This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a(1) = A045537(0) = 2. a(2) = A045537(2) = 5. a(3) = A045537(11) = 11. a(4) = A045537(12) = 14. a(5) = A045537(15) = 26. a(6) = A045537(102) = 28.
2^8 = 256 begins with 2, and 2^k does not begin with 2 for any smaller integer k > 1, so a(2) = 8.
import Data.List (isPrefixOf) a051248 n = 2 + length (takeWhile (not . (show n `isPrefixOf`) . show) $ iterate (* n) (n^2)) -- Reinhard Zumkeller, Sep 29 2011
f[n_]:=Module[{i=2},While[Take[IntegerDigits[n^i],IntegerLength[n]] != IntegerDigits[n],i++];i]; Array[f,70] (* Harvey P. Dale, Oct 04 2011 *)
from itertools import count
def a(n):
s = str(n)
return next(e for e in count(2) if str(n**e).startswith(s))
print([a(n) for n in range(1, 33)]) # Michael S. Branicky, Feb 23 2025
from itertools import count
def a(n):
s = str(n)
return next(n**e for e in count(2) if s in str(n**e))
print([a(n) for n in range(1, 22)]) # Michael S. Branicky, Feb 23 2025
from itertools import count
def a(n):
s = str(n)
return next(n**e for e in count(2) if str(n**e).startswith(s))
print([a(n) for n in range(2, 16)]) # Michael S. Branicky, Feb 23 2025
def a(n):
s = str(n)
return next(k for k in range(2, 11) if s in str(n*k))
print([a(n) for n in range(71)]) # Michael S. Branicky, Feb 23 2025
a(12) = 21 because 12^21 is the smallest power (>1) of 12 that ends in 12 (that, is 12 is a 21-morphic number); a(14) = -1 because there is no power (>1) of 14 that ends in 14 (that is, 14 cannot be any p-morphic number).
SelectFirst[Range[2, 120], Function[k, Mod[#^k, 10^IntegerLength@ #] == #]] & /@ Range@ 200 /. n_ /; MissingQ@ n -> -1 (* Michael De Vlieger, Dec 02 2015, Version 10 *)
For n=12, a(12)=3 because 12^3=1728 contains all decimal digits of n. Compare to A253600(12)=2 because 12^2=144 contains any digit of n.
seq={}; Do[k=1;Until[ContainsAll[IntegerDigits[n^k],IntegerDigits[n] ],k++];AppendTo[seq,k] ,{n,0,80}];seq
a(n) = my(k=2, d=Set(digits(n))); while(setintersect(Set(digits(n^k)), d) != d, k++); k; \\ Michel Marcus, Jun 01 2024
from itertools import count
def a(n):
s = set(str(n))
return next(k for k in count(2) if s <= set(str(n**k)))
print([a(n) for n in range(81)]) # Michael S. Branicky, May 27 2024
n=6: 6='110' is not contained in 6^2='100100', but in 6^3='11011000', therefore a(6)=3.
5^1 = 5 does not contain a 1 but 5^2 = 25 does contain a 2 so a(5) = 2. 7^1 = 7 does not contain a 1, 7^2 = 49 does not contain a 2, but 7^3 = 343 does contain a 3 so a(7) = 3.
f:= proc(n) local k;
for k from 1 to 9 do
if member(k,convert(n^k,base,10)) then return k fi
od;
FAIL
end proc:
map(f, [$1..100]); # Robert Israel, Sep 16 2024
a[n_] := Block[{k=1}, While[{} == StringPosition[ ToString[n^k], ToString[k]], k++]; k]; Array[a, 84] (* Giovanni Resta, Mar 11 2014 *)
sk[n_]:=Module[{k=1},While[SequenceCount[IntegerDigits[n^k],IntegerDigits[k]] == 0,k++];k]; Array[sk,90] (* Harvey P. Dale, May 12 2022 *)
def Sub(x):
for n in range(10**3):
if str(x**n).find(str(n)) > -1:
return n
x = 1
while x < 10**3:
print(Sub(x))
x += 1
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