A034705 -id:A034705 - cp's OEIS frontend

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

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

A319184 Numbers that are sums of consecutive pentagonal numbers.

Original entry on oeis.org

0, 1, 5, 6, 12, 17, 18, 22, 34, 35, 39, 40, 51, 57, 69, 70, 74, 75, 86, 92, 108, 117, 120, 121, 125, 126, 145, 156, 162, 176, 178, 190, 195, 196, 209, 210, 213, 247, 248, 262, 270, 279, 282, 287, 288, 321, 330, 354, 365, 376, 386, 387, 399, 404, 405, 424, 425, 438, 457, 475

Offset: 1

Ilya Gutkovskiy, Dec 21 2018

nonn

Eric Weisstein's World of Mathematics, Pentagonal Number

Cf. A000326, A002411, A034705, A034706, A319185, A322636, A322637.

anmax = 1000; nmax = Floor[Sqrt[2*anmax/3]] + 1; Select[Union[Flatten[Table[Sum[k*(3*k-1)/2, {k, i, j}], {i, 0, nmax}, {j, i, nmax}]]], # <= anmax &] (* Vaclav Kotesovec, Dec 21 2018 *)
Module[{nn=20,pn},pn=PolygonalNumber[5,Range[0,nn]];Take[Union[Flatten[Table[Total/@Partition[pn,d,1],{d,nn}]]],60]] (* Harvey P. Dale, Jun 22 2025 *)

A322611 Numbers that are sums (of a nonempty sequence) of consecutive centered square numbers.

Original entry on oeis.org

1, 5, 6, 13, 18, 19, 25, 38, 41, 43, 44, 61, 66, 79, 84, 85, 102, 113, 127, 140, 145, 146, 181, 187, 198, 212, 221, 225, 230, 231, 258, 259, 265, 300, 313, 325, 326, 338, 343, 344, 365, 402, 404, 421, 439, 445, 470, 481, 483, 486, 488, 489, 524, 545, 547, 578, 585, 613, 626, 651, 660

Offset: 1

Ilya Gutkovskiy, Dec 20 2018

nonn

Eric Weisstein's World of Mathematics, Centered Square Number

Cf. A001844, A005900, A034705, A320728, A322479, A322610, A322638, A322640.

anmax = 1000; nmax = Floor[Sqrt[anmax/2]] + 1; Select[Union[Flatten[Table[Sum[k^2 + (k + 1)^2, {k, i, j}], {i, 0, nmax}, {j, i, nmax}]]], # <= anmax &] (* Vaclav Kotesovec, Dec 21 2018 *)

A319185 Numbers that are sums of consecutive hexagonal numbers (A000384).

Original entry on oeis.org

0, 1, 6, 7, 15, 21, 22, 28, 43, 45, 49, 50, 66, 73, 88, 91, 94, 95, 111, 120, 139, 153, 154, 157, 160, 161, 190, 202, 211, 230, 231, 245, 251, 252, 273, 276, 277, 322, 325, 343, 350, 364, 365, 371, 372, 378, 421, 430, 435, 463, 475, 496, 503, 507, 518, 524, 525, 554, 561

Offset: 1

Ilya Gutkovskiy, Dec 21 2018

nonn

Eric Weisstein's World of Mathematics, Hexagonal Number

Cf. A000384, A002412, A034705, A034706, A319184, A322636, A322637.

anmax = 1000; nmax = Floor[Sqrt[anmax/2]] + 1; Select[Union[Flatten[Table[Sum[k*(2*k-1), {k, i, j}], {i, 0, nmax}, {j, i, nmax}]]], # <= anmax &] (* Vaclav Kotesovec, Dec 21 2018 *)

A320728 Numbers that are sums of consecutive odd squares (or centered octagonal numbers).

Original entry on oeis.org

0, 1, 9, 10, 25, 34, 35, 49, 74, 81, 83, 84, 121, 130, 155, 164, 165, 169, 202, 225, 251, 276, 285, 286, 289, 290, 361, 371, 394, 420, 441, 445, 454, 455, 514, 515, 529, 596, 625, 645, 650, 670, 679, 680, 683, 729, 802, 804, 841, 875, 885, 934, 959, 961, 968, 969, 970, 1044, 1089

Offset: 1

Ilya Gutkovskiy, Dec 21 2018

nonn

Index entries for sequences related to sums of squares

Cf. A000447, A016754, A034705, A152043, A322610, A322611, A322637, A322638, A322640.

ok(n)={my(i=sqrtint(n)); i=i-(i%2==0); while(i>0, my(a=i^2, j=i); while(j>0 && a<=n, if(a==n, return(1)); j-=2; a=a+j^2); i-=2); 0}
concat([0], select(ok, [1..1200])) \\ Antonio Roldán, Mar 12 2020

A180436 Palindromic numbers which are sum of consecutive squares.

