

A347164


Positive integers k such that the decimal representation of 2^k ends with some permutation of the string "0123456789".


2



7386, 11061, 15176, 16054, 19950, 24493, 26749, 29160, 33902, 42207, 43013, 44233, 45627, 52235, 54727, 56186, 59228, 59229, 59230, 60883, 62823, 63468, 65404, 69960, 71225, 71804, 75176, 78392, 89416, 96576, 96682, 97723, 98085, 98561, 102735, 104125, 105301, 105760
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OFFSET

1,1


COMMENTS

If k is a term of the sequence then some nonzero digit must occur more than once in the decimal representation of 2^k because 1+2+3+4+5+6+7+8+9=45, and 2^k is not divisible by 9. Thus 2^k>10^10 and therefore k>33 for any term k.
A positive integer k is a term of the sequence iff a decimal representation of the remainder of 2^k modulo 10^10 (possibly containing a leading zero) is a permutation of the string "0123456789".
Let d = 7812500 = 4*5^9 = phi(5^10) where phi is Euler's totient function. The remainders of the powers of 2 modulo 10^10 form an eventually periodic sequence with period d: if k >= 10 then 2^(k+d)  2^k is divisible by 10^10 since 2^(k+d)  2^k = 2^k*(2^d1) and 10^10 = 2^10*5^10. Hence if k >= 10 then k + d is a term iff k is a term.
Actually the above equivalence holds for all positive integers k because neither k nor k + d is a term of the sequence for k < 10 (the decimal representations of the numbers 2^(k + d) with k = 1, 2, ..., 9 end, respectively, with the following strings: 3574218752, 7148437504, 4296875008, 8593750016, 7187500032, 4375000064, 8750000128, 7500000256, 5000000512).
There are 2795 terms not exceeding d. The last of them is 7808304, with decimal representation of the corresponding power of 2 ending with 9745238016.


LINKS



FORMULA

a(n+c) = a(n) + d with c=2795 and d as above.


EXAMPLE

7386, 11061 and 15176 are in the sequence because the decimal representations of the corresponding powers of 2 end with 9815307264, 4706813952 and 0294875136, respectively.


MAPLE

q:= n> (l> is({l[], `if`(nops(l)<10, 0, [][])}=
{$0..9}))(convert(2&^n mod 10^10, base, 10)):


MATHEMATICA

Select[Range[10^5], Union[If[Length[s=IntegerDigits@PowerMod[2, #, 10^10]]==9, Join[{0}, s], s]]==0~Range~9&] (* Giorgos Kalogeropoulos, Sep 03 2021 *)


PROG

(Python)
k, r, n=1, 2, 1
while n<=6000:
s, t=set(), r
for i in range(10):
s.add(t%10)
t=t//10
if len(s)==10:
print(n, k)
n=n+1
k, r=k+1, 2*r%10**10
(PARI) isok(k) = my(d=digits(lift(Mod(2, 10^10)^k))); if (#d<10, d = concat(d, 0)); #Set(d) == 10; \\ Michel Marcus, Oct 01 2021


CROSSREFS



KEYWORD

nonn,base


AUTHOR



STATUS

approved



