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User: Robert Dawson

Robert Dawson has authored 1 sequences.

A301912 Numbers k such that the decimal representation of k ends that of the sum of the first k cubes.

0, 1, 5, 25, 76, 376, 500, 625, 876, 1876, 2500, 5001, 5625, 9376, 15625, 25001, 40625, 50001, 62500, 65625, 71876, 75001, 90625, 109376, 171876, 265625, 375001, 390625, 500001, 765625, 875001, 890625, 1171876, 2265625, 2890625, 4062500, 4375001, 5000001

Offset: 1

Robert Dawson, Mar 28 2018

For j >= 3, 1 + 5*10^j = A199685(j) is in the sequence, so the sequence is infinite. - Vaclav Kotesovec, Mar 29 2018
From Robert Dawson, Apr 12 2018: (Start)
This sequence is the union of the following ten subsequences.
Terms in  have fewer than d digits: they are always terms of the sequence, and always appear elsewhere, as an earlier term of the same subsequence or a related subsequence. (However, the d-th terms of the subsequences are always distinct for any d > 4.) Dashes replace certain solutions to the congruences for small values of d for which certain other divisibility criteria are not met. The integers n_0(d) and n_1(d) are the even and odd zeros of n^2+3n+4 (mod 2^d) (note that by Hensel's Lemma these always exist and each is unique).
(i)   p(d) satisfying 2^d| p(d) - n_0(d), 5^d |p(d):
  (0,<0>,500,2500,62500,62500,4062500,14062500,...)
(ii)  q(d) satisfying 2^{d-1}|q(d)-1, 5^d|q(d) for d != 3:
  (0,25,-,<625>,40625,390625,2890625,12890625,...)
(iii) q(d) + 5x10^{d-1} for d != 2:
  (5,-, 625,5625,90625, 890625,7890625, 62890625,...)
(iv)  q'(d) satisfying 2^{d-1}|q'(d) - n_1(d), 5^d|q'(d), for d != 1,3:
  (-,25,-,<625>,15625,265625,2265625,47265625,...)
(v)   q'(d) + 5x10^{d-1} for d != 2:
  (5,-,625,5625,65625,765625,7265625,97265625,...)
(vi)  r(d) satisfying 2^d|r(d), 5^d|r(d)-1 for d >= 2
  (-,76,376,9376,<9376>,109376,7109376,87109376,...) = A016090(d)
(vii) r'(d) satisfying 2^d|r'(d) - n_0(d), 5^d|r'(d)-1 for d >= 2:
  (-,76,876,1876,71876,171876,1171876,<1171876>,...)
(viii)s(d) := 5x10^{d-1}+1 for d >= 4:
  (-,-,-,5001,50001,500001,5000001,50000001,...) = A199685(d-1)
(ix)  t(d) satisfying 2^{d-1}|t(d)-n_0(d), 5^d|t(d)-1:
  (1,<1>,<1>,<1>,25001,375001,4375001,34375001,...)
(x)   t(d) + 5x10^{d-1} for d >= 4:
  (-,-,-,5001,75001,875001,9375001,84375001,...)
For d > 4, the sequence A301912 has at most 10 and at least 5 terms with d digits. The maximum is first attained for d=7. The minimum is first attained for d=168. (End)

			The sum of the first five cubes is 225, which ends in 5, so 5 is in the sequence.

Vaclav Kotesovec, Table of n, a(n) for n = 1..61
Robert Dawson, On Some Sequences Related to Sums of Powers, J. Int. Seq., Vol. 21 (2018), Article 18.7.6.
Wikipedia, Hensel's lemma

Cf. A000537, A199685.

seq = {}; Do[If[StringTake[ToString[k^2*(k+1)^2/4], -StringLength[ToString[k]]] == ToString[k], seq = Join[seq, {k}]], {k, 0, 1000000}]; seq (* Vaclav Kotesovec, Mar 29 2018 *)

A301912_list, k, n = [], 1, 1
while len(A301912_list) < 100:
    if n % 10**(len(str(k))) == k:
        A301912_list.append(k)
    k += 1
    n += k**3 # Chai Wah Wu, Mar 30 2018

Corrected and extended by Vaclav Kotesovec, Mar 29 2018

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User: Robert Dawson

A301912 Numbers k such that the decimal representation of k ends that of the sum of the first k cubes.

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