Original entry on oeis.org

1, 4, 5, 9, 55, 77, 121, 181, 313, 434, 484, 505, 545, 595, 636, 676, 818, 1001, 1111, 1441, 1771, 4334, 6446, 10201, 12321, 14641, 17371, 17871, 19691, 21712, 40804, 41214, 42924, 44444, 44944, 46564, 51015, 65756, 69696, 81818, 94249, 97679, 99199

Offset: 1

Zhining Yang, Jan 19 2011

In more than one way: 554455, 9343439, ... (A267600) - Robert G. Wilson v, May 28 2012

			1001 is in the sequence because 1001 is palindromic and it can be written as sum of consecutive squares (1001 = 4^2 + 5^2 + 6^2 + ... + 13^2 + 14^2).

Giovanni Resta, Table of n, a(n) for n = 1..10400 (terms < 10^18, first 228 terms from Robert G. Wilson v)

Cf. A002113, A002779, A034705, A216446, A267600.

palQ[n_Integer] := Block[{idn = IntegerDigits[n]}, idn == Reverse[idn]]; lst = {}; k = 1; While[k < 1000, AppendTo[lst, Select[ Accumulate[ Range[k, 1000]^2], palQ]]; lst = Union@ Flatten@ lst; k++]; Select[lst, # < 10^6 &] (* Robert G. Wilson v, May 28 2012 *)

A216446 Palindromic numbers which can be written as the sum of two or more consecutive squares.

Original entry on oeis.org

5, 55, 77, 181, 313, 434, 505, 545, 595, 636, 818, 1001, 1111, 1441, 1771, 4334, 6446, 17371, 17871, 19691, 21712, 41214, 42924, 44444, 46564, 51015, 65756, 81818, 97679, 99199, 108801, 127721, 137731, 138831, 139931, 148841, 161161, 166661, 171171, 188881

Offset: 1

V. Raman, Sep 07 2012

			636 is in the sequence because it is a palindrome and 636 = 4^2+5^2+6^2+7^2+8^2+9^2+10^2+11^2+12^2.

V. Raman and Giovanni Resta, Table of n, a(n) for n = 1..10306 (terms < 10^18, first 298 terms from V. Raman)

Cf. A034705, A180436, A267600 (terms with more than one representation).

palQ[n_] := Block[{d = IntegerDigits@ n}, d == Reverse@ d]; upto = 10^6; Union[ Reap[ For[i=1, s=i^2 + (i+1)^2; s < upto, i++, For[j=i+1, s < upto, j++; s += j^2, If[palQ[s], Sow@ s]]]][[2, 1]]] (* Giovanni Resta, Jun 14 2018 *)
With[{nn=200},Select[Union[Flatten[Table[Total/@Partition[Range[nn]^2,n,1],{n,2,nn}]]],PalindromeQ]] (* Harvey P. Dale, Oct 17 2021 *)

Errors in previous b-file noticed by Riley Waugh, Jun 13 2018

A368570 Numbers k such that both k and k+1 are the sums of consecutive squares.

Original entry on oeis.org

0, 4, 13, 29, 49, 54, 85, 90, 139, 144, 203, 255, 284, 365, 384, 505, 509, 649, 676, 728, 729, 818, 924, 960, 1013, 1014, 1201, 1210, 1225, 1239, 1454, 1495, 1784, 1854, 2108, 2214, 2469, 2665, 2779, 2813, 2814, 2869, 3025, 3135, 3309, 3310, 3794, 4230, 4323, 4324, 4705

Offset: 1

Allan C. Wechsler, Dec 30 2023

nonn

A subsequence of A034705, identifying the lower of each pair of consecutive integers belonging to that sequence.

