A034705 -id:A034705 - cp's OEIS frontend

A217849 Numbers which are the sums of consecutive ninth powers.

0, 1, 512, 513, 19683, 20195, 20196, 262144, 281827, 282339, 282340, 1953125, 2215269, 2234952, 2235464, 2235465, 10077696, 12030821, 12292965, 12312648, 12313160, 12313161, 40353607, 50431303, 52384428, 52646572, 52666255, 52666767, 52666768, 134217728

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 = 10^9; t = {0}; Do[k = n; s = 0; While[s = s + k^9; s <= nMax, AppendTo[t, s]; k++], {n, nMax^(1/9)}]; t = Union[t]

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]

A307608 Number of partitions of n^2 into consecutive positive squares.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1

Offset: 1

Ilya Gutkovskiy, Apr 18 2019

nonn

			29^2 = 20^2 + 21^2, so a(29) = 2.

Index entries for sequences related to sums of squares

Cf. A000290, A030273, A034705, A037444, A097812, A151557, A296338.

a(n) = [x^(n^2)] Sum_{i>=1} Sum_{j>=i} Product_{k=i..j} x^(k^2).
a(n) = A296338(A000290(n)).
a(n) >= 2 for n in A097812.

A217847 Numbers which are the sums of consecutive seventh powers.

Original entry on oeis.org

0, 1, 128, 129, 2187, 2315, 2316, 16384, 18571, 18699, 18700, 78125, 94509, 96696, 96824, 96825, 279936, 358061, 374445, 376632, 376760, 376761, 823543, 1103479, 1181604, 1197988, 1200175, 1200303, 1200304, 2097152, 2920695, 3200631, 3278756, 3295140

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 = 10^7; t = {0}; Do[k = n; s = 0; While[s = s + k^7; s <= nMax, AppendTo[t, s]; k++], {n, nMax^(1/7)}]; t = Union[t]

A217848 Numbers which are the sums of consecutive eighth powers.

Original entry on oeis.org

0, 1, 256, 257, 6561, 6817, 6818, 65536, 72097, 72353, 72354, 390625, 456161, 462722, 462978, 462979, 1679616, 2070241, 2135777, 2142338, 2142594, 2142595, 5764801, 7444417, 7835042, 7900578, 7907139, 7907395, 7907396, 16777216, 22542017, 24221633, 24612258

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 = 10^8; t = {0}; Do[k = n; s = 0; While[s = s + k^8; s <= nMax, AppendTo[t, s]; k++], {n, nMax^(1/8)}]; t = Union[t]

A256677 Primes whose nearest neighbors are both the sum of consecutive squares.

Original entry on oeis.org

1291, 37813, 48091, 76091, 100829, 120091, 167911, 193549, 271639, 281291, 436621, 847727, 847741, 1188529, 1705549, 2933083, 3126173, 5451643, 7703653, 9623533, 9755149, 10468261, 11971373, 17424301, 20892871, 20967883

Offset: 1

Torlach Rush, Jun 26 2015

nonn

			a(1) = 1291; 1290 = 14^2 + ... + 18^2; 1292 = 9^2 + ... + 16^2.
a(2) = 37813; 37812 = 29^2 + ... + 51^2; 37814 = 32^2 + ... + 52^2.

Chai Wah Wu, Table of n, a(n) for n = 1..47
Index entries for sequences related to sums of squares

Cf. A034705.

A322135 Table of truncated square pyramid numbers, read by antidiagonals.

