A298467 -id:A298467 - cp's OEIS frontend

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.

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.

A296338 a(n) = number of partitions of n into consecutive positive squares.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Jan 14 2018

nonn

			   1 = 1^2,                   so  a(1) = 1.
   4 = 2^2,                   so  a(4) = 1.
   5 = 1^2 + 2^2,             so  a(5) = 1.
   9 = 3^2,                   so  a(9) = 1.
  13 = 2^2 + 3^2,             so a(13) = 1.
  14 = 1^2 + 2^2 + 3^2,       so a(14) = 1.
  16 = 4^2,                   so a(16) = 1.
  25 = 3^2 + 4^2 = 5^2,       so a(25) = 2.
  29 = 2^2 + 3^2 + 4^2,       so a(29) = 1.
  30 = 1^2 + 2^2 + 3^2 + 4^2, so a(30) = 1.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000290, A001227, A034705, A130052, A234304, A297199, A298467, A299173.

nMax = 100; t = {0}; Do[k = n; s = 0; While[s = s + k^2; s <= nMax, AppendTo[t, s]; k++], {n, 1, nMax}]; tt = Tally[t]; a[] = 0; Do[a[tt[[i, 1]]] = tt[[i, 2]], {i, 1, Length[tt]}]; Table[a[n], {n, 1, nMax}] (* _Jean-François Alcover, Feb 04 2018, using T. D. Noe's program for A034705 *)

def A296338(n)
  m = Math.sqrt(n).to_i
  ary = Array.new(n + 1, 0)
  (1..m).each{|i|
    sum = i * i
    ary[sum] += 1
    i += 1
    sum += i * i
    while sum <= n
      ary[sum] += 1
      i += 1
      sum += i * i
    end
  }
  ary[1..-1]
end
p A296338(100)

a(A034705(n)) >= 1 for n > 1.

G.f.: Sum_{i>=1} Sum_{j>=i} Product_{k=i..j} x^(k^2). - Ilya Gutkovskiy, Apr 18 2019

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

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]

A343777 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero n-gonal numbers in exactly n ways.

Original entry on oeis.org

2180, 554503705

Offset: 3

Ilya Gutkovskiy, Apr 30 2021

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

Cf. A298467, A334007, A334008, A334010, A334011, A334012, A338105.

A329236 a(n) is the least integer that can be expressed as the sum of one or more consecutive centered triangular numbers in exactly n ways.

Original entry on oeis.org

1, 64, 1789760

Offset: 1

Ilya Gutkovskiy, Apr 13 2020

If it exists, a(4) > 10^18. - Bert Dobbelaere, Apr 17 2020

Eric Weisstein's World of Mathematics, Centered Triangular Number

Cf. A005448, A054859, A068314, A298467, A322610, A334007.

cp's OEIS Frontend

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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A296338 a(n) = number of partitions of n into consecutive positive squares.

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

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A343777 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero n-gonal numbers in exactly n ways.

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A329236 a(n) is the least integer that can be expressed as the sum of one or more consecutive centered triangular numbers in exactly n ways.

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