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 1-8 of 8 results.

A070811 Nonprime numbers k such that phi(k-phi(k)) = A054571(k) is not a power of 2.

Original entry on oeis.org

1, 15, 21, 26, 27, 30, 33, 34, 35, 45, 49, 51, 52, 54, 57, 60, 63, 66, 68, 69, 70, 74, 75, 78, 81, 82, 85, 86, 87, 90, 91, 93, 95, 98, 99, 102, 104, 105, 106, 108, 110, 111, 114, 115, 117, 119, 120, 121, 122, 123, 125, 126, 129, 130, 132, 133, 135, 136, 138, 140
Offset: 1

Views

Author

Labos Elemer, May 08 2002

Keywords

Examples

			For k = 30: phi(30) = 8, cototient(30) = 22, phi(22) = 10 is not a power of 2.

Links

Crossrefs

Cf. A000010, A054571, A051953, A070556, A070806, A070807, A070809, A070810, A070812.

Programs

  • Mathematica
    Do[s=EulerPhi[n-EulerPhi[n]]; If[ !IntegerQ[Log[2, s]]&&!PrimeQ[n], Print[n]], {n, 1, 256}]
  • PARI
    is(k) = if(k == 1, 1, if(isprime(k), 0, my(m = eulerphi(k - eulerphi(k))); m >> valuation(m, 2) > 1)); \\ Amiram Eldar, Nov 08 2024

A070810 Nonprime numbers k such that phi(k-phi(k)) = A054571(k) is a power of 2.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 25, 28, 32, 36, 38, 39, 40, 42, 44, 46, 48, 50, 55, 56, 58, 62, 64, 65, 72, 76, 77, 80, 84, 88, 92, 94, 96, 100, 112, 116, 118, 124, 128, 134, 144, 152, 158, 160, 165, 168, 176, 184, 188, 192, 200, 202, 224, 232, 235, 236
Offset: 1

Views

Author

Labos Elemer, May 08 2002

Keywords

Examples

			For k = 168: 168 - phi(168) = 168-48 = 120, phi(120) = 32, a power of 2.

Links

Crossrefs

Cf. A000010, A054571, A051953, A070556, A070806, A070807, A070809, A070811, A070812.

Programs

  • Mathematica
    Do[s=EulerPhi[n-EulerPhi[n]]; If[IntegerQ[Log[2, s]]&&!PrimeQ[n], Print[n]], {n, 1, 256}]
  • PARI
    is(k) = if(k == 1 || isprime(k), 0, my(m = eulerphi(k - eulerphi(k))); m >> valuation(m, 2) == 1); \\ Amiram Eldar, Nov 08 2024

A070556 a(n) = cototient(totient(n)).

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 4, 2, 4, 2, 6, 2, 8, 4, 4, 4, 8, 4, 12, 4, 8, 6, 12, 4, 12, 8, 12, 8, 16, 4, 22, 8, 12, 8, 16, 8, 24, 12, 16, 8, 24, 8, 30, 12, 16, 12, 24, 8, 30, 12, 16, 16, 28, 12, 24, 16, 24, 16, 30, 8, 44, 22, 24, 16, 32, 12, 46, 16, 24, 16, 46, 16, 48
Offset: 1

Views

Author

N. J. A. Sloane, May 06 2002

Keywords

Links

Crossrefs

Cf. A000010, A051953, A054571.

Programs

  • Magma
    [EulerPhi(n)-EulerPhi(EulerPhi(n)): n in [1..100]]; // Vincenzo Librandi, Aug 06 2015
  • Maple
    A051953 := proc(n)
        n-numtheory[phi](n) ;
end proc:
A070556 := proc(n)
        A051953(numtheory[phi](n)) ;
end proc: # R. J. Mathar, Oct 13 2011
  • Mathematica
    Table[EulerPhi[n] - EulerPhi[EulerPhi[n]], {n, 80}] (* Vincenzo Librandi, Aug 06 2015 *)

Formula

a(n) = A051953(A000010(n)).

A290087 a(1) = 0; for n > 1, a(n) = A289626(A051953(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 5, 1, 5, 4, 5, 1, 5, 1, 5, 4, 5, 1, 8, 3, 4, 4, 8, 1, 6, 1, 8, 7, 4, 6, 13, 1, 8, 8, 13, 1, 8, 1, 13, 11, 13, 1, 17, 4, 8, 10, 11, 1, 11, 8, 17, 11, 8, 1, 18, 1, 17, 10, 17, 9, 12, 1, 11, 14, 12, 1, 21, 1, 10, 19, 21, 9, 10, 1, 21, 10, 11, 1, 21, 11, 18, 16, 21, 1, 18, 10, 21, 18, 21, 12, 25, 1, 28, 19, 21, 1, 19, 1, 28, 29
Offset: 1

Views

Author

Antti Karttunen, Aug 07 2017

Keywords

Links

Crossrefs

Cf. A051953, A054571, A289626, A290086, A290085, A290086, A290088, A290089.

Formula

a(1) = 0; for n > 1, a(n) = A289626(A051953(n)).

A070806 Primes p such that cototient(totient(p)) = A070556(p) is a power of 2.

