A358792 Numbers k such that for some r we have d(1) + ... + d(k - 1) = d(k + 1) + ... + d(k + r), where d(i) = A000005(i).

3, 10, 16, 23, 24, 27, 42, 43, 45, 46, 49, 57, 60, 62, 67, 82, 92, 113, 117, 119, 122, 146, 151, 152, 157, 158, 159, 182, 188, 192, 193, 197, 198, 222, 223, 226, 228, 235, 242, 268, 270, 272, 274, 286, 288, 320, 323, 328, 334, 337, 361, 372, 373, 378, 381, 386

Offset

1,1

Comments

These numbers might be called "divisor sequence balancing numbers", after the Behera and Panda link.

Numbers k such that A006218(k-1) + A006218(k) is a term of A006218. - Robert Israel, Dec 01 2022

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

A. Behera and G. K. Panda, On the square roots of triangular numbers, The Fibonacci Quarterly, 37.2 (1999), 98-105.

Example

k = 3:
d(1) + d(2) = d(4) = 3.
Thus the balancing number k = 3 is a term. The balancer r is 1.
k = 10:
d(1) + ... + d(9) = d(11) + ... + d(16) = 23.
Thus the balancing number k = 10 is a term. The balancer r is 6.
d(i) = A000005(i).

Maple

Tau:= map(numtheory:-tau, [$1..1000]):
S:= ListTools:-PartialSums(Tau):
A:=select(t -> member(S[t-1]+S[t], S), [$2..1000]); # Robert Israel, Dec 01 2022

Mathematica

With[{m = 720}, d = DivisorSigma[0, Range[m]]; s = Accumulate[d]; Select[Range[2, m], MemberQ[s, 2*s[[#]] - d[[#]]] &]] (* Amiram Eldar, Dec 01 2022 *)

Prog

(Python)
from sympy import divisor_count
from itertools import count, islice
def agen(): # generator of terms
    d, s, sset, i = [0, 1, 2], [0, 1, 3], set(), 3
    for k in count(2):
        target = s[k-1] + s[k]
        while s[-1] < target:
            di = divisor_count(i); nexts = s[-1] + di; i += 1
            d.append(di); s.append(nexts); sset.add(nexts)
        if target in sset: yield k
print(list(islice(agen(), 56))) # Michael S. Branicky, Dec 04 2022

Crossrefs

Cf. A000005, A001109, A006218, A358797.

Sequence in context: A242203 A093516 A246302 * A278041 A063209 A063109

Adjacent sequences: A358789 A358790 A358791 * A358793 A358794 A358795

Keyword

nonn,changed

Author

Ctibor O. Zizka, Dec 01 2022

Extensions

More terms from Michael S. Branicky, Dec 01 2022

Status

approved