Original entry on oeis.org

1, 4, 5, 9, 13, 14, 16, 25, 29, 30, 25, 41, 50, 54, 55, 36, 61, 77, 86, 90, 91, 49, 85, 110, 126, 135, 139, 140, 64, 113, 149, 174, 190, 199, 203, 204, 81, 145, 194, 230, 255, 271, 280, 284, 285, 100, 181, 245, 294, 330, 355, 371, 380, 384, 385, 121, 221, 302

Offset: 1

Allan C. Wechsler, Nov 27 2018

The n-th row contains n numbers: n^2, n^2 + (n-1)^2, ..., n^2 + (n-1)^2 + ... + 1^2.
All numbers that appear in the table are listed in ascending order at A034705.
All numbers that appear twice or more are listed at A130052.
The left column is A000290 (the squares).
The top row is A000330 (the square pyramidal numbers).
The columns are A000290, A099776 (or a tail of A001844), a tail of A005918 or A120328, a tail of A027575, a tail of A027578, a tail of A027865, ...
The first two rows are A000330 and a tail of A168599, but subsequent rows are not currently in the OEIS, and are all tails of A000330 minus various constants.
The main diagonal is A050410.

			The 17th term is entry 2 on antidiagonal 6, so we sum two terms: 6^2 + 5^2 = 61.
Table begins:
   1   5  14  30  55  91 140 204 ...
   4  13  29  54  90 139 203 ...
   9  25  50  86 135 199 ...
  16  41  77 126 190 ...
  25  61 110 174 ...
  36  85 149 ...
  49 113 ...
  64 ...
  ...

See comments; also cf. A000330, A059255.

T[n_,k_] = Sum[(n+i)^2, {i,0,k-1}]; Table[T[n-k+1, k], {n,1,10},  {k,1,n}] // Flatten (* Amiram Eldar, Nov 28 2018 *)
f[n_] := Table[SeriesCoefficient[-((y (y (1 + y) + x (1 - 2 y - 3 y^2) + x^2 (1 - 3 y + 4 y^2)))/((-1 + x)^3 (-1 + y)^4)) , {x, 0,
i + 1 - j}, {y, 0, j}], {i, n, n}, {j, 1, n}]; Flatten[Array[f, 10]] (* Stefano Spezia, Nov 28 2018 *)

T(n,k) = n^2 + (n+1)^2 + ... + (n+k-1)^2 = A000330(n + k - 1) - A000330(n - 1) = T(n, k) = k*n^2 + (k^2 - k)*n + (1/3*k^3 - 1/2*k^2 + 1/6*k)

G.f.: -y*(y*(1 + y) + x*(1 - 2*y - 3*y^2) + x^2*(1 - 3*y + 4*y^2))/((- 1 + x)^3*(- 1 + y)^4). - Stefano Spezia, Nov 28 2018

A368590 Numbers k such that all of k, k+1 and k+2 are the sums of consecutive squares.

Original entry on oeis.org

728, 1013, 2813, 3309, 4323, 4899, 12438, 21259, 23113, 31394, 35719, 37812, 38023, 111894, 143449, 194053, 418613, 418614, 487368, 535309, 2232593, 2452644, 2490669, 9226854, 17367998, 19637644, 20341453, 28553671, 33406839, 174398434, 468936719, 1468970139, 2136314464

Offset: 1

David A. Corneth, Dec 31 2023

nonn

418613 is the smallest k such that k through k + 3 are the sums of consecutive squares.
After an idea by Allan C. Wechsler.
a(30)-a(33) were calculated using the b-file at A368570.

			728 is in the sequence via 728 = 7^2 + 8^2 + 9^2 + 10^2 + 11^2 + 12^2 + 13^2, 729 = 27^2 and 730 = 10^2 + 11^2 + 12^2 + 13^2 + 14^2.

Michael S. Branicky, Table of n, a(n) for n = 1..94 (terms 1..74 from Frank A. Stevenson)
David A. Corneth, PARI program

Subsequence of A034705 and of A368570.

PARI
```
\\ See PARI program
```

import heapq
from itertools import islice
def agen(): # generator of terms
    m = 1; h = [(m, 1, 1)]; nextcount = 2
    v1 = v2 = -1
    while True:
        (v, s, l) = heapq.heappop(h)
        if v != v1:
            if v2 + 2 == v1 + 1 == v: yield v2
            v2, 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(), 33))) # Michael S. Branicky, Jan 01 2024

A273874 Least positive integer k such that k^2 + (k+1)^2 + ... + (k+n-2)^2 + (k+n-1)^2 is the sum of two nonzero squares. a(n) = 0 if no solution exists.

