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Numbers k such that the decimal representation of k ends that of the sum of the first k cubes.
3

%I #58 Dec 05 2018 15:42:48

%S 0,1,5,25,76,376,500,625,876,1876,2500,5001,5625,9376,15625,25001,

%T 40625,50001,62500,65625,71876,75001,90625,109376,171876,265625,

%U 375001,390625,500001,765625,875001,890625,1171876,2265625,2890625,4062500,4375001,5000001

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

%C For j >= 3, 1 + 5*10^j = A199685(j) is in the sequence, so the sequence is infinite. - _Vaclav Kotesovec_, Mar 29 2018

%C From _Robert Dawson_, Apr 12 2018: (Start)

%C This sequence is the union of the following ten subsequences.

%C Terms in <angle brackets> 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).

%C (i) p(d) satisfying 2^d| p(d) - n_0(d), 5^d |p(d):

%C (0,<0>,500,2500,62500,62500,4062500,14062500,...)

%C (ii) q(d) satisfying 2^{d-1}|q(d)-1, 5^d|q(d) for d != 3:

%C (0,25,-,<625>,40625,390625,2890625,12890625,...)

%C (iii) q(d) + 5x10^{d-1} for d != 2:

%C (5,-, 625,5625,90625, 890625,7890625, 62890625,...)

%C (iv) q'(d) satisfying 2^{d-1}|q'(d) - n_1(d), 5^d|q'(d), for d != 1,3:

%C (-,25,-,<625>,15625,265625,2265625,47265625,...)

%C (v) q'(d) + 5x10^{d-1} for d != 2:

%C (5,-,625,5625,65625,765625,7265625,97265625,...)

%C (vi) r(d) satisfying 2^d|r(d), 5^d|r(d)-1 for d >= 2

%C (-,76,376,9376,<9376>,109376,7109376,87109376,...) = A016090(d)

%C (vii) r'(d) satisfying 2^d|r'(d) - n_0(d), 5^d|r'(d)-1 for d >= 2:

%C (-,76,876,1876,71876,171876,1171876,<1171876>,...)

%C (viii)s(d) := 5x10^{d-1}+1 for d >= 4:

%C (-,-,-,5001,50001,500001,5000001,50000001,...) = A199685(d-1)

%C (ix) t(d) satisfying 2^{d-1}|t(d)-n_0(d), 5^d|t(d)-1:

%C (1,<1>,<1>,<1>,25001,375001,4375001,34375001,...)

%C (x) t(d) + 5x10^{d-1} for d >= 4:

%C (-,-,-,5001,75001,875001,9375001,84375001,...)

%C 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.

%C (End.)

%H Vaclav Kotesovec, <a href="/A301912/b301912.txt">Table of n, a(n) for n = 1..61</a>

%H Robert Dawson, <a href="https://www.emis.de/journals/JIS/VOL21/Dawson/dawson6.html">On Some Sequences Related to Sums of Powers</a>, J. Int. Seq., Vol. 21 (2018), Article 18.7.6.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hensel%27s_lemma">Hensel's lemma</a>

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

%t 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 *)

%o (Python)

%o A301912_list, k, n = [], 1, 1

%o while len(A301912_list) < 100:

%o if n % 10**(len(str(k))) == k:

%o A301912_list.append(k)

%o k += 1

%o n += k**3 # _Chai Wah Wu_, Mar 30 2018

%Y Cf. A000537, A199685.

%K nonn,base

%O 1,3

%A _Robert Dawson_, Mar 28 2018

%E Corrected and extended by _Vaclav Kotesovec_, Mar 29 2018