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A358797
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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).
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2
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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
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OFFSET
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1,2
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COMMENTS
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These numbers might be called "divisor sequence balancers" after Behera and Panda.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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