A068314 -id:A068314 - cp's OEIS frontend

A257414 Values of n such that there are exactly 7 solutions to x^2 - y^2 = n with x > y >= 0.

768, 1280, 1792, 2816, 3328, 3645, 4352, 4864, 5103, 5832, 5888, 7424, 7936, 8019, 9472, 9477, 10496, 11008, 12032, 12393, 13568, 13851, 14580, 15104, 15616, 16384, 16767, 17152, 18176, 18688, 20224, 20412, 21141, 21248, 22599, 22784, 24832, 25856, 26368

Offset: 1

Colin Barker, Apr 22 2015

nonn

			768 is in the sequence because there are 7 solutions to x^2 - y^2 = 768, namely (x,y) = (28,4), (32,16), (38,26), (52,44), (67,61), (98,94), (193,191).

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 120 terms from Colin Barker)

Cf. A257408, A257409, A257410, A257411, A257412, A257413, A257415, A257416, A257417.

Cf. A034178, A068314.

nn = 30000;
t = Table[0, {nn}];
Do[n = x^2 - y^2; If[n <= nn, t[[n]]++], {x, nn}, {y, 0, x - 1}];
Position[t, 7] // Flatten (* Jean-François Alcover, Jun 18 2020, after T. D. Noe in A034178 *)

is_A257414(n)={A034178(n)==7} \\ M. F. Hasler, Apr 22 2015

A334007 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero triangular numbers in exactly n ways.

Original entry on oeis.org

1, 10, 2180, 10053736, 13291443468940

Offset: 1

Ilya Gutkovskiy, Apr 12 2020

			Let S(k, m) denote the sum of m triangular numbers starting from k(k+1)/2. We have
a(1) = S(1, 1);
a(2) = S(4, 1) = S(1, 3);
a(3) = S(31, 4) = S(27, 5) = S(9, 15);
a(4) = S(945, 22) = S(571, 56) = S(968, 21) = S(131, 266);
a(5) = S(4109, 38947) = S(25213, 20540) = S(10296, 32943) = S(32801, 15834) = S(31654, 16472).

Eric Weisstein's World of Mathematics, Triangular Number
Index to sequences related to polygonal numbers

Cf. A000217, A054859, A068314, A034706, A050943, A186337, A298467, A307666, A309783, A334008, A334010, A334011, A334012.

a(5) from Giovanni Resta, Apr 13 2020

A334012 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero octagonal numbers in exactly n ways.

Original entry on oeis.org

1, 1045, 5985

Offset: 1

Ilya Gutkovskiy, Apr 12 2020

			From _Seiichi Manyama_, May 16 2021: (Start)
Let S(k, m) denote the sum of m octagonal numbers starting from k*(3*k-2). We have
a(1) = S(1, 1);
a(2) = S(19, 1) = S(1, 10);
a(3) = S(45, 1) = S(11, 9) = S(1, 18). (End)

Eric Weisstein's World of Mathematics, Octagonal Number
Index to sequences related to polygonal numbers

Cf. A000567, A054859, A068314, A186337, A298467, A322637, A334007, A334008, A334010, A334011, A344376.

A334008 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero pentagonal numbers in exactly n ways.

Original entry on oeis.org

1, 287, 472320, 89051435880

Offset: 1

Ilya Gutkovskiy, Apr 12 2020

			Let S(k, m) denote the sum of m pentagonal numbers starting from the k-th. We have
a(1) = S(1, 1);
a(2) = S(14, 1) = S(2, 7);
a(3) = S(103, 24) = S(19, 80) = S(67, 41);
a(4) = S(10833, 484) = S(4542, 1936) = S(9153, 660) = S(2817, 3036);

Eric Weisstein's World of Mathematics, Pentagonal Number
Index to sequences related to polygonal numbers

Cf. A000326, A054859, A068314, A186337, A298467, A319184, A334007, A334010, A334011, A334012.

a(4) from Giovanni Resta, Apr 13 2020

A334010 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero hexagonal numbers in exactly n ways.

