A111044 -id:A111044 - cp's OEIS frontend

A298467 Smallest positive integer that can be expressed as the sum of consecutive positive squares in exactly n ways.

1, 25, 20449, 554503705

Offset: 1

Seiichi Manyama, Jan 19 2018

a(5) > 10^9. - Rémy Sigrist, Jan 19 2018

a(5) > 10^15. - Michael S. Branicky, Feb 18 2023

			a(2) = 25 because 3^2 + 4^2 = 5^2 = 25,
a(3) = 20449 because 7^2 + 8^2 + ... + 39^2 = 38^2 + 39^2 + ... + 48^2 = 143^2 = 20449.
a(4) = 554503705 because 480^2 + 481^2 + ... + 1210^2 = 3570^2 + 3571^2 + ... + 3612^2 = 3613^2 + 3614^2 + ... + 3654^2 = 7442^2 + 7443^2 ... + 7451^2 = 554503705. - _Rémy Sigrist_, Jan 19 2018

Cf. A038547, A111044, A130052, A234304, A234311.

a(n) <= A234311(n).

a(4) from Rémy Sigrist, Jan 19 2018

A062681 Numbers that are sums of 2 or more consecutive squares in more than 1 way.

Original entry on oeis.org

365, 1405, 1730, 2030, 3281, 3655, 3740, 4510, 4705, 4760, 5244, 5434, 5915, 7230, 7574, 8415, 9385, 11055, 11900, 12325, 12524, 14905, 16745, 17484, 18879, 19005, 19855, 20449, 20510, 21790, 22806, 23681, 25580, 25585, 27230, 27420, 28985

Offset: 1

Erich Friedman, Jul 04 2001

A subsequence of A174069; A111044 is the subsequence of numbers allowing at least 3 representations of the given form. - M. F. Hasler, Dec 23 2013

			365 = 10^2 + 11^2 + 12^2 = 13^2 + 14^2.

David W. Wilson, Table of n, a(n) for n = 1..10000
G. Campbell et al., Sums of consecutive squares in two or more ways, discussion in Number Theory group on LinkedIn, Dec 2013

Cf. A174069, A111044.

{N=200;A174069=[];A062681=[];for(i=2,N,for(k=1,i-1,setsearch(A174069,t=sum(j=i-k,i,j^2))&&(A062681=setunion(A062681,Set(t)))&&next;t>N^2*2&&break;A174069=setunion(A174069,Set(t))));A062681} \\ M. F. Hasler, Dec 22 2013

A234311 Least number which can be written as a sum of at least 2 consecutive squares in at least n different ways.

Original entry on oeis.org

5, 365, 147441, 554503705

Offset: 1

M. F. Hasler, Dec 23 2013

No more terms < 10^18. [Lars Blomberg, Jun 02 2014]

			a(1) = 5 = 1^2 + 2^2.
a(2) = 365 = 10^2 + 11^2 + 12^2 = 13^2 + 14^2.
a(3) = 147441 = 85^2+...+101^2 = 29^2+...+77^2 = 18^2+...+76^2.
a(4) = 554503705 = 480^2+...+1210^2 = 3570^2+...+3612^2 = 3613^2+...+3654^2 = 7442^2+...+7451^2.

Cf. A174069, A062681, A111044.

A299173 a(n) is the maximum number of squared consecutive positive integers into which the integer n can be partitioned.

Original entry on oeis.org

1, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 2, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 4, 5, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 2, 4, 0, 0, 0, 5, 6, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Jean-François Alcover, Feb 04 2018

a(k^2)>=1, the inequality being strict if k is in A097812.

			25 = 5^2 = 3^2 + 4^2 and no such partition is longer, so a(25) = 2.
30 = 1^2 + 2^2 + 3^2 + 4^2 and no such partition is longer, so a(30) = 4.
2018 = 7^2 + 8^2 + 9^2 + 10^2 + 11^2 + 12^2 + 13^2 + 14^2 + 15^2 + 16^2 + 17^2 + 18^2 and no such partition is longer, so a(2018) = 12. (This special example is due to _Seiichi Manyama_.) - _Jean-François Alcover_, Feb 05 2018

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A034705, A097812, A130052, A111044, A234304, A234311, A296338, A298467.

N:= 200: # to get a(1)..a(N)
A:= Vector(N):
S:= n -> n*(n+1)*(2*n+1)/6:
M:= floor(sqrt(N)):
for d from 1 to M do
  for b from d to M do
    s:= S(b) - S(b-d);
    if s > N then break fi;
    A[s]:= d
od od:
convert(A,list); # Robert Israel, Feb 04 2018

terms = 100; jmax = Ceiling[Sqrt[terms]]; kmax = Ceiling[(3*terms)^(1/3)]; Clear[a]; a[_] = 0; Do[r = Range[j, j + k - 1]; n = r . r; If[k > a[n], a[n] = k], {j, jmax}, {k, kmax}]; Array[a, terms]

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A298467 Smallest positive integer that can be expressed as the sum of consecutive positive squares in exactly n ways.

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A062681 Numbers that are sums of 2 or more consecutive squares in more than 1 way.

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PARI

A234311 Least number which can be written as a sum of at least 2 consecutive squares in at least n different ways.

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A299173 a(n) is the maximum number of squared consecutive positive integers into which the integer n can be partitioned.

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