A100253 -id:A100253 - cp's OEIS frontend

A100252 Least square n-gonal number greater than 1, or 0 if none exists.

36, 4, 9801, 1225, 81, 225, 9, 0, 196, 64, 36, 441, 3025, 16, 17689, 100, 484, 0, 2601, 729, 68121, 225, 25, 7225, 25921, 81, 1225, 203401, 441, 1089, 4761, 196, 15376, 36, 1936, 511225, 784, 576, 55071241, 47089, 1156, 256, 529046001, 2916, 1134225

Offset: 3

Charlie Marion, Nov 21 2004

nonn

Also, let j be the smallest integer for which 1+(1+1*n)+(1+2*n)+... +(1+j*n)=k^2=s. Then a(n)=s; if no such j exists, then a(n)=0. Basis for sequence is shortest arithmetic series with initial term 1 and difference n that sums to a perfect square.

See A100251 and A188898 for the corresponding indices of these terms. Note that a(n) is zero for n = 10, 20, 52 (numbers in A188896). Although the Mathematica program searches only the first 25000 square numbers for n-gonal numbers, the Reduce function can show that there are no square n-gonal numbers (other than 0 and 1) for these n. - T. D. Noe, Apr 19 2011

			a(3)=9801 since 1 + 4 + 7 +...+ (1+80*3)= 99^2 = 9801 and no other arithmetic series with initial term 1, difference 3 and fewer terms sums to a perfect square.

Vincenzo Librandi, Table of n, a(n) for n = 3..100
Eric W. Weisstein, MathWorld: Polygonal Number
Eric W. Weisstein, MathWorld: Square Number

Cf. A000290 (squares), A188891 (similar sequence for triangular numbers).

NgonIndex[n_, v_] := (-4 + n + Sqrt[16 - 8*n + n^2 - 16*v + 8*n*v])/(n - 2)/2; Table[k = 2; While[sqr = k^2; i = NgonIndex[n, sqr]; k < 25000 && ! IntegerQ[i], k++]; If[k == 25000, k = sqr = i = 0]; sqr, {n, 3, 64}] (* T. D. Noe, Apr 19 2011 *)

1+(1+1*n)+(1+2*n)+...+(1+A100254(n)*n) = 1+(1+1*n)+(1+2*n)+...+A100253(n) = A100251(n)^2 = a(n).

A101157 Let j be the smallest integer for which n+(n+1)+...+(n+j) is a square, say k^2; then a(n)=k.

Original entry on oeis.org

1, 3, 5, 2, 9, 11, 13, 15, 3, 19, 6, 5, 25, 27, 29, 4, 33, 10, 37, 39, 14, 43, 45, 7, 5, 9, 53, 55, 57, 59, 61, 18, 65, 67, 15, 6, 18, 75, 22, 9, 81, 83, 15, 87, 21, 26, 12, 95, 7, 99, 101, 33, 30, 107, 109, 111, 22, 25, 117, 11, 121, 42, 125, 8, 129

Offset: 1

Charlie Marion, Dec 29 2004

nonn

Basis for sequence is shortest arithmetic sequence with initial term n and difference 1 that sums to a perfect square. Cf. A100251, A100252, A100253, A100254.

a(n) is the least k>0 such that triangular(n-1) + k^2 is a triangular number. - Alex Ratushnyak, May 17 2013

			a(11)=6 since 11+12+13 = 6^2.

Shawn A. Broyles, Table of n, a(n) for n = 1..1000

Cf. A101158, A101159, A101160.

a(n) = {j = 0; while(! issquare(v=sum(k=0, j, n+k)), j++); sqrtint(v);} \\ Michel Marcus, Sep 01 2013

n+(n+1)+...+(n+A101160(n)) = n+(n+1)+...+A101159(n) = a(n)^2 = A101158(n).

a(n^2) = n. - Michel Marcus, Jun 28 2013

More terms from Michel Marcus, Jun 28 2013

A100251 The square root of A100252; the index of the least square number greater than 1 that is also an n-gonal number, or 0 if none exists.

