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.

Previous Showing 11-13 of 13 results.

A342364 The primes associated with A239727.

Original entry on oeis.org

3, 17, 73, 191, 709, 1289, 3181, 5449, 7681, 17477, 33889, 87961, 437389, 2290573, 7160227, 10429681, 19196227, 24504049, 47577857, 70513979, 82605937, 156671243, 271785793, 328939937, 568119509, 1125978241, 1534657963, 1710749497, 4936728373, 7647104183
Offset: 1

Views

Author

Jianing Song, Mar 09 2021

Keywords

Examples

			The smallest prime of the form 2*A239727(3)*k + 1 = 24*k+1 is 73, hence a(3) = 73.
The smallest prime of the form 2*A239727(4)*k + 1 = 38*k+1 is 191, hence a(4) = 191.

Links

Crossrefs

Cf. A239727, A239746, A016014, A070846, A342365.

Programs

  • PARI
    A016014(n)=my(k); while(!isprime(2*n*k+++1), ); k
r=0; for(n=1, 1e8, t=A016014(n); if(t>r, r=t; print1(2*n*r+1", "))) \\ based on the program of A239727

Formula

a(n) = 2*A239727(n)*A239746(n) + 1.

A342365 The primes associated with A239800 (1 if A239800(n) = 0).

Original entry on oeis.org

3, 17, 73, 337, 191, 709, 1289, 3137, 3313, 3181, 7349, 5449, 8243, 11621, 7681, 16673, 17477, 28657, 27893, 74441, 71023, 87869, 94439, 33889, 250301, 298013, 205957, 131489, 327179, 87961, 1178993, 354689, 1769791, 595817, 1304591, 417169, 2343359
Offset: 1

Views

Author

Jianing Song, Mar 09 2021

Keywords

Examples

			The smallest m such that 2*m*i + 1 is not prime until i = 3 is m = 12, and the corresponding prime is 2*12*3 + 1 = 73 = a(3).
The smallest m such that 2*m*i + 1 is not prime until i = 4 is m = 42, and the corresponding prime is 2*42*4 + 1 = 337 = a(4).
The smallest m such that 2*m*i + 1 is not prime until i = 5 is m = 19, and the corresponding prime is 2*19*5 + 1 = 191 = a(5).

Links

Crossrefs

Cf. A239800, A016014, A070846, A342364.

Programs

  • PARI
    isok(n, m) = for(i=1, n-1, if(isprime(2*m*i+1), return(0))); if(isprime(2*m*n+1), 1, 0)
a(n) = for(m=1, oo, if(isok(n, m), return(2*n*m+1))) \\ based on the conjecture that all numbers occur in A016014

Formula

a(n) = 2*n*A239800(n)+1.

A341845 a(n) = A061026(2n): smallest k such that 2n divides phi(k), phi = A000010.

Original entry on oeis.org

3, 5, 7, 15, 11, 13, 29, 17, 19, 25, 23, 35, 53, 29, 31, 51, 103, 37, 191, 41, 43, 69, 47, 65, 101, 53, 81, 87, 59, 61, 311, 85, 67, 137, 71, 73, 149, 229, 79, 123, 83, 129, 173, 89, 181, 141, 283, 97, 197, 101, 103, 159, 107, 109, 121, 113, 229, 177, 709, 143
Offset: 1

Views

Author

Jianing Song, Feb 21 2021

Keywords

Comments

A061026(n) = A061026(2n) for odd n > 1 since phi(m) is even for m >= 3. In this sequence the redundant values are omitted.
We have the obvious inequality A070846(n) >= A307437(n) >= a(n). For odd p = prime(k), A307437(p) = a(p), and if A341861(k) > 0 we have A070846(p) = a(p).
The smallest n such that A070846(n) > A307437(n) > a(n) is n = 40, where A070846(40) = 241, A307437(40) = 187 and a(40) = 123.

Examples

			a(12) = 35 since phi(35) = 24 is divisible by 2*12, and there is no m < 12 such that phi(m) is divisible by 2*12.
a(16) = 51 since phi(51) = 32 is divisible by 2*16, and there is no m < 16 such that phi(m) is divisible by 2*16.

Links

Crossrefs

Cf. A061026, A000010, A070846, A307437, A341861.

Programs

  • PARI
    a(n) = for(m=1, (2*n)^2, if(eulerphi(m)%(2*n)==0, return(m)))
  • Python
    from sympy import totient as phi
def a(n):
  k = 1
  while phi(k)%(2*n) != 0: k += 1
  return k
print([a(n) for n in range(1, 61)]) # Michael S. Branicky, Feb 21 2021
Previous Showing 11-13 of 13 results.

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

Content is available under The OEIS End-User License Agreement.