A252491 -id:A252491 - cp's OEIS frontend

A138940 Indices n such that A019328(n) = Phi(n,10) is prime, where Phi is a cyclotomic polynomial.

2, 4, 10, 12, 14, 19, 23, 24, 36, 38, 39, 48, 62, 93, 106, 120, 134, 150, 196, 317, 320, 385, 586, 597, 654, 738, 945, 1031, 1172, 1282, 1404, 1426, 1452, 1521, 1752, 1812, 1836, 1844, 1862, 2134, 2232, 2264, 2667, 3750, 3903, 3927, 4274, 4354, 5877, 6022

Offset: 1

M. F. Hasler, Apr 03 2008

nonn

Unique period primes (A040017) are often of the form Phi(k,10) or Phi(k,-10).

Terms of this sequence which are the square of a prime, a(n)=p^2, are such that A252491(p) is prime. Apart from a(2)=2^2, there is no such term up to 26570. - M. F. Hasler, Jan 09 2015

Ray Chandler, Table of n, a(n) for n = 1..102 (first 50 terms from Robert Price, terms 92-93 from Serge Batalov, others from Kamada link)
Chris Caldwell, Unique Primes.
Makoto Kamada, Factorizations of Phi_n(10) (including prime members up to 200000).
Index entries for cyclotomic polynomials, values at X

Cf. A019328, A040017, A085035, A252491.

Cf. Subsequence of A007498, contains A004023.

Select[Range[1000], PrimeQ[Cyclotomic[#, 10]] &] (* T. D. Noe, Mar 03 2012 *)

for( i=1,999, isprime( polcyclo(i,10)) && print1( i","))

a(28)-a(43) from Robert Price, Mar 03 2012
a(44)-a(50) from Robert Price, Apr 14 2012
a(51)-a(91) from Ray Chandler, Maksym Voznyy et al. (cf. Phi_n(10) link), ca. 2009
a(92)-a(93) from Serge Batalov, Mar 28 2015

A333237 Numbers k such that 1/k contains at least one '9' in its decimal expansion.

Original entry on oeis.org

11, 13, 17, 19, 21, 23, 29, 31, 34, 38, 41, 42, 43, 46, 47, 49, 51, 52, 53, 57, 58, 59, 61, 62, 67, 68, 69, 71, 73, 76, 77, 81, 82, 83, 84, 85, 86, 87, 89, 91, 92, 94, 95, 97, 98, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 118

Offset: 1

Andrew Slattery, Mar 12 2020

Almost every prime appears in this sequence.
Among the first 10000 primes, only 2, 3, 5, 7, 37, 79, 239, 4649, and 62003 do not appear in the sequence. - Giovanni Resta, Mar 13 2020
The next primes not in the sequence are 538987, 35121409, and 265371653. - Robert Israel, Mar 18 2020

			5 is not in the sequence because 1/5 = 0.2 does not contain any 9s.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to decimal expansion of 1/n

Cf. A333236.
Subsequences (for terms > 1): A000533, A002275, A135577, A252491.
Cf. A216664 (a subsequence).
Cf. A187614.

f:= proc(n) local m,S,r;
   m:= 1; S:= {1};
   do
     r:= floor(m/n);
     if r = 9 then return true fi;
     m:= (m - r*n)*10;
     if member(m,S) then return false fi;
     S:= S union {m};
   od
end proc:
select(f, [$1..1000]); # Robert Israel, Mar 18 2020

Select[Range[120], MemberQ[ Flatten@ RealDigits[1/#][[1]], 9] &] (* Giovanni Resta, Mar 12 2020 *)

from itertools import count, islice
from sympy import n_order, multiplicity
def A333237_gen(startvalue=1): # generator of terms
    for m in count(max(startvalue,1)):
        m2, m5 = multiplicity(2,m), multiplicity(5,m)
        if max(str(10**(max(m2,m5)+n_order(10,m//2**m2//5**m5))//m)) == '9':
            yield m
A333237_list = list(islice(A333237_gen(), 10)) # Chai Wah Wu, Feb 07 2022

A333236(a(n)) = 9.

More terms from Giovanni Resta, Mar 12 2020

A087403 a(n) = smallest prime of the form 10*K(n) + 1, where K is a number obtained by concatenation of n with itself, or 0 if no such prime exists.

Original entry on oeis.org

11, 2221, 31, 41, 555555555551, 61, 71, 881, 991, 101, 1111111111111111111, 1212121, 131, 14141414141, 151, 1616161, 1717171717171717171717171717171, 181, 191, 20201, 211

Offset: 1

Amarnath Murthy, Sep 10 2003

Conjecture: No term is zero.
Next term a(22) is too large (121 digits) to include in sequence. - Ray Chandler, Sep 23 2003
From Farideh Firoozbakht, Jan 07 2015: (Start)
The conjecture is not true. There exist many numbers n such that a(n)=0.
By using the theorem and its corollary mentioned in the comments lines of the sequence A086766, we can prove that for m = 2, 3, ..., 275 a(10^m)=0.
What is the smallest odd prime p, such that (10^(p^2)-1)/(10^p-1) is a prime number (a(10^(p-1)) is nonzero)?
What is the smallest integer m, such that m > 1 and a(10^m) is nonzero?
Conjecture: If n is not of the form 10^m then a(n) is nonzero.
(End)

			a(2) = 2221 is a prime but 21 and 221 are composite.

Cf. A086766, A252491.

cp's OEIS Frontend

A138940 Indices n such that A019328(n) = Phi(n,10) is prime, where Phi is a cyclotomic polynomial.

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A333237 Numbers k such that 1/k contains at least one '9' in its decimal expansion.

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A087403 a(n) = smallest prime of the form 10*K(n) + 1, where K is a number obtained by concatenation of n with itself, or 0 if no such prime exists.

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