Original entry on oeis.org

6, 2, 99, 35, 9, 15, 3, 0, 14, 8, 6, 21, 55, 4, 133, 10, 22, 0, 51, 27, 261, 15, 5, 85, 161, 9, 35, 451, 21, 33, 69, 14, 124, 6, 44, 715, 28, 24, 7421, 217, 34, 16, 23001, 54, 1065, 36, 7, 76, 156, 0, 245

Offset: 3

Charlie Marion, Nov 21 2004

nonn

Let j be the smallest integer for which 1 + (1+1*n) + (1+2*n) + ... + (1+j*n) = k^2 = s. Then a(n)=k; if no such j exists, then a(n)=0. Basis for sequence is shortest arithmetic series with initial term 1 and difference n that sums to a perfect square.

			a(3)=99 since 1 + 4 + 7 + ... + (1+80*3) = 99^2 = 9801 and no other arithmetic series with initial term 1, difference 3 and fewer terms sums to a perfect square.

Cf. A100252, A100253.

NgonIndex[n_, v_] := (-4 + n + Sqrt[16 - 8*n + n^2 - 16*v + 8*n*v])/(n - 2)/2; Table[k = 2; While[sqr = k^2; i = NgonIndex[n, sqr]; k < 25000 && ! IntegerQ[i], k++]; If[k == 25000, k = sqr = i = 0]; k, {n, 3, 64}] (* T. D. Noe, Apr 19 2011 *)

a(n)^2 = 1 + (1+1*n) + (1+2*n) + ... + (1+A100254(n)*n) = 1 + (1+1*n) +(1+2*n) + ... + A100253(n).

a(n)^2 = A100252(n).

A100254 Let j be the smallest integer for which 1 + (1+1n) + (1+2n) + ... + (1+j*n) = k^2 = s. Then a(n)=j; if no such j exists, then a(n)=0.

Original entry on oeis.org

7, 1, 80, 24, 5, 8, 1, 0, 6, 3, 2, 8, 21, 1, 48, 3, 7, 0, 16, 8, 80, 4, 1, 24, 45, 2, 9, 120, 5, 8, 17, 3, 30, 1, 10, 168, 6, 5, 1680, 48, 7, 3, 4960, 11, 224, 7, 1, 15, 31, 0, 48, 24, 16, 12, 288, 8, 48, 6, 26, 80, 117, 1, 136160, 195, 13, 3, 9, 840, 1520, 24, 49, 8, 70, 2, 1680

Offset: 1

Charlie Marion, Nov 21 2004

nonn

Basis for sequence is shortest arithmetic series with initial term 1 and difference n that sums to a perfect square.

			a(3)=80 since 1 + 4 + 7 +...+ (1+80*3) = 99^2 = 9801 and no other arithmetic series with initial term 1, difference 3 and fewer terms sums to a perfect square.

a[n_] := Block[{k = 1}, While[ !IntegerQ[ Sqrt[(k + 1)(1 + k*n/2)]], k++ ]; k]; a[18] = a[50] = 0; Table[ a[n], {n, 75}] (* Robert G. Wilson v, Nov 27 2004 *)

1 + (1+1*n) + (1+2*n) + ... + (1+a(n)*n) = 1 + (1+1*n) + (1+2*n) + ... + A100253(n) = A100251(n)^2 = A100252(n).

More terms from Robert G. Wilson v, Nov 27 2004

A101160 a(n) is the smallest integer j for which n+(n+1)+...+(n+j) is a square.

Original entry on oeis.org

0, 2, 4, 0, 8, 10, 12, 14, 0, 18, 2, 1, 24, 26, 28, 0, 32, 4, 36, 38, 7, 42, 44, 1, 0, 2, 52, 54, 56, 58, 60, 8, 64, 66, 5, 0, 7, 74, 10, 1, 80, 82, 4, 86, 8, 12, 2, 94, 0, 98, 100, 17, 14, 106, 108, 110, 7, 9, 116, 1, 120, 23, 124, 0, 128

Offset: 1

Charlie Marion, Dec 29 2004

nonn

Basis for sequence is shortest arithmetic sequence with initial term n and difference 1 that sums to a perfect square. Cf. A100251, A100252, A100253, A100254.

0 <= a(n) <= 2*(n - 1). - Ctibor O. Zizka, Oct 05 2023

			a(11)=2 since j=2 is the smallest integer for which 11+...+11+j = 6^2 = 36 is a perfect square.

Shawn A. Broyles, Table of n, a(n) for n = 1..1000

Cf. A101157, A101158, A101159.