Original entry on oeis.org

1, 703, 274550, 11132303325

Offset: 1

Ilya Gutkovskiy, Apr 12 2020

			Let S(k, m) denote the sum of m hexagonal numbers starting from the k-th. We have
a(1) = S(1, 1);
a(2) = S(19, 1) = S(13, 2);
a(3) = S(62, 25) = S(184, 4) = S(25, 51);
a(4) = S(3065, 505) = S(22490, 11) = S(1215, 1430) = S(1938, 946).

Eric Weisstein's World of Mathematics, Hexagonal Number
Index to sequences related to polygonal numbers

Cf. A000384, A054859, A068314, A186337, A298467, A319185, A334007, A334008, A334011, A334012.

a(4) from Giovanni Resta, Apr 13 2020

A334011 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero heptagonal numbers in exactly n ways.

Original entry on oeis.org

1, 872, 8240232, 263346158075

Offset: 1

Ilya Gutkovskiy, Apr 12 2020

			Let S(k, m) denote the sum of m heptagonal numbers starting from the k-th. We have
a(1) = S(1, 1);
a(2) = S(13, 2) = S(3, 8);
a(3) = S(133, 98) = S(479, 14) = S(168, 77);
a(4) = S(6773, 1785) = S(810, 6006) = S(7467, 1547) = S(38758, 70).

Eric Weisstein's World of Mathematics, Heptagonal Number
Index to sequences related to polygonal numbers

Cf. A000566, A054859, A068314, A186337, A298467, A322636, A334007, A334008, A334010, A334012.

a(4) from Giovanni Resta, Apr 14 2020

A334034 a(n) is the least integer that can be expressed as the difference of two pentagonal numbers in exactly n ways.

Original entry on oeis.org

1, 22, 70, 715, 1330, 4025, 6370, 14014, 17290, 25025, 45815, 73150, 121030, 95095, 85085, 256025, 350350, 432250, 1179178, 425425, 575575, 734825, 950950, 1926925, 3751930, 2187185, 1616615, 1956955, 3148145, 3658655, 4029025, 2977975, 4352425, 6656650, 13918450

Offset: 1

Ilya Gutkovskiy, Apr 12 2020

nonn

The least integer that can be expressed as the sum of one or more consecutive numbers congruent to 1 mod 3 in exactly n ways.

Index of first occurrence of n in A333815.

Chai Wah Wu, Table of n, a(n) for n = 1..72
Eric Weisstein's World of Mathematics, Pentagonal Number
Index to sequences related to polygonal numbers

Cf. A000326, A016777, A038547, A068314, A333815, A334008, A334035, A334036, A334037.

More terms from Jinyuan Wang, Apr 13 2020

A334035 a(n) is the least integer that can be expressed as the difference of two hexagonal numbers in exactly n ways.

Original entry on oeis.org

1, 45, 225, 585, 2415, 4725, 9945, 10395, 31185, 28665, 45045, 58905, 143325, 257985, 135135, 225225, 329175, 487305, 405405, 831285, 1091475, 675675, 1396395, 1576575, 2927925, 3132675, 2436525, 2027025, 2567565, 2297295, 6235515, 5360355, 4729725, 3828825, 10503675

Offset: 1

Ilya Gutkovskiy, Apr 12 2020

nonn

The least integer that can be expressed as the sum of one or more consecutive numbers congruent to 1 mod 4 in exactly n ways.

Index of first occurrence of n in A333816.

Eric Weisstein's World of Mathematics, Hexagonal Number
Index to sequences related to polygonal numbers

Cf. A000384, A016813, A038547, A068314, A333816, A334010, A334034, A334036, A334037.

nmax = 10000; A333816 = Rest[CoefficientList[Series[Sum[x^(k*(2*k - 1))/(1 - x^(4*k)), {k, 1, 1 + Sqrt[nmax/2]}], {x, 0, nmax}], x]]; Flatten[Table[FirstPosition[A333816, k], {k, 1, Max[A333816]}]] (* Vaclav Kotesovec, Apr 19 2020 *)

More terms from Jinyuan Wang, Apr 13 2020

A334037 a(n) is the least integer that can be expressed as the difference of two octagonal numbers in exactly n ways.

Original entry on oeis.org

1, 133, 560, 1729, 4160, 10640, 14560, 22400, 44800, 58240, 138320, 98560, 123200, 203840, 246400, 394240, 320320, 492800, 800800, 640640, 1047200, 1823360, 1724800, 1281280, 2094400, 1601600, 2475200, 2722720, 4484480, 3203200, 5532800, 6697600, 5445440, 7958720

Offset: 1

Ilya Gutkovskiy, Apr 12 2020

nonn

The least integer that can be expressed as the sum of one or more consecutive numbers congruent to 1 mod 6 in exactly n ways.

Index of first occurrence of n in A333818.

Eric Weisstein's World of Mathematics, Octagonal Number
Index to sequences related to polygonal numbers

Cf. A000567, A016921, A038547, A068314, A333818, A334012, A334034, A334035, A334036.

More terms from Jinyuan Wang, Apr 13 2020

A094191 a(n) = smallest positive number that occurs exactly n times as a difference between two positive squares.

Original entry on oeis.org

3, 15, 45, 96, 192, 240, 576, 480, 720, 960, 12288, 1440, 3600, 3840, 2880, 3360, 20736, 5040, 147456, 6720, 11520, 14400, 50331648, 10080, 25920, 245760, 25200, 26880, 3221225472, 20160, 57600, 30240, 184320, 3932160, 103680, 40320, 129600, 2985984, 737280, 60480, 13194139533312, 80640, 9663676416, 430080, 100800, 251658240, 84934656, 110880, 921600, 181440

Offset: 1

Johan Claes, Jun 02 2004

nonn

Related to A005179, "Smallest number with exactly n divisors", with which it shares a lot of common terms (in different positions).

It appears that, for entries having prime index p > 3, the minimal solution is 2^(p+1)*3 for Sophie Germain primes p. The number 43 is not such a prime, and we have the smaller solution 2^30*3^2. - T. D. Noe, Mar 14 2018

			a(1)=3 because there is only one difference of positive squares which equals 3 (2^2-1^1).
a(2)=15 because 15 = 4^2-1^2 = 8^2-7^2.
a(3)=45 because 45 = 7^2-2^2 = 9^2-6^2 = 23^2-22^2.

T. D. Noe, Table of n, a(n) for n = 1..999
Johan Claes, homepage. [Broken link (unknown server) replaced with link to current user's "homepage". - _M. F. Hasler_, Mar 14 2018]

Cf. A068314.

s = Split[ Sort[ Flatten[ Table[ Select[ Table[ b^2 - c^2, {c, b - 1}], # < 500000 &], {b, 250000}]]]]; f[s_, p_] := Block[{l = Length /@ s}, If[ Position[l, p, 1, 1] != {}, d = s[[ Position[l, p, 1, 1][[1, 1]] ]] [[1]], d = 0]; d]; t = Table[ f[s, n], {n, 36}] (* Robert G. Wilson v, Jun 04 2004 *)

{occurrences(d)=local(c,n,a);c=0;for(n=1,(d-1)\2,if(issquare(a=n^2+d),c++));c} {m=50;z=30000;v=vector(m,n,-1);for(d=1,z,k=occurrences(d);if(0Klaus Brockhaus

Edited by Don Reble and Klaus Brockhaus, Jun 04 2004
Further terms from Johan Claes, Jun 07 2004
a(43) corrected by T. D. Noe, Mar 14 2018

cp's OEIS Frontend

A257414 Values of n such that there are exactly 7 solutions to x^2 - y^2 = n with x > y >= 0.

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A334007 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero triangular numbers in exactly n ways.

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A334012 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero octagonal numbers in exactly n ways.

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A334008 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero pentagonal numbers in exactly n ways.

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A334010 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero hexagonal numbers in exactly n ways.

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A334011 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero heptagonal numbers in exactly n ways.

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A334034 a(n) is the least integer that can be expressed as the difference of two pentagonal numbers in exactly n ways.

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A334035 a(n) is the least integer that can be expressed as the difference of two hexagonal numbers in exactly n ways.

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A334037 a(n) is the least integer that can be expressed as the difference of two octagonal numbers in exactly n ways.

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A094191 a(n) = smallest positive number that occurs exactly n times as a difference between two positive squares.

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