Lorenzo Sillari - cp's OEIS frontend

A358719 A sequence of primes starting with p_1 = 2, p_2 = 3, p_3 = 5, p_4 = 11, p_5 = 13, p_6 = 23, such that, for i >= 7, (p_i + 1)/2 divides the product p_1p_2...p_(i-1) of the earlier primes and each prime factor of (p_i-1)/2 is a prime factor of the product p_1p_2...p_(i-1).

2, 3, 5, 11, 13, 23, 19, 37, 73, 109, 131, 229, 457, 571, 1459, 1481, 2179, 2621, 2917, 2963, 4357, 8713, 49921, 1318901, 3391489, 6782977, 13565953

Offset: 1

Lorenzo Sillari, Nov 28 2022

The sequence was used, together with A358717 and A358718, by Ferrari and Sillari (Preprint-2022) to prove that there are at least three solutions n to phi(n+k) = 2*phi(n) for all even k <= 4*10^58.

Prime a(28) does not exist, which can be established by going over the divisors d of the product a(1)*...*a(27) and testing 2*d-1 as a candidate for a(28). - Max Alekseyev, Feb 19 2024

M. Ferrari and L. Sillari, On the minimal number of solutions of the equation phi(n+k) = M*phi(n), M=1,2, arXiv:2110.05401 [math.NT], 2021.

Similar to A001259.
The sequence is a slight modification of A358717.
Cf. A358718.

s = {2, 3, 5, 11, 13, 23}; step[s_] := Module[{p = 7, r = Times @@ s}, While[MemberQ[s, p] || ! Divisible[r, (p + 1)/2] || ! Divisible[r, Times @@ FactorInteger[(p - 1)/2][[;; , 1]]], p = NextPrime[p]]; Join[s, {p}]]; Nest[step, s, 21] (* Amiram Eldar, Dec 01 2022 *)

Keywords 'full' and 'fini' added by Max Alekseyev, Feb 19 2024

A358718 A sequence of sorted primes p_1 = 2, p_2 = 3, p_3 = 5, p_4 =7, p_5 < ... < p_m such that, for i >= 5, (p_i + 1)/2 divides the product p_1p_2...p_(i-1) of the earlier primes and each prime factor of (p_i-1)/2 is a prime factor of the product p_1p_2...p_(i-1).

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 19, 29, 37, 41, 43, 59, 73, 83, 109, 113, 131, 163, 173, 181, 227, 257, 331, 347, 353, 379, 419, 491, 523, 571, 601, 653, 661, 677, 739, 757, 769, 811, 859, 1091, 1201, 1217, 1297, 1307, 1321, 1459, 1481, 1621, 1721, 2029, 2081, 2089, 2179

Offset: 1

Lorenzo Sillari, Nov 28 2022

nonn

The sequence was used, together with A358717 and A358719, by Ferrari and Sillari (Preprint-2022) to prove that there are at least three solutions n to phi(n+k) = 2*phi(n) for all even k <= 4*10^58.
Similar to A001259.
The sequence is a slight modification of A358717.

Max Alekseyev, Table of n, a(n) for n = 1..1000
M. Ferrari and L. Sillari, On the minimal number of solutions of the equation phi(n+k) = M*phi(n), M=1,2, arXiv:2110.05401 [math.NT], 2021.

Similar to A001259. See also A358717 and A358719.

s = {2, 3, 5, 7}; step[s_] := Module[{p = NextPrime[s[[-1]]], r = Times @@ s}, While[! Divisible[r, (p + 1)/2] || ! Divisible[r, Times @@ FactorInteger[(p - 1)/2][[;; , 1]]], p = NextPrime[p]]; Join[s, {p}]]; Nest[step, s, 55] (* Amiram Eldar, Dec 01 2022 *)

A358717 A sequence of sorted primes 2 = p_1 < p_2 < ... < p_m such that (p_i + 1)/2 divides the product p_1p_2...*p_(i-1) of the earlier primes and each prime factor of (p_i-1)/2 is a prime factor of the product.