Cf. A100251, A100252, A100253, A100254.

a(n) = {j = 0; while(! issquare(sum(k=0, j, n+k)), j++); j;} \\ Michel Marcus, Sep 01 2013

n+(n+1)+...+(n+a(n)) = n+(n+1)+...+A101159(n) = A101157(n)^2 = A101158(n).

a(n^2) = 0. - Michel Marcus, Jun 28 2013

More terms from Michel Marcus, Jun 28 2013

A101158 Let j be the smallest integer for which n+(n+1)+...+(n+j) is a square; sequence gives the squares.

Original entry on oeis.org

1, 9, 25, 4, 81, 121, 169, 225, 9, 361, 36, 25, 625, 729, 841, 16, 1089, 100, 1369, 1521, 196, 1849, 2025, 49, 25, 81, 2809, 3025, 3249, 3481, 3721, 324, 4225, 4489, 225, 36, 324, 5625, 484, 81, 6561, 6889, 225, 7569, 441, 676, 144, 9025, 49, 9801

Offset: 1

Charlie Marion, Dec 29 2004

nonn

Basis for sequence is shortest arithmetic sequence with initial term n and difference 1 that sums to a perfect square. Cf. A100251, A100252, A100253, A100254.

			a(11)=36 since 11+12+13 = 36.

Shawn A. Broyles, Table of n, a(n) for n = 1..1000

Cf. A101157, A101159, A101160.

n+(n+1)+...+(n+A101160(n)) = n+(n+1)+...+A101159(n) = A101157(n)^2 = a(n).

a(n^2) = n^2. - Michel Marcus, Jun 28 2013

a(21) corrected by Michel Marcus, Jun 29 2013

A101159 Let j be the smallest integer for which n+(n+1)+...+(n+j) is a square; then a(n) = n+j.

Original entry on oeis.org

1, 4, 7, 4, 13, 16, 19, 22, 9, 28, 13, 13, 37, 40, 43, 16, 49, 22, 55, 58, 28, 64, 67, 25, 25, 28, 79, 82, 85, 88, 91, 40, 97, 100, 40, 36, 44, 112, 49, 41, 121, 124, 47, 130, 53, 58, 49, 142, 49, 148, 151, 69, 67, 160, 163, 166, 64, 67, 175, 61

Offset: 1

Charlie Marion, Dec 29 2004

nonn

Basis for sequence is shortest arithmetic sequence with initial term n and difference 1 that sums to a perfect square. Cf. A100251, A100252, A100253, A100254.

			a(11)=13 since j=13 is the smallest integer such that 11+...+j=6^2=36 is a perfect square.

Shawn A. Broyles, Table of n, a(n) for n = 1..1000

Cf. A101157, A101158, A101160, A108269.

a(n) = {my(j = 0); while(! issquare(sum(k=0, j, n+k)), j++); n+j;} \\ Michel Marcus, May 02 2018

a(n) = my(s = n, t = 0); while(!issquare(s), s += n + t++); n + t \\ David A. Corneth, May 05 2018

n+(n+1)+...+(n+A101160(n)) = n+(n+1)+...+a(n) = A101157(n)^2 = A101158(n).
a(n^2) = n^2. - Michel Marcus, Jun 28 2013
a(n) <= 3*n - 2. - David A. Corneth, May 03 2018

More terms from Michel Marcus, Jun 28 2013

cp's OEIS Frontend

A100252 Least square n-gonal number greater than 1, or 0 if none exists.

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A101157 Let j be the smallest integer for which n+(n+1)+...+(n+j) is a square, say k^2; then a(n)=k.

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A100251 The square root of A100252; the index of the least square number greater than 1 that is also an n-gonal number, or 0 if none exists.

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A100254 Let j be the smallest integer for which 1 + (1+1*n) + (1+2*n) + ... + (1+j*n) = k^2 = s. Then a(n)=j; if no such j exists, then a(n)=0.

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A101160 a(n) is the smallest integer j for which n+(n+1)+...+(n+j) is a square.

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A101158 Let j be the smallest integer for which n+(n+1)+...+(n+j) is a square; sequence gives the squares.

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A101159 Let j be the smallest integer for which n+(n+1)+...+(n+j) is a square; then a(n) = n+j.

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PARI

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A100254 Let j be the smallest integer for which 1 + (1+1n) + (1+2n) + ... + (1+j*n) = k^2 = s. Then a(n)=j; if no such j exists, then a(n)=0.