A110581 -id:A110581 - cp's OEIS frontend

A217199 Odd primes p such that 2p-1 is prime and no p is equal to 2q-1 with q in the sequence.

3, 7, 19, 31, 79, 97, 139, 199, 211, 229, 271, 307, 331, 337, 367, 379, 439, 499, 547, 577, 601, 607, 619, 691, 727, 811, 829, 937, 967, 1009, 1069, 1171, 1279, 1297, 1399, 1429, 1459, 1531, 1609, 1627, 1759, 1867, 2011, 2029, 2089, 2131, 2179, 2221, 2281

Offset: 1

Michel Marcus, Sep 27 2012

nonn

At each step, the smallest possible p is chosen.
These are the primes described in lemma 2 of the paper by Holt. - T. D. Noe, Sep 28 2012
This sequence was used by Holt (2003) to prove that there are at least two solutions k to phi(n+k) = phi(k) for all even n <= 1.38*10^26595411. - Amiram Eldar, Mar 19 2021

Michel Marcus, Table of n, a(n) for n = 1..1000
Jeffery J. Holt, The minimal number of solutions to phi(n)=phi(n+k), Math. Comp., 72 (2003), 2059-2061.
A. Schinzel and Andrzej Wakulicz, Sur l'équation phi(x+k)=phi(x), I., Acta Arith. 4 (1958), 181-184.

Cf. A007015, A110581, A217198, A342701.

t = {}; p = 2; Do[p = NextPrime[p]; If[PrimeQ[2*p - 1] && ! MemberQ[2*t - 1, p], AppendTo[t, p]], {PrimePi[2281]}]; t

intab(val, tab) = {for (ii=1, length(tab),if (tab[ii] == val, return (1);););return(0);}
lista(nn) = {tab = []; for (i=1, nn, len = length(tab); if (len == 0, p = 3, p = nextprime(tab[len]+1)); while (! isprime(2*p-1) || intab((p+1)/2, tab) , p = nextprime(p+1);); tab = concat(tab, p); print1(p, ", "););}

A217198 A sequence of odd primes p such that 2p-1 is prime and no p is equal to any 2q-1 with q in the sequence.

Original entry on oeis.org

3, 7, 31, 37, 97, 139, 157, 199, 211, 229, 271, 307, 331, 337, 367, 379, 439, 499, 547, 577, 601, 607, 619, 691, 727, 811

Offset: 1

Michel Marcus, Sep 27 2012

nonn

Sequence used in conjunction with A001259 in the referenced articles.

It is not clear that this sequence is correct -- it certainly does not match the title. See A217199 for the correct version. - T. D. Noe, Sep 28 2012

A. Schinzel and Andrzej Wakulicz, Sur l'équation phi(x+k)=phi(x), I., Acta Arith. 4 (1958), 181-184.
A. Schinzel and Andrzej Wakulicz, Sur l'équation phi(x+k)=phi(x). II. Acta Arith. 5 1959 425-426.

Cf. A001259, A110581, A217199.

A379144 a(n) is the number of iterations of the function x --> 2*x - 1 such that x remains prime, starting from A005382(n).

Original entry on oeis.org

2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1

Offset: 1

Ctibor O. Zizka, Dec 16 2024

nonn

Cunningham chain of the second kind of length i is a sequence of prime numbers (p_1, ..., p_i) such that p_(r + 1) = 2*p_r - 1 for all 1 =< r < i. This sequence tells the length of the Cunningham chain of the second kind for primes from A005382.

			n = 1: A005382(1) = 2 --> 3 --> 5 --> 9, 9 is not a prime, thus a(1) = 2.
n = 3: A005382(3) = 7 --> 13 --> 25, 25 is not a prime, thus a(3) = 1.

Cf. A000040, A005382, A005383, A057326, A064812, A110581, A307390.

s[n_] := -2 + Length[NestWhileList[2*# - 1 &, n, PrimeQ[#] &]]; Select[Array[s, 5000], # > 0 &] (* Amiram Eldar, Dec 16 2024 *)

a(A110581(n)) = 1.

a(A057326(n)) = 2.

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A217199 Odd primes p such that 2p-1 is prime and no p is equal to 2q-1 with q in the sequence.

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A217198 A sequence of odd primes p such that 2p-1 is prime and no p is equal to any 2q-1 with q in the sequence.

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A379144 a(n) is the number of iterations of the function x --> 2*x - 1 such that x remains prime, starting from A005382(n).

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