Original entry on oeis.org

2, 3, 5, 11, 19, 37, 73, 109, 1459, 2179, 2917, 4357, 8713

Offset: 1

Lorenzo Sillari, Nov 28 2022

The sequence was used, together with A358718 and A358719, by Ferrari and Sillari (Preprint-2022) to prove that there are at least three solutions n to phi(n+k) = 2* phi(n) for all even k <= 4*10^58.
I have checked up to 10^8 and found no more terms.
Prime a(14) does not exist, which can be established by going over the divisors d of the product a(1)*...*a(13) and testing 2*d-1 as a candidate for a(14). - Max Alekseyev, Feb 19 2024

M. Ferrari and L. Sillari, On the minimal number of solutions of the equation phi(n+k) = M*phi(n), M=1,2, arXiv:2110.05401 [math.NT], 2021.

Similar to A001259.

cp's OEIS Frontend

User: Lorenzo Sillari

A358719 A sequence of primes starting with p_1 = 2, p_2 = 3, p_3 = 5, p_4 = 11, p_5 = 13, p_6 = 23, such that, for i >= 7, (p_i + 1)/2 divides the product p_1p_2...p_(i-1) of the earlier primes and each prime factor of (p_i-1)/2 is a prime factor of the product p_1p_2...p_(i-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A358718 A sequence of sorted primes p_1 = 2, p_2 = 3, p_3 = 5, p_4 =7, p_5 < ... < p_m such that, for i >= 5, (p_i + 1)/2 divides the product p_1p_2...p_(i-1) of the earlier primes and each prime factor of (p_i-1)/2 is a prime factor of the product p_1p_2...p_(i-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A358717 A sequence of sorted primes 2 = p_1 < p_2 < ... < p_m such that (p_i + 1)/2 divides the product p_1p_2...*p_(i-1) of the earlier primes and each prime factor of (p_i-1)/2 is a prime factor of the product.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

cp's OEIS Frontend

User: Lorenzo Sillari

A358719 A sequence of primes starting with p_1 = 2, p_2 = 3, p_3 = 5, p_4 = 11, p_5 = 13, p_6 = 23, such that, for i >= 7, (p_i + 1)/2 divides the product p_1*p_2*...*p_(i-1) of the earlier primes and each prime factor of (p_i-1)/2 is a prime factor of the product p_1*p_2*...*p_(i-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A358718 A sequence of sorted primes p_1 = 2, p_2 = 3, p_3 = 5, p_4 =7, p_5 < ... < p_m such that, for i >= 5, (p_i + 1)/2 divides the product p_1*p_2*...*p_(i-1) of the earlier primes and each prime factor of (p_i-1)/2 is a prime factor of the product p_1*p_2*...*p_(i-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A358717 A sequence of sorted primes 2 = p_1 < p_2 < ... < p_m such that (p_i + 1)/2 divides the product p_1*p_2*...*p_(i-1) of the earlier primes and each prime factor of (p_i-1)/2 is a prime factor of the product.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A358719 A sequence of primes starting with p_1 = 2, p_2 = 3, p_3 = 5, p_4 = 11, p_5 = 13, p_6 = 23, such that, for i >= 7, (p_i + 1)/2 divides the product p_1p_2...p_(i-1) of the earlier primes and each prime factor of (p_i-1)/2 is a prime factor of the product p_1p_2...p_(i-1).

A358718 A sequence of sorted primes p_1 = 2, p_2 = 3, p_3 = 5, p_4 =7, p_5 < ... < p_m such that, for i >= 5, (p_i + 1)/2 divides the product p_1p_2...p_(i-1) of the earlier primes and each prime factor of (p_i-1)/2 is a prime factor of the product p_1p_2...p_(i-1).

A358717 A sequence of sorted primes 2 = p_1 < p_2 < ... < p_m such that (p_i + 1)/2 divides the product p_1p_2...*p_(i-1) of the earlier primes and each prime factor of (p_i-1)/2 is a prime factor of the product.