A001259 -id:A001259 - cp's OEIS frontend

A342701 a(n) is the second smallest k such that phi(n+k) = phi(k), or 0 if no such solution exists.

3, 7, 5, 14, 9, 34, 7, 16, 15, 26, 11, 68, 39, 28, 15, 32, 33, 72, 25, 40, 35, 56, 17, 101, 45, 37, 45, 56, 29, 152, 31, 61, 39, 56, 35, 144, 37, 61, 39, 74, 41, 128, 35, 88, 45, 161, 47, 192, 49, 82, 51, 74, 95, 216, 43, 97, 75, 203, 59, 304, 91, 88, 63, 122

Offset: 1

Amiram Eldar, Mar 18 2021

nonn

Sierpiński (1956) proved that there is at least one solution for all n>=1.
Schinzel (1958) proved that there are at least two solutions k to phi(n+k) = phi(k) for all n <= 8*10^47. Schinzel and Wakulicz (1959) increased this bound to 2*10^58.
Schinzel (1958) observed that under the prime k-tuple conjecture there is a second solution for all even n.
Holt (2003) proved that there is a second solution for all even n <= 1.38 * 10^26595411.

			a(1) = 3 since the solutions to the equation phi(1+k) = phi(k) are k = 1, 3, 15, 104, 164, ... (A001274), and 3 is the second solution.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, section B36, page 138-142.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 3, p. 217-219.
Wacław Sierpiński, Sur une propriété de la fonction phi(n), Publ. Math. Debrecen, Vol. 4 (1956), pp. 184-185.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jeffery J. Holt, The minimal number of solutions to phi(n)=phi(n+k), Math. Comp., Vol. 72, No. 244 (2003), pp. 2059-2061.
Andrzej Schinzel, Sur l'équation phi(x + k) = phi(x), Acta Arith., Vol. 4, No. 3 (1958), pp. 181-184.
Andrzej Schinzel and Andrzej Wakulicz, Sur l'équation phi(x+k)=phi(x). II, Acta Arith., Vol. 5, No. 4 (1959), pp. 425-426.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture.

Cf. A000010, A001259, A001274, A007015, A217199.

f[n_, 0] = 0; f[n_, k0_] := Module[{k = f[n, k0 - 1] + 1}, While[EulerPhi[n + k] != EulerPhi[k], k++]; k]; Array[f[#, 2] &, 100]

a(n) = my(k=1, nb=0); while ((nb += (eulerphi(n+k)==eulerphi(k))) != 2, k++); k; \\ Michel Marcus, Mar 19 2021

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.

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).

Original entry on oeis.org

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

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.

cp's OEIS Frontend

A342701 a(n) is the second smallest k such that phi(n+k) = phi(k), or 0 if no such solution exists.

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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).

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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.

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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).

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Programs

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Extensions

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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Author

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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.

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).