cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A376338 Numbers k such that phi(k)/2 - 1 = phi(k + 1) where phi = A000010.

Original entry on oeis.org

11, 19, 43, 49, 67, 163, 211, 283, 331, 523, 547, 691, 787, 907, 1051, 1123, 1171, 1531, 1723, 1867, 2011, 2083, 2251, 2347, 2371, 2467, 2707, 2731, 2803, 2971, 3187, 3307, 3547, 3643, 3907, 3931, 4051, 4243, 4363, 4603, 4651, 4723, 5107, 5227, 5443, 5923
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Sep 20 2024

Keywords

Comments

Conjecture: this sequence is the union {49} and the primes of the form 4*p - 1 where p odd prime.

Examples

			Number 49 is in this sequence because phi(49)/2 - 1 = 42/2 - 1 = 21 - 1 = 20 is equal to phi(49 + 1) = phi(50) = 20.

Crossrefs

Cf. A000010, A162857, A376337.

Programs

  • Magma
    [k: k in [3..6000] | ((EulerPhi(k) div 2) - 1) eq EulerPhi(k + 1)];
  • Mathematica
    Select[Range[6000], EulerPhi[#]/2-1==EulerPhi[#+1] &] (* Stefano Spezia, Sep 22 2024 *)

A377142 Numbers m such that phi(2*m-1)/2 = phi(2*m) - 1, where phi = A000010.

Original entry on oeis.org

2, 4, 5, 16, 64, 4096, 65536, 262144
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Oct 19 2024

Keywords

Comments

Conjecture 1: each term has the form p^(q-1), where p, q both some primes.
Conjecture 2: sequence is infinite.
Presumably the sequence of numbers of the form (exponent of a(n)) + (smallest divisor of a(n)) is a supersequence of Mersenne exponents.
If 2*m-1 is a Mersenne prime (A000668), then phi(2*m-1)/2 = m-1 = phi(2*m) - 1, so m is a term. - Robert Israel, Oct 20 2024

Examples

			2 is a term because phi(2*2-1)/2 = phi(3)/2 = 2/2 = 1 is equal to phi(2*2)-1 = phi(4)-1 = 2-1 = 1;
5 is a term because phi(2*5-1)/2 = phi(9)/2 = 6/2 = 3 is equal to phi(2*5)-1 = phi(10)-1 = 4-1 = 3.

Crossrefs

Supersequence of A019279 and A061652.
Cf. A000010, A000043, A000668, A090748, A291901, A376337.

Programs

  • Magma
    [m: m in [2..2*10^6] | EulerPhi(2*m-1)/2 eq EulerPhi(2*m)-1];
  • Maple
    filter:= m -> numtheory:-phi(2*m-1)/2 = numtheory:-phi(2*m)-1:
select(filter, [$1..10^7]); # Robert Israel, Oct 20 2024
  • Mathematica
    Select[Range[300000], EulerPhi[2*# - 1]/2 == EulerPhi[2*#] - 1 &] (* Amiram Eldar, Oct 30 2024 *)
  • PARI
    isok(m) = eulerphi(2*m-1)/2 == eulerphi(2*m) - 1; \\ Michel Marcus, Oct 30 2024

Formula

a(n) = (A376337(n) + 1)/2.
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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