Original entry on oeis.org

3, 5, 7, 13, 17, 29, 97, 113, 193, 257, 449, 509, 769, 7937, 12289, 65537, 114689, 520193, 786433, 7340033, 8388593, 33292289, 33550337, 469762049, 2130706433, 3221225473, 8588886017, 137438691329, 206158430209
Offset: 1

Views

Author

Labos Elemer, May 08 2002

Keywords

Examples

			Powers of 2 observable in A070556[this sequence] = {1, 2, 4, 8, 16, 64, 128, 256, 512, 4096, 8192, 32768, 65536, 262144, 524288, ...}. For F(m), Fermat prime:phi[F(m)]=2^m, cototient[2^m]=2^(m-1); if n=113: phi[113]=112, cototient[112]=112-48=64, so 113 is in this sequence.

Crossrefs

Cf. A070556, A051953, A054571, A070807, A070809-A070811.

Programs

  • Mathematica
    Do[s= EulerPhi[n]-EulerPhi[EulerPhi[n]]; If[IntegerQ[Log[2, s]]&&PrimeQ[n], Print[n]], {n, 1, 10000000}]
  • PARI
    ispow2(n)=n==1<Charles R Greathouse IV, May 17 2011

Extensions

a(20)-a(27) from Donovan Johnson, Feb 06 2010
a(28)-a(29) from Charles R Greathouse IV, May 17 2011

A070807 Composite numbers n such that Cototient(totient(n))=A070556(n) is power of 2.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 26, 28, 30, 32, 34, 35, 36, 39, 40, 42, 45, 48, 51, 52, 56, 58, 60, 64, 65, 68, 70, 72, 78, 80, 84, 85, 87, 90, 96, 102, 104, 105, 112, 116, 119, 120, 128, 130, 136, 140, 144, 145, 153, 156, 160, 168, 170, 174, 180, 192
Offset: 1

Views

Author

Labos Elemer, May 08 2002

Keywords

Examples

			n=87=3.29:phi[87]=56,56-phi[56]=56-24=32

Crossrefs

Cf. A070556, A051953, A054571, A070806, A070809-A070811.

Programs

  • Mathematica
    Do[s= EulerPhi[n]-EulerPhi[EulerPhi[n]]; If[IntegerQ[Log[2, s]]&&!PrimeQ[n], Print[n]], {n, 1, 10000000}]

A070809 Cototient(totient(n))=A070556(n) is not a power of 2 and n is not a prime number.

Original entry on oeis.org

1, 22, 25, 27, 33, 38, 44, 46, 49, 50, 54, 55, 57, 62, 63, 66, 69, 74, 75, 76, 77, 81, 82, 86, 88, 91, 92, 93, 94, 95, 98, 99, 100, 106, 108, 110, 111, 114, 115, 117, 118, 121, 122, 123, 124, 125, 126, 129, 132, 133, 134, 135, 138, 141, 142, 143, 146, 147, 148
Offset: 1

Views

Author

Labos Elemer, May 08 2002

Keywords

Examples

			n=95: Phi[95]=72,cototient[72]=72-phi[72]=72-24-=48 is not a power of 2.

Crossrefs

Cf. A070556, A051953, A054571, A070806, A070807, A070810, A070811.

Programs

  • Mathematica
    Do[s= EulerPhi[n]-EulerPhi[EulerPhi[n]]; If[ !IntegerQ[Log[2, s]]&&!PrimeQ[n], Print[n]], {n, 1, 1000}]

A346692 a(n) = phi(n) - phi(n-phi(n)), a(1) = 1.

Original entry on oeis.org

1, 0, 1, 1, 3, 0, 5, 2, 4, 2, 9, 0, 11, 2, 2, 4, 15, 2, 17, 4, 6, 6, 21, 0, 16, 6, 12, 4, 27, -2, 29, 8, 8, 10, 14, 4, 35, 10, 16, 8, 39, 4, 41, 12, 12, 14, 45, 0, 36, 12, 14, 12, 51, 6, 32, 8, 24, 20, 57, -4, 59, 14, 18, 16, 32, -2, 65, 20, 24, 2, 69, 8, 71, 18, 16, 20, 44, 6, 77, 16
Offset: 1

Views

Author

Bernard Schott, Jul 29 2021

Keywords

Comments

P. Erdős conjectured that a(n) > 0 on a set of asymptotic density 1, then Luca and Pomerance proved this conjecture (see link).

References

  • Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B42, p. 150.

Links

Crossrefs

Cf. A000010, A054571.
Cf. A051487 (a(n)=0), A051488 (a(n)<0).

Programs

  • Maple
    with(numtheory):
A := seq(phi(n) - phi(n-phi(n)), n=1..100);
  • Mathematica
    a[n_] := (phi = EulerPhi[n]) - EulerPhi[n - phi]; Array[a, 100] (* Amiram Eldar, Jul 29 2021 *)
  • PARI
    a(n) = if (n==1, 1, eulerphi(n) - eulerphi(n-eulerphi(n))); \\ Michel Marcus, Jul 29 2021
  • Python
    from sympy import totient as phi
def a(n):
    if n == 1: return 1
    phin = phi(n)
    return phin - phi(n - phin)
print([a(n) for n in range(1, 81)]) # Michael S. Branicky, Jul 29 2021

Formula

a(n) = A000010(n) - A054571(n).
If p prime, a(p) = p-2, and for k >= 2, a(p^k) = (p-1)^2 * p^(k-2).
Showing 1-8 of 8 results.

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

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