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

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

1,2

Comments

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

Links

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

David A. Corneth, PARI program

Jon E. Schoenfield, Magma program

Example

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

Mathematica

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 *)

Prog

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

Crossrefs

Cf. A034705.

Sequence in context: A155387 A155366 A340002 * A135039 A168559 A212247

Adjacent sequences: A368567 A368568 A368569 * A368571 A368572 A368573

Keyword

nonn

Author

Allan C. Wechsler, Dec 30 2023

Extensions

More terms from Jon E. Schoenfield, 30 Dec 2023

Status

approved