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A343594
Numbers k that, when written in all bases from base 2 to base 10, are a substring of k^k when written in the same base.
0
1, 5, 17, 25, 31, 41, 63, 92, 151, 170, 202, 221, 263, 266, 278, 322, 327, 347, 364, 401, 404, 412, 421, 423, 437, 467, 470, 482, 490, 498, 501, 515, 519, 543, 558, 578, 590, 612, 623, 636, 646, 647, 671, 683, 685, 705, 707, 717, 718, 726, 764, 785, 795, 859, 867, 872, 875, 881, 890, 892, 897
OFFSET
1,2
EXAMPLE
5 is a term. See below table:
.
base | 5 in base | 5^5 in base
---------+-------------+-------------
10 5 3125
9 5 4252
8 5 6065
7 5 12053
6 5 22245
5 10 100000
4 11 300311
3 12 11021202
2 101 110000110101
.
5^5 in all bases contains 5 in that base as a substring.
PROG
(Python)
from sympy.ntheory import digits
def nstr(n, b): return "".join(map(str, digits(n, b=b)[1:]))
def ok(k): return all(nstr(k, b) in nstr(k**k, b) for b in range(10, 1, -1))
print(list(filter(ok, range(900)))) # Michael S. Branicky, Apr 25 2021
(PARI) str(v) = my(s=""); for (k=1, #v, s = concat(s, Str(v[k]))); s;
isok(k) = {for (b=2, 10, my(kb = digits(k, b), kkb = digits(k^k, b)); if (#strsplit(str(kkb), str(kb)) <=1 , return (0)); ); return (1); } \\ Michel Marcus, Apr 26 2021
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Scott R. Shannon, Apr 21 2021
STATUS
approved