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).

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

Offset

1,2

Comments

These numbers might be called "divisor sequence balancers" after Behera and Panda.

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

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

Cf. A000005, A001109, A006218, A358792.

Sequence in context: A141352 A276973 A248351 * A190552 A305408 A315437

Adjacent sequences: A358794 A358795 A358796 * A358798 A358799 A358800

Keyword

nonn

Author

Ctibor O. Zizka, Dec 01 2022

Extensions

More terms from Michael S. Branicky, Dec 01 2022

Status

approved