A131629 -id:A131629 - cp's OEIS frontend

A131613 Numbers k such that the decimal expansion of 3^k contains no 9.

0, 1, 3, 4, 5, 7, 8, 11, 12, 16, 19, 20, 21, 29, 32, 56

Offset: 1

Shyam Sunder Gupta, Sep 01 2007

I conjecture that 56 is the last term.

Numbers k such that the decimal expansion of 3^k contains no m: A030700 (m=0), A131627 (m=1), A131625 (m=2), A131629 (m=3), A131618 (m=4), A131617 (m=5), A131616 (m=6), A131615 (m=7), A131614 (m=8), this sequence (m=9).

Cf. A007377.

[n: n in [0..1000] | not 9 in Intseq(3^n) ]; // Vincenzo Librandi, May 06 2015

Join[{0}, Select[ Range@10000, FreeQ[ IntegerDigits[3^# ], 9] &]]

Adapted Mma and initial 0 added by Vincenzo Librandi, May 06 2015

A136291 Array read by rows: each row is a sequence of numbers k such that n^k does not contain the digit n.

Original entry on oeis.org

0, 2, 3, 4, 6, 12, 14, 16, 20, 22, 23, 26, 34, 35, 36, 39, 42, 46, 54, 64, 74, 83, 168, 0, 2, 3, 4, 6, 7, 8, 10, 11, 14, 19, 27, 28, 34, 40, 50, 55, 84, 0, 2, 4, 8, 12, 20, 0, 0, 0, 2, 3, 4, 7, 16, 22, 24, 39, 0, 2, 3, 4, 6, 7, 8, 26, 0, 2, 4, 6, 8, 10, 16, 28

Offset: 0

Zak Seidov, Mar 20 2008

Last values in each row are 168,84,20,0,0,39,26,28 (all terms highly probably correct).

			Each row is sequence of numbers k such that n^k does not contain the digit n (all rows probably finite, checked up to k=10^4), n=2..9:
0,2,3,4,6,12,14,16,20,22,23,26,34,35,36,39,42,46,54,64,74,83,168 (A034293)
0,2,3,4,6,7,8,10,11,14,19,27,28,34,40,50,55,84 (A131629)
0,2,4,8,12,20
0,
0,
0,2,3,4,7,16,22,24,39
0,2,3,4,6,7,8,26
0,2,4,6,8,10,16,28.

Cf. A034293, A131629.

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

A378557 Powers of 3 that do not contain the digit 3.

Original entry on oeis.org

1, 9, 27, 81, 729, 2187, 6561, 59049, 177147, 4782969, 1162261467, 7625597484987, 22876792454961, 16677181699666569, 12157665459056928801, 717897987691852588770249, 174449211009120179071170507, 11972515182562019788602740026717047105681

Offset: 1

Erich Friedman, Nov 30 2024

Any additional terms have exponent at least 10^5.

Cf. A000244, A131629.

Select[3^Range[0,100000],Not[MemberQ[IntegerDigits[#],3]]&]

a(n) = 3^A131629(n).

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A131613 Numbers k such that the decimal expansion of 3^k contains no 9.

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A136291 Array read by rows: each row is a sequence of numbers k such that n^k does not contain the digit n.

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

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A378557 Powers of 3 that do not contain the digit 3.

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