A276517 -id:A276517 - cp's OEIS frontend

A001422 Numbers which are not the sum of distinct squares.

2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44, 47, 48, 60, 67, 72, 76, 92, 96, 108, 112, 128

Offset: 1

N. J. A. Sloane, Jeff Adams (jeff.adams(AT)byu.net)

This is the complete list (Sprague).

S. Lin, Computer experiments on sequences which form integral bases, pp. 365-370 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.
Harry L. Nelson, The Partition Problem, J. Rec. Math., 20 (1988), 315-316.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 222.

R. E. Dressler and T. Parker, 12,758, Math. Comp. 28 (1974), 313-314.
T. Sillke, Not the sum of distinct squares
R. Sprague, Über Zerlegungen in ungleiche Quadratzahlen, Math. Z. 51, (1948), 289-290.
Eric Weisstein's World of Mathematics, Square Number.
Index entries for sequences related to sums of squares

Complement of A003995. Subsequence of A004441.
Cf. A025524 (number of numbers not the sum of distinct n-th-order polygonal numbers)
Cf. A007419 (largest number not the sum of distinct n-th-order polygonal numbers)
Cf. A053614, A121405 (corresponding sequences for triangular and pentagonal numbers)
Cf. A033461, A276517.
Cf. A001476, A046039, A194768, A194769 for 3rd, 4th, 5th, 6th powers.
Cf. A001661.

nn=50; t=Rest[CoefficientList[Series[Product[(1+x^(k*k)), {k,nn}], {x,0,nn*nn}], x]]; Flatten[Position[t,0]] (* T. D. Noe, Jul 24 2006 *)

select( is_A001422(n,m=n)={m^2>n&& m=sqrtint(n); n!=m^2&&!while(m>1,isSumOfSquares(n-m^2,m--)&&return)}, [1..128]) \\ M. F. Hasler, Apr 21 2020

A276516 Expansion of Product_{k>=1} (1-x^(k^2)).

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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, 1, -1, 1, 1, 0, -1, 0, 0, 0, 0, 0, 0, -2, 1, 1, 1, 0, 0, -1, -1, 1, 1, -1, 0, 0, -1, 1, -1, 2, -1, 0, 1, -2, 0, 1, 0, 1, 0, -1, 0, -2, 2, 0

Offset: 0

Vaclav Kotesovec, Dec 12 2016

The difference between the number of partitions of n into an even number of distinct squares and the number of partitions of n into an odd number of distinct squares. - Ilya Gutkovskiy, Jan 15 2018

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.

Cf. A001156, A033461, A279360, A279368, A276517, A341040.

nn = 15; CoefficientList[Series[Product[(1-x^(k^2)), {k, nn}], {x, 0, nn^2}], x]
nmax = 200; nn = Floor[Sqrt[nmax]]+1; poly = ConstantArray[0, nn^2 + 1]; poly[[1]] = 1; poly[[2]] = -1; poly[[3]] = 0; Do[Do[poly[[j + 1]] -= poly[[j - k^2 + 1]], {j, nn^2, k^2, -1}];, {k, 2, nn}]; Take[poly, nmax+1]

a(n) = Sum_{k>=0} (-1)^k * A341040(n,k). - Alois P. Heinz, Feb 03 2021

a(n) = A033461(n) - 2*A339367(n). - R. J. Mathar, Jul 29 2025

A279486 Indices k such that A279484(k) = 0.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 66, 67, 68, 69, 70, 71, 74, 75, 76, 77

Offset: 1

Vaclav Kotesovec, Dec 13 2016

nonn

This is different from A001476, first difference: a(450) = 540, A001476(450) = 542.

Conjecture: for k > 353684 there are no more terms in this sequence (tested for k < 1000000).

