A347164 Positive integers k such that the decimal representation of 2^k ends with some permutation of the string "0123456789".

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

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^d-1) 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

Dimiter Skordev, Table of n, a(n) for n = 1..6000

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)):
select(q, [$1..120000])[];  # Alois P. Heinz, Aug 23 2021

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

Cf. A090493, A291926. Subsequence of A130694.

Sequence in context: A022255 A189504 A028540 * A206202 A364182 A034279

Adjacent sequences: A347161 A347162 A347163 * A347165 A347166 A347167

Keyword

nonn,base

Author

Dimiter Skordev, Aug 20 2021

Status

approved