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 1-3 of 3 results.

A067544 Numbers k such that phi(k*(k+1)*(k+2)/6)/phi(k*(k+1)/2) is not an integer where phi(k) = A000010(k) is the Euler totient function.

Original entry on oeis.org

8, 9, 18, 27, 44, 45, 53, 62, 80, 81, 90, 98, 99, 108, 116, 126, 134, 135, 143, 162, 171, 189, 198, 224, 225, 233, 234, 251, 260, 261, 270, 279, 287, 288, 296, 315, 332, 350, 351, 387, 413, 422, 423, 441, 449, 450, 458, 459, 468, 476, 477, 503, 504, 512, 522
Offset: 1

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Author

Benoit Cloitre, Jan 28 2002

Keywords

Comments

Often a(k) == 0 (mod 3) and if a(k+1) - a(k) = 1 then a(k+1) == 0 (mod 3).

Links

Crossrefs

Cf. A000010, A000217, A000292, A086700.

Programs

  • Mathematica
    Select[Range[500], ! Divisible[EulerPhi[#*(# + 1)*(# + 2)/6], EulerPhi[#*(# + 1)/2]] &] (* Amiram Eldar, Jul 17 2022 *)

A291199 Primes p such that phi(p*(p+1)/2) is a triangular number (A000217).

Original entry on oeis.org

2477, 44287823, 58192759, 110369351, 664009019, 2574106333, 6870260119, 7423240007, 60370077539, 188271042191, 235399729007, 236767359977, 305214702643, 717724689959
Offset: 1

Views

Author

Altug Alkan, Aug 20 2017

Keywords

Comments

a(15) > 10^12. - Giovanni Resta, Aug 21 2017

Examples

			Prime number 2477 is a term since phi(2477*2478/2) = 1856*1857/2.

Crossrefs

Cf. A000010, A000217, A034953, A086700.

Programs

  • PARI
    isok(n) = isprime(n) && ispolygonal(eulerphi(n*(n+1)/2), 3);
  • PARI
    is(n) = ispolygonal(eulerphi(n\2+1)*(n-1), 3) && isprime(n) \\ Charles R Greathouse IV, Aug 22 2017
  • Python
    from _future_ import division
from sympy.ntheory.primetest import is_square
from sympy import totient, nextprime
A291199_list, p = [], 3
while p < 10**8:
    if is_square(8*(p-1)*totient((p+1)//2)+1):
        A291199_list.append(p)
    p = nextprime(p) # Chai Wah Wu, Aug 22 2017

Extensions

a(5)-a(14) from Giovanni Resta, Aug 21 2017

A347167 Numbers k such that phi(binomial(k,2)) is a power of 2.

Original entry on oeis.org

2, 3, 4, 5, 6, 16, 17, 256, 257, 65536, 65537, 4294967296
Offset: 1

Views

Author

Arkadiusz Wesolowski, Aug 20 2021

Keywords

Comments

Every Fermat prime appears in this sequence.
A number greater than 2^32 is in this sequence if and only if it is a Fermat prime.

References

  • M. Krizek, F. Luca and L. Somer, 17 Lectures on Fermat Numbers: From Number Theory to Geometry, CMS Books in Mathematics, vol. 9, Springer-Verlag, New York, 2001, p. 86.
  • F. Luca, Pascal's triangle and constructible polygons, Util. Math. 58 (2000d), pp. 209-214.

Crossrefs

Cf. A019434, A086700.

Programs

  • Magma
    r:=7; IsInteger:=func; lst:=[k: k in [2..6] | IsInteger(Log(2, EulerPhi(Binomial(k, 2))))]; t:=1; for x in [1..r] do m:=4^(2^x); if t eq 1 then Append(~lst, m); end if; if IsPrime(m+1) then Append(~lst, m+1); else t:=0; end if; end for; lst;
  • Mathematica
    Select[Range[10^5],IntegerQ@Log2[EulerPhi@Binomial[#,2]]&] (* Giorgos Kalogeropoulos, Sep 08 2021 *)

Formula

For n >= 13, a(n) = A019434(n-7) (if it exists).
Showing 1-3 of 3 results.

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

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