login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

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).
2

%I #48 Jan 05 2025 19:51:42

%S 3,10,16,23,24,27,42,43,45,46,49,57,60,62,67,82,92,113,117,119,122,

%T 146,151,152,157,158,159,182,188,192,193,197,198,222,223,226,228,235,

%U 242,268,270,272,274,286,288,320,323,328,334,337,361,372,373,378,381,386

%N 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).

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

%C Numbers k such that A006218(k-1) + A006218(k) is a term of A006218. - _Robert Israel_, Dec 01 2022

%H Michael S. Branicky, <a href="/A358792/b358792.txt">Table of n, a(n) for n = 1..10000</a>

%H A. Behera and G. K. Panda, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/37-2/behera.pdf">On the square roots of triangular numbers</a>, The Fibonacci Quarterly, 37.2 (1999), 98-105.

%e k = 3:

%e d(1) + d(2) = d(4) = 3.

%e Thus the balancing number k = 3 is a term. The balancer r is 1.

%e k = 10:

%e d(1) + ... + d(9) = d(11) + ... + d(16) = 23.

%e Thus the balancing number k = 10 is a term. The balancer r is 6.

%e d(i) = A000005(i).

%p Tau:= map(numtheory:-tau, [$1..1000]):

%p S:= ListTools:-PartialSums(Tau):

%p A:=select(t -> member(S[t-1]+S[t], S), [$2..1000]); # _Robert Israel_, Dec 01 2022

%t 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 *)

%o (Python)

%o from sympy import divisor_count

%o from itertools import count, islice

%o def agen(): # generator of terms

%o d, s, sset, i = [0, 1, 2], [0, 1, 3], set(), 3

%o for k in count(2):

%o target = s[k-1] + s[k]

%o while s[-1] < target:

%o di = divisor_count(i); nexts = s[-1] + di; i += 1

%o d.append(di); s.append(nexts); sset.add(nexts)

%o if target in sset: yield k

%o print(list(islice(agen(), 56))) # _Michael S. Branicky_, Dec 04 2022

%Y Cf. A000005, A001109, A006218, A358797.

%K nonn

%O 1,1

%A _Ctibor O. Zizka_, Dec 01 2022

%E More terms from _Michael S. Branicky_, Dec 01 2022