Original entry on oeis.org

5, 1, 2, 0, 2, 0, 0, 0, 0, 2, 5, 1, 12, 0, 3, 0, 3, 0, 0, 0, 0, 0, 53, 1, 1, 1, 2, 0, 4, 0, 0, 0, 5, 2, 0, 0, 2, 0, 3, 0, 5, 0, 0, 5, 0, 0, 73, 1, 3, 1, 2, 0, 2, 0, 5, 0, 0, 2, 97, 1, 4, 0, 0, 0, 2, 5, 0, 0, 30, 0, 0, 0, 1, 1, 4, 0, 0, 0, 0, 0, 0, 2, 26, 0, 6

Offset: 1

Altug Alkan, Jun 02 2016

nonn

Least positive integer k such that Sum_{i=0..n-1} (k+i)^2 = n*(6*k^2 + 6*k*n - 6*k + 2*n^2 - 3*n + 1)/6 is the sum of two nonzero squares. a(n) = 0 if no k exists for corresponding n.

			a(1) = 5 because 5^2 = 3^2 + 4^2.
a(3) = 2 because 2^2 + 3^2 + 4^2 = 2^2 + 5^2.

Cf. A000404, A034705.

a(7)-a(50) from Giovanni Resta, Jun 02 2016

More terms from Jinyuan Wang, May 02 2021

A364143 a(n) is the minimal number of consecutive squares needed to sum to A216446(n).

Original entry on oeis.org

2, 5, 3, 2, 2, 3, 10, 2, 7, 9, 12, 11, 6, 11, 14, 3, 11, 29, 14, 7, 23, 4, 49, 8, 24, 5, 17, 12, 38, 46, 27, 34, 6, 14, 22, 66, 11, 66, 14, 11, 6, 77, 36, 63, 96, 11, 50, 3, 19, 96, 52, 41, 66, 33, 11, 3, 14, 121, 66, 89, 34, 127, 51, 2, 86, 54, 181, 48, 8

Offset: 1

Darío Clavijo, Jul 10 2023

			a(8) = 7 is because 7 consecutive squares are needed to sum to A216446(8) = 595 = 6^2 + 7^2 + 8^2 + 9^2 + 10^2 + 11^2 + 12^2.

Project Euler, Problem 125: Palindromic Sums.

Cf. A216446, A034705, A180436, A267600.

is_palindrome = lambda n: str(n) == str(n)[::-1]
def g(L):
  L2, squares, D = L*L, [x*x for x in range(0, L + 1)], {}
  for i in range(1, L + 1):
    for j in range(i + 1, L + 1):
      candidate = sum(squares[i:j+1])
      if candidate < L2 and is_palindrome(candidate):
        if candidate in D:
          D[candidate]= min(D[candidate], j-i-1)
        else:
          D[candidate] = j-i+1
  return [D[k] for k in sorted(D.keys())]
print(g(1000))

cp's OEIS Frontend

A217849 Numbers which are the sums of consecutive ninth powers.

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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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Mathematica

A307608 Number of partitions of n^2 into consecutive positive squares.

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A217847 Numbers which are the sums of consecutive seventh powers.

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Mathematica

A217848 Numbers which are the sums of consecutive eighth powers.

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Mathematica

A256677 Primes whose nearest neighbors are both the sum of consecutive squares.

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A322135 Table of truncated square pyramid numbers, read by antidiagonals.

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Mathematica

Formula

A368590 Numbers k such that all of k, k+1 and k+2 are the sums of consecutive squares.

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PARI

Python

A273874 Least positive integer k such that k^2 + (k+1)^2 + ... + (k+n-2)^2 + (k+n-1)^2 is the sum of two nonzero squares. a(n) = 0 if no solution exists.

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