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.

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

Links

Table of n, a(n) for n=1..61.

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

Cf. A049329, A045537, A051248.

Sequence in context: A106012 A230527 A242178 * A063588 A307286 A339955

Adjacent sequences: A343591 A343592 A343593 * A343595 A343596 A343597

Keyword

nonn,base

Author

Scott R. Shannon, Apr 21 2021

Status

approved