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

Author

Allan C. Wechsler, Dec 30 2023

Keywords

• nonn

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

Examples

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

Links

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

Crossrefs

Cf. A034705.

Programs

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 *)

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

Extensions

More terms from Jon E. Schoenfield, 30 Dec 2023