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

Author

Dimiter Skordev, Aug 20 2021

Keywords

• nonn
• base

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A090493, A291926. Subsequence of A130694.

Programs

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 *)

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

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

Formula

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