 2516, 2592, 3106, 3557

Offset: 1

Zhi-Wei Sun, Sep 05 2009

On Sep 04 2009, Zhi-Wei Sun conjectured that the sequence A160324 contains every positive integer, i.e., for each positive integer n there exists a positive integer s which can be written in exactly n ways as the sum of a square, a pentagonal number and a hexagonal number. Based on this conjecture we create the current sequence. It seems that 0.9 < a(n)/n^2 < 1.6 for n > 33. Zhi-Wei Sun conjectured that a(n)/n^2 has a limit c with 1.1 < c < 1.2. On Sun's request, his friend Qing-Hu Hou produced a list of a(n) for n = 1..913 (see the b-file).

			For n=5 the a(5)=16 solutions are 0^2+1+15 = 1^2+0+15 = 2^2+12+0 = 3^2+1+6 = 4^2+0+0 = 16.

Robert G. Wilson v, Table of n, a(n) for n = 1..2658(terms 1..913 from Quig-Hu Hou)
F. Ge and Z. W. Sun, On some universal sums of generalized polygonal numbers, preprint, arXiv:0906.2450 [math.NT], 2009-2016.
M. B. Nathanson, A short proof of Cauchy's polygonal number theorem, Proc. Amer. Math. Soc. 99(1987), 22-24.
Zhi-Wei Sun, A challenging conjecture on sums of polygonal number (a message to Number Theory List), 2009.
Zhi-Wei Sun, Polygonal numbers, primes and ternary quadratic forms (a talk given at a number theory conference), 2009.
Zhi-Wei Sun, Mixed Sums of Primes and Other Terms (a webpage).
Zhi-Wei Sun, On universal sums of polygonal numbers, preprint, arXiv:0905.0635 [math.NT], 2009-2015.

Cf. A160324, A000290, A000326, A000384, A160325, A160326.

SQ[x_] := x>-1 && IntegerPart[Sqrt[x]]^2==x;
RN[n_] := RN[n] = Sum[If[SQ[n - (3 y^2 - y)/2 - (2 z^2 - z)], 1, 0], {y, 0, (1 + Sqrt[1 + 24 n])/6}, {z, 0, (1 + Sqrt[8 (n - (3 y^2 - y)/2) + 1])/4}] (* iterators modified by Robert G. Wilson v, Apr 05 2025 *);
Do[Do[If[RN[m]==n,Print[n," ", m];Goto[aa]],{m,1,1000000}];Label[aa];Continue,{n,1,100}]
f = Compile[{{n, Integer}}, Block[{cnt = 0, pentind, hex, hexind = Floor[(1 + Sqrt[8n +1])/4]}, While[ hexind > -1, hex = hexind (2 hexind - 1); pentind = Floor[(1 + Sqrt[1 + 24 (n - hex)])/6]; While[pentind > -1, If[ Floor[ Sqrt[ n - hex - pentind (3 pentind - 1)/2]]^2 == n - hex - pentind (3 pentind - 1)/2, cnt++]; pentind--]; hexind--]; cnt]]; t[] := 0; k = 1; While[k < 100001, If[ t[f[k]] == 0, t[f[k]] = k]; k++]; t /@ Range@ 250 (* Robert G. Wilson v, Apr 04 2025 *)

a(n) = min{m>0: m=x^2+(3y^2-y)/2+(2z^2-z) has exactly n solutions with x,y,z=0,1,2,...}.

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}]

A254668 Number of ways to write n as the sum of a square, a second pentagonal number, and a hexagonal number.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Feb 04 2015

nonn

Conjecture: a(n) > 0 for all n. Also, a(n) = 1 only for n = 0, 5, 13, 14, 20, 112, 125.
Compare this conjecture with the conjecture in A160324.
The conjecture that a(n) > 0 for all n = 0,1,2,... appeared in Conjecture 1.2(ii) of the author's JNT paper in the links. - Zhi-Wei Sun, Oct 03 2016

			a(20) = 1 since 20 = 2^2 + 3*(3*3+1)/2 + 1*(2*1-1).
a(112) = 1 since 112 = 7^2 + 6*(3*6+1)/2 + 2*(2*2-1).
a(125) = 1 since 125 = 5^2 + 8*(3*8+1)/2 + 0*(2*0-1).

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, On universal sums of polygonal numbers, arXiv:0905.0635 [math.NT], 2009-2015.
Zhi-Wei Sun, On x(ax+1)+y(by+1)+z(cz+1) and x(ax+b)+y(ay+c)+z(az+d), J. Number Theory 171(2017), 275-283.

Cf. A000290, A000384, A005449, A160324, A160325, A254661.

SQ[n_]:=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-y(3y+1)/2-z(2z-1)],r=r+1],{y,0,(Sqrt[24n+1]-1)/6},{z,0,(Sqrt[8(n-y(3y+1)/2)+1]+1)/4}];
Print[n," ",r];Continue,{n,0,100}]

A254680 Least positive integer m with A254661(m) = n.

Original entry on oeis.org

1, 3, 7, 17, 21, 51, 66, 72, 157, 147, 121, 136, 246, 297, 332, 367, 402, 506, 547, 577, 677, 796, 892, 731, 926, 1216, 1116, 976, 1181, 1402, 1556, 1416, 1507, 1496, 2287, 1622, 1977, 2112, 1942, 2131, 2017, 2882, 2767, 2501, 3162, 3671, 3097, 3187, 3047, 3762, 3867, 2952, 4356, 4111, 4826, 5112, 5211, 4811, 4686, 5461

Offset: 1

Zhi-Wei Sun, Feb 05 2015

nonn

Conjecture: a(n) exists for any n > 0. Moreover, no term is divisible by 5, and no term with n>2 is congruent to 3 modulo 5.

Zhi-Wei Sun, Table of n, a(n) for n = 1..170
Zhi-Wei Sun, On universal sums of polygonal numbers, arXiv:0905.0635 [math.NT], 2009-2015.

Cf. A000217, A000290, A005449, A160325, A254661, A254677.

a(3) = 7 since 7 is the first positive integer that can be written as x*(x+1)/2 + (2y)^2 + z*(3*z+1)/2 (with x,y,z nonnegative integers) in exactly 3 ways. In fact, 7 = 0*1/2 + 0^2 +2*(3*2+1)/2 = 1*2/2 + 2^2 +1*(3*1+1)/2 = 2*3/2 + 2^2 + 0*(3*0+1)/2.

TQ[n_]:=IntegerQ[Sqrt[8n+1]]
Do[Do[m=0;Label[aa];m=m+1;r=0;Do[If[TQ[m-4y^2-z(3z+1)/2],r=r+1;If[r>n,Goto[aa]]],{y,0,Sqrt[m/4]}, {z,0,(Sqrt[24(m-4y^2)+1]-1)/6}];
If[r==n,Print[n," ",m];Goto[bb],Goto[aa]]]; Label[bb];Continue,{n,1,60}]

cp's OEIS Frontend

A165141 The least positive integer that can be written in exactly n ways as the sum of a square, a pentagonal number and a hexagonal number.

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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).

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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).

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A254668 Number of ways to write n as the sum of a square, a second pentagonal number, and a hexagonal number.

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A254680 Least positive integer m with A254661(m) = n.

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