OFFSET
1,3
COMMENTS
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 <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).
(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.)
LINKS
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
EXAMPLE
The sum of the first five cubes is 225, which ends in 5, so 5 is in the sequence.
MATHEMATICA
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 *)
PROG
(Python)
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
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Robert Dawson, Mar 28 2018
EXTENSIONS
Corrected and extended by Vaclav Kotesovec, Mar 29 2018
STATUS
approved