A131625 -id:A131625 - 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

A294088 Least prime p_k such that (p_k)^n has p_{k-1} as substring.

Original entry on oeis.org

3701, 3, 43, 3, 3, 3, 5, 5, 7, 11, 11, 3, 3, 5, 3, 3, 3, 3, 5, 3, 5, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 5, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 2

Paolo P. Lava, Feb 09 2018

It appears that a(n) = 3 for n>59. In other words, for n>59, 2 is always a substring of 3^n. Is there any proof? See A131625.

			3701^2 = 13697401 and 3697 is the prime before 3701.
3^3 = 27 and 2 is the prime before 3.
43^4 = 3418801 and 41 is the prime before 43.

Cf. A052073, A052075, A131625, A294087.

P:=proc(q) local a,b,h,k,n,ok; for h from 2 to q do ok:=1; for n from 1 to q do
if ok=1 then a:=ithprime(n); b:=prevprime(a); for k from 1 to ilog10(a^h)-ilog10(b)+1 do
if b=trunc(a^h/10^(k-1)) mod 10^(ilog10(b)+1) then print(a); ok:=0; break;
fi; od; fi; od; od; end: P(10^6);

cp's OEIS Frontend

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

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