Similarly, two consecutive integers in this sequence, a(n+1) = a(n)+1, such as 1013 and 1014, or 3309 and 3310, correspond to three consecutive integers in A034705, and so on. - M. F. Hasler, Jan 02 2024

			85 = 6^2 + 7^2, and 86 = 3^2 + 4^2 + 5^2 + 6^2, so 85 is in the list.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000
David A. Corneth, PARI program
Jon E. Schoenfield, Magma program

Cf. A034705.

a[n_] := Module[{v, r = {}, s = 1, t, ul = 100, pr = 1}, While[Length[r] < n, v = ConstantArray[0, ul + 1]; Do[t = 0; Do[t += j^2; If[t <= ul + 1, v[[t]] = 1, Break[]], {j, i, 1, -1}], {i, 1, Sqrt[ul + 1]}]; Do[If[v[[i]] == 1, s++; If[s >= 2 && Not[MemberQ[r, i - 1]], AppendTo[r, i - 1]], s = 0], {i, pr, ul + 1}]; pr = ul + 1; ul *= 2; ]; Take[r, n]];
a[49] (* Robert P. P. McKone, Dec 30 2023 *)

PARI
```
\\ See PARI link
```

is_A368570(n)=is_A034705(n)&&is_A034705(n+1) \\ M. F. Hasler, Jan 02 2024

import heapq
from itertools import islice
def agen(): # generator of terms
    m = 0; h = [(m, 0, 0)]; nextcount = 1; v1 = -2
    while True:
        (v, s, l) = heapq.heappop(h)
        if v != v1:
            if v1 + 1 == v: yield v1
            v1 = v
        if v >= m:
            m += nextcount*nextcount
            heapq.heappush(h, (m, 1, nextcount))
            nextcount += 1
        v -= s*s; s += 1; l += 1; v += l*l
        heapq.heappush(h, (v, s, l))
print(list(islice(agen(), 51))) # Michael S. Branicky, Jan 01 2024

More terms from Jon E. Schoenfield, 30 Dec 2023

A217845 Numbers which are the sums of consecutive fifth powers.

Original entry on oeis.org

0, 1, 32, 33, 243, 275, 276, 1024, 1267, 1299, 1300, 3125, 4149, 4392, 4424, 4425, 7776, 10901, 11925, 12168, 12200, 12201, 16807, 24583, 27708, 28732, 28975, 29007, 29008, 32768, 49575, 57351, 59049, 60476, 61500, 61743, 61775, 61776, 91817, 100000, 108624

Offset: 1

T. D. Noe, Oct 23 2012

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A034705, A217843-A217850.

nMax = 200000; t = {0}; Do[k = n; s = 0; While[s = s + k^5; s <= nMax, AppendTo[t, s]; k++], {n, nMax^(1/5)}]; t = Union[t]

A217846 Numbers which are the sums of consecutive sixth powers.

Original entry on oeis.org

0, 1, 64, 65, 729, 793, 794, 4096, 4825, 4889, 4890, 15625, 19721, 20450, 20514, 20515, 46656, 62281, 66377, 67106, 67170, 67171, 117649, 164305, 179930, 184026, 184755, 184819, 184820, 262144, 379793, 426449, 442074, 446170, 446899, 446963, 446964, 531441

Offset: 1

T. D. Noe, Oct 23 2012

T. D. Noe, Table of n, a(n) for n = 1..1000

Subsequences include A001014 and A008516.

Cf. A034705, A217843-A217850.

nMax = 10^6; t = {0}; Do[k = n; s = 0; While[s = s + k^6; s <= nMax, AppendTo[t, s]; k++], {n, nMax^(1/6)}]; t = Union[t]

list(lim)=my(v=List(apply(n->n^6, [0..sqrtnint(lim\=1,6)])),s); for(n=2,lim, s=n*(n-1)*(2*n-1)*(3*n^4-6*n^3+3*n+1)/42; if(s>lim,break); for(k=n,lim, s+=k^6-(k-n)^6; if(s>lim,break); listput(v,s))); Set(v) \\ Charles R Greathouse IV, Apr 22 2020

cp's OEIS Frontend

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

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A319184 Numbers that are sums of consecutive pentagonal numbers.

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A322611 Numbers that are sums (of a nonempty sequence) of consecutive centered square numbers.

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A319185 Numbers that are sums of consecutive hexagonal numbers (A000384).

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Mathematica

A320728 Numbers that are sums of consecutive odd squares (or centered octagonal numbers).

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A180436 Palindromic numbers which are sum of consecutive squares.

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A216446 Palindromic numbers which can be written as the sum of two or more consecutive squares.

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A368570 Numbers k such that both k and k+1 are the sums of consecutive squares.

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Mathematica

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A217845 Numbers which are the sums of consecutive fifth powers.

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