			3 is in the sequence because A279484(3) = 0
8 is not in the sequence because A279484(8) = -1
344739 is in the sequence because A279484(344739) = 0
353684 is in the sequence because A279484(353684) = 0

Vaclav Kotesovec, Table of n, a(n) for n = 1..5216

Cf. A001476, A001661, A276517, A279484.

nn = 10; A279484 = Rest[CoefficientList[Series[Product[(1-x^(k^3)), {k, nn}], {x, 0, nn^3}], x]]; Select[Range[nn^3], A279484[[#]]==0&]
nmax = 1000; nn = Floor[nmax^(1/3)]+1; poly = ConstantArray[0, nn^3 + 1]; poly[[1]] = 1; poly[[2]] = -1; poly[[3]] = 0; Do[Do[poly[[j + 1]] -= poly[[j - k^3 + 1]], {j, nn^3, k^3, -1}];, {k, 2, nn}]; A279484 = Take[poly, {2, nmax+1}]; Select[Range[nmax], A279484[[#]]==0&]

A279487 Indices k such that A279485(k) = 0.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Vaclav Kotesovec, Dec 13 2016

nonn

This is different from A046039, first difference: a(14328) = 14979, A046039(14328) = 14981.
Conjecture: Last term is a(1040799) = 64674419. For k > 64674419 there are no more terms in this sequence (tested for k < 150000000).
Last terms are: 30082710, 30345655, 30358709, 30530388, 30982210, 31463972, 32369456, 32374194, 32594966, 32658048, 32780596, 32875172, 32997892, 33135812, 33440935, 33647428, 34086978, 34112787, 34629875, 35535908, 35638453, 36081828, 36140868, 36945332, 39218566, 39581363, 40364547, 40491526, 41235157, 43853600, 47973011, 57353782, 57767766, 64674419

			3 is in the sequence because A279485(3) = 0
16 is not in the sequence because A279485(16) = -1
57767766 is in the sequence because A279485(57767766) = 0
64674419 is in the sequence because A279485(64674419) = 0

Vaclav Kotesovec, Table of n, a(n) for n = 1..100000

Cf. A001661, A046039, A279485.

Cf. A276517, A279486.

nn = 10; A279485 = Rest[CoefficientList[Series[Product[(1-x^(k^4)), {k, nn}], {x, 0, nn^4}], x]]; Select[Range[nn^4], A279485[[#]]==0&]
nmax = 10000; nn = Floor[nmax^(1/4)]+1; poly = ConstantArray[0, nn^4 + 1]; poly[[1]] = 1; poly[[2]] = -1; poly[[3]] = 0; Do[Do[poly[[j + 1]] -= poly[[j - k^4 + 1]], {j, nn^4, k^4, -1}];, {k, 2, nn}]; A279485 = Take[poly, {2, nmax+1}]; Select[Range[nmax], A279485[[#]]==0&]

A292740 Indices k such that A292547(k) = 0.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Vaclav Kotesovec, Sep 22 2017

nonn

Complement of A290276.

Conjecture: for k > 212594 there are no more terms in this sequence (tested for k < 63000000).

			3 is in the sequence because A292547(3) = 0
8 is not in the sequence because A292547(8) = -1
201254 is in the sequence because A292547(201254) = 0
212594 is in the sequence because A292547(212594) = 0

Vaclav Kotesovec, Table of n, a(n) for n = 1..54766

Cf. A276517, A279329, A279486, A279529, A290276, A292547.

Cf. A001476, A022555, A031980.

With[{nn = 200}, -1 + Position[#, 0][[All, 1]] &@ CoefficientList[ Series[Product[1 + x^((2 k - 1)^3), {k, 1, Floor[nn^(1/3)/2] + 1}], {x, 0, nn}], x]] (* Michael De Vlieger, Sep 22 2017, after Vaclav Kotesovec at A292547 *)

A279529 a(n) is a largest m such that coefficient [x^m] in Product_{k>=1} (1-x^(k^n)) is equal to zero.

