A368570 -id:A368570 - cp's OEIS frontend

A034705 Numbers that are sums of consecutive squares.

0, 1, 4, 5, 9, 13, 14, 16, 25, 29, 30, 36, 41, 49, 50, 54, 55, 61, 64, 77, 81, 85, 86, 90, 91, 100, 110, 113, 121, 126, 135, 139, 140, 144, 145, 149, 169, 174, 181, 190, 194, 196, 199, 203, 204, 221, 225, 230, 245, 255, 256, 265, 271, 280, 284, 285, 289, 294, 302

Offset: 1

Erich Friedman

nonn

Also, differences of any pair of square pyramidal numbers (A000330). These could be called "truncated square pyramidal numbers". - Franklin T. Adams-Watters, Nov 29 2006

If n is the sum of d consecutive squares up to m^2, n = A000330(m) - A000330(m-d) = d*(m^2 - (d-1)*m + (d-1)*(2*d-1)/6) <=> m^2 - (d-1)*m = c := n/d - (d-1)*(2*d-1)/6 <=> m = (d-1)/2 + sqrt((d-1)^2/4 + c) which must be an integer. Moreover, A000330(x) >= x^3/3, so m and d can't be larger than (3*n)^(1/3). - M. F. Hasler, Jan 02 2024

			All squares (A000290: 0, 1, 4, 9, ...) are in this sequence, since "consecutive" in the definition means a subsequence without interruption, so a single term qualifies.
5 = 1^2 + 2^2 = A000330(2) is in this sequence, and similarly 13 = 2^2 + p3^2  = A000330(3) - A000330(1) and 14 = 1^2 + 2^2 + 3^2 = A000330(3), etc.

Cf. A000290, A000330, A034706.
Cf. A217843-A217850 (sums of consecutive powers 3 to 10).
Cf. A368570 (first of each pair of consecutive integers in this sequence).

import Data.Set (deleteFindMin, union, fromList); import Data.List (inits)
a034705 n = a034705_list !! (n-1)
a034705_list = f 0 (tail $ inits $ a000290_list) (fromList [0]) where
   f x vss'@(vs:vss) s
     | y < x = y : f x vss' s'
     | otherwise = f w vss (union s $ fromList $ scanl1 (+) ws)
     where ws@(w:_) = reverse vs
           (y, s') = deleteFindMin s
-- Reinhard Zumkeller, May 12 2015

nMax = 1000; t = {0}; Do[k = n; s = 0; While[s = s + k^2; s <= nMax, AppendTo[t, s]; k++], {n, Sqrt[nMax]}]; t = Union[t] (* T. D. Noe, Oct 23 2012 *)

{is_A034705(n)= for(d=1,sqrtnint(n*3,3), my(b = (d-1)/2, s = n/d - (d-1)*(d*2-1)/6 + b^2); denominator(s)==denominator(b)^2 && issquare(s, &s) && return(b+s)); !n} \\ Return the index of the largest square of the sum (or 1 for n = 0) if n is in the sequence, else 0. - 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 = None
    while True:
        (v, s, l) = heapq.heappop(h)
        if v != v1: yield v; 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(), 60))) # Michael S. Branicky, Jan 06 2024

Terms a(1..10^4) double-checked with independent code by M. F. Hasler, Jan 02 2024

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

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A034705 Numbers that are sums of consecutive squares.

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