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A358797 Numbers r such that for some k we have d(1) + ... + d(k - 1) = d(k + 1) + ... + d(k + r), where d(i) = A000005(i). 2
1, 6, 11, 16, 17, 19, 31, 32, 34, 34, 37, 43, 45, 47, 52, 63, 72, 89, 92, 92, 97, 117, 120, 120, 126, 126, 126, 146, 150, 154, 156, 158, 159, 178, 179, 182, 184, 190, 197, 217, 219, 221, 222, 232, 234, 260, 264, 267, 272, 276, 298, 304, 306, 310, 314, 317, 317 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
These numbers might be called "divisor sequence balancers" after Behera and Panda.
LINKS
A. Behera and G. K. Panda, On the square roots of triangular numbers, The Fibonacci Quarterly, 37.2 (1999), 98-105.
EXAMPLE
r = 1:
d(1) + d(2) = d(4) = 3.
Thus the balancer r = 1 is a term. The balancing number k = 3.
r = 6:
d(1) + ... + d(9) = d(11) + ... + d(16) = 23.
Thus the balancer r = 6 is a term. The balancing number k = 10.
d(i) = A000005(i).
MATHEMATICA
With[{m = 720}, d = DivisorSigma[0, Range[m]]; s = Accumulate[d]; e = 2*s - d; i = Select[Range[2, m], MemberQ[s, e[[#]]] &]; Position[s, #][[1, 1]] & /@ e[[i]] - i] (* Amiram Eldar, Dec 01 2022 *)
PROG
(Python)
from sympy import divisor_count
from itertools import count, islice
def agen(): # generator of terms
d, s, sdict, i = [0, 1, 2], [0, 1, 3], dict(), 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); sdict[nexts] = i-1
if target in sdict: yield sdict[target] - k
print(list(islice(agen(), 57))) # Michael S. Branicky, Dec 04 2022
CROSSREFS
Sequence in context: A141352 A276973 A248351 * A190552 A305408 A315437
KEYWORD
nonn
AUTHOR
Ctibor O. Zizka, Dec 01 2022
EXTENSIONS
More terms from Michael S. Branicky, Dec 01 2022
STATUS
approved

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Last modified February 26 06:43 EST 2024. Contains 370335 sequences. (Running on oeis4.)