Original entry on oeis.org

7169, 353684, 64674419

Offset: 2

Vaclav Kotesovec, Dec 14 2016

			a(2) = 7169 because A276516(7169) = 0 and A276516(m) <> 0 for m > 7169.
a(3) = 353684 because A279484(353684) = 0 and A279484(m) <> 0 for m > 353684.
a(4) = 64674419 because A279485(64674419) = 0 and A279485(m) <> 0 for m > 64674419.
a(2) = A276517(173) = 7169.
a(3) = A279486(5216) = 353684.
a(4) = A279487(1040799) = 64674419.

Cf. A001661.
Cf. A276517, A279486, A279487.
Cf. A276516, A279484, A279485.
Cf. A001422, A001476, A046039.

A304028 Numbers k such that A033461(k) is divisible by k.

Original entry on oeis.org

1, 2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44, 47, 48, 60, 67, 72, 76, 92, 96, 108, 112, 128, 2229, 2929, 3022, 4481, 34542, 34951, 36996, 58091, 292949, 437728, 438237, 2103581, 2237158, 3215950, 3375578

Offset: 1

Vaclav Kotesovec, May 04 2018

nonn

A001422 is a finite subsequence.

			2229 is in the sequence because A033461(2229) = 51267 = 23 * 2229.

Cf. A001422, A033461, A051177, A056848, A276517.

max = 100; p = ConstantArray[0, max^2 + 1]; p[[1]] = 1; p[[2]] = 1; Do[Do[p[[j + 1]] += p[[j - k^2 + 1]], {j, max^2, k^2, -1}];, {k, 2, max}]; Select[Range[1, max^2], Divisible[p[[# + 1]], #] &]

A305440 Indices k such that A292518(k) = 0.

Original entry on oeis.org

2, 5, 8, 12, 21, 22, 23, 24, 33, 36, 45, 46, 48, 50, 67, 72, 75, 78, 85, 86, 88, 91, 92, 107, 111, 112, 121, 130, 139, 149, 156, 158, 161, 170, 180, 189, 193, 196, 200, 205, 224, 225, 230, 242, 270, 291, 305, 341, 369, 386, 387, 394, 397, 403, 426, 459, 475, 493, 521, 603, 666, 750

Offset: 1

Seiichi Manyama, Jun 01 2018

nonn

Conjecture: for k > 3957 there are no more terms in this sequence.

			   2 is in the sequence because A292518(   2) = 0.
3957 is in the sequence because A292518(3957) = 0.

Seiichi Manyama, Table of n, a(n) for n = 1..72

Cf. A276517, A292518, A305441.

A305441 Indices k such that A305355(k) = 0.

Original entry on oeis.org

2, 3, 4, 7, 8, 9, 10, 11, 14, 15, 16, 19, 20, 21, 24, 25, 26, 29, 30, 31, 32, 33, 37, 38, 42, 43, 44, 45, 46, 49, 50, 52, 54, 55, 57, 59, 60, 61, 65, 66, 67, 69, 70, 72, 74, 75, 77, 80, 81, 84, 89, 93, 94, 95, 96, 97, 100, 101, 102, 107, 112, 114, 116, 121, 124, 128

Offset: 1

Seiichi Manyama, Jun 01 2018

nonn

Conjecture: for k > 10316 there are no more terms in this sequence.

			    2 is in the sequence because A305355(    2) = 0.
10316 is in the sequence because A305355(10316) = 0.

Seiichi Manyama, Table of n, a(n) for n = 1..227

Cf. A276517, A305355, A305440.

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A001422 Numbers which are not the sum of distinct squares.

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A276516 Expansion of Product_{k>=1} (1-x^(k^2)).

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A279486 Indices k such that A279484(k) = 0.

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A279487 Indices k such that A279485(k) = 0.

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A292740 Indices k such that A292547(k) = 0.

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A279529 a(n) is a largest m such that coefficient [x^m] in Product_{k>=1} (1-x^(k^n)) is equal to zero.

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A304028 Numbers k such that A033461(k) is divisible by k.

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A305440 Indices k such that A292518(k) = 0.

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A305441 Indices k such that A305355(k) = 0.

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