OFFSET
1,1
COMMENTS
These numbers might be called "divisor sequence balancing numbers", after the Behera and Panda link.
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
KEYWORD
nonn
AUTHOR
Ctibor O. Zizka, Dec 01 2022
EXTENSIONS
More terms from Michael S. Branicky, Dec 01 2022
STATUS
approved