A136291 -id:A136291 - cp's OEIS frontend

A034293 Numbers k such that 2^k does not contain the digit 2 (probably finite).

0, 2, 3, 4, 6, 12, 14, 16, 20, 22, 23, 26, 34, 35, 36, 39, 42, 46, 54, 64, 74, 83, 168

Offset: 1

Is 168 the last term?
First row of A136291. - R. J. Mathar Apr 29 2008
Equivalently, indices of zeros in A065710. - M. F. Hasler, Feb 10 2023

			Here is 2^168, conjecturally the largest power of 2 that does not contain a 2: 374144419156711147060143317175368453031918731001856. - _N. J. A. Sloane_, Feb 10 2023

Cf. A007377.
See also similar sequences listed in A035064.
Cf. A065710 (number of '2's in 2^n), A094776.

[n: n in [0..1000] | not 2 in Intseq(2^n) ]; // Vincenzo Librandi, May 07 2015

isA034293 := proc(n) RETURN(not 2 in convert(2^n,base,10)) ; end: for n from 0 to 100000 do if isA034293(n) then print(n) ; fi ; od: # R. J. Mathar, Oct 04 2007

Join[{0}, Select[ Range@10000, FreeQ[ IntegerDigits[2^# ], 2] &]] (* Shyam Sunder Gupta, Sep 01 2007 *)(* adapted by Vincenzo Librandi, May 07 2015 *)
Select[Range[0, 10^4], DigitCount[2^#][[2]] == 0 &] (* Michael De Vlieger, Apr 29 2016 *)

is(n)=setsearch(Set(digits(2^n)),2)==0 \\ Charles R Greathouse IV, May 10 2016

is_A034293(n)=!foreach(digits(2^n),d,d==2&&return) \\ M. F. Hasler, Feb 10 2023

def is_A034293(n): return'2'not in str(2**n)
[n for n in range(199) if is_A034293(n)] # M. F. Hasler, Feb 10 2023

The last term is A094776(2), by definition. - M. F. Hasler, Feb 10 2023

Edited by N. J. A. Sloane, Oct 03 2007

Removed keyword "fini" since it is only a conjecture that this sequence contains only finitely many terms. - Altug Alkan, May 07 2016

A131629 Numbers k such that the decimal expansion of 3^k contains no 3.

Original entry on oeis.org

0, 2, 3, 4, 6, 7, 8, 10, 11, 14, 19, 27, 28, 34, 40, 50, 55, 84

Offset: 1

Shyam Sunder Gupta, Sep 01 2007

I conjecture that 84 is the last term.

Cf. similar sequences listed in A131613.

Cf. A007377, A136291.

[n: n in [0..1000] | not 3 in Intseq(3^n) ]; // Vincenzo Librandi, May 06 2015

Join[{0}, Select[ Range@10000, FreeQ[ IntegerDigits[3^# ], 3] &]]

Initial 0 added and Mathematica code adapted by Vincenzo Librandi, May 06 2015

A185186 Numbers divisible by at least one of their digits other than 1.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 12, 15, 20, 22, 24, 25, 26, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 50, 52, 55, 60, 62, 63, 64, 65, 66, 70, 72, 75, 77, 80, 82, 84, 85, 88, 90, 92, 93, 95, 96, 99, 102, 104, 105, 112, 115, 120, 122, 123, 124, 125, 126, 128, 132

Offset: 1

Zak Seidov, Mar 11 2011

The only primes in the sequence are 2, 3, 5, 7. No repunits are eligible.
Also, an interesting class of non-eligible integers consists of some powers of 2, 3 and 7:
"2, 4, 8-less" powers of 2, 2^m = 1, 16, 65536 with m = 0, 4, 16 (a subsequence of A034293);
"3, 9-less" powers of 3, 3^m = {1, 27, 81, 177147, 1162261467}, with m = {0, 3, 4, 11, 19} (a subsequence of A131629);
"seven-less" powers of 7, 7^m, with m = 0, 2, 3, 4, 7, 16, 22, 24, 39 (see 6th row of A136291 Array read by rows: each row is a sequence of numbers k such that n^k does not contain the digit n).
Asymptotic density 27/35 = 0.771... - Charles R Greathouse IV, Mar 11 2011
The asymptotic density of numbers having a prime digit is 1 for each prime digit. The asymptotic density of numbers being divisible by 2, 3, 5 or 7 is -Sum_{d|210, d>1}((-1)^omega(d) / d) = 27/35. Also, the asymptotic density of numbers divisible by the first n primes is r(n) where r(1) = 1/2 and r(n) = r(n - 1) + (1 - r(n - 1)) / prime(n). - David A. Corneth, May 28 2017

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A187398, A187516, A187238, A187533, A187534, A187551.

digDivQ[n_] := AnyTrue[IntegerDigits[n], # > 1 && Mod[n, #] == 0 &]; Select[Range[200], digDivQ] (* Giovanni Resta, May 27 2017 *)

is(n) = my(d = vecsort(digits(n), , 8), t = 1); while(t<=#d&&d[t] < 2, t++); sum(i=t, #d, n%d[i]==0) > 0 \\ David A. Corneth, May 27 2017

Name edited by Alonso del Arte, May 16 2017

cp's OEIS Frontend

A034293 Numbers k such that 2^k does not contain the digit 2 (probably finite).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Python

Formula

Extensions

A131629 Numbers k such that the decimal expansion of 3^k contains no 3.

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Mathematica

Extensions

A185186 Numbers divisible by at least one of their digits other than 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions