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

Author

Scott R. Shannon, Apr 21 2021

Keywords

• nonn
• base

Examples

			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.
		

Crossrefs

Cf. A049329, A045537, A051248.

Programs

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

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