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 1-7 of 7 results.

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}]

A054571 a(n) = phi(n - phi(n)), a(1) = 0.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 4, 1, 4, 6, 4, 1, 4, 1, 4, 6, 4, 1, 8, 4, 6, 6, 8, 1, 10, 1, 8, 12, 6, 10, 8, 1, 8, 8, 8, 1, 8, 1, 8, 12, 8, 1, 16, 6, 8, 18, 12, 1, 12, 8, 16, 12, 8, 1, 20, 1, 16, 18, 16, 16, 22, 1, 12, 20, 22, 1, 16, 1, 18, 24, 16, 16, 18, 1, 16, 18
Offset: 1

Views

Author

J. Sandor (mstaicu(AT)dualnet.ro), Mar 09 2002

Keywords

Links

Crossrefs

Cf. A000010, A051953, A070556, A290087.

Programs

Formula

a(1) = 0, and for n > 1, a(n) = totient(cototient(n)) = A000010(A051953(n)). - Antti Karttunen, Aug 07 2017

Extensions

Description clarified with a(1) = 0 explicitly set by convention. - Antti Karttunen, Aug 07 2017

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

A293516 a(n) = phi(n) - 2*phi(phi(n)), where phi = Euler totient function, A000010.

Original entry on oeis.org

-1, -1, 0, 0, 0, 0, 2, 0, 2, 0, 2, 0, 4, 2, 0, 0, 0, 2, 6, 0, 4, 2, 2, 0, 4, 4, 6, 4, 4, 0, 14, 0, 4, 0, 8, 4, 12, 6, 8, 0, 8, 4, 18, 4, 8, 2, 2, 0, 18, 4, 0, 8, 4, 6, 8, 8, 12, 4, 2, 0, 28, 14, 12, 0, 16, 4, 26, 0, 4, 8, 22, 8, 24, 12, 8, 12, 28, 8, 30, 0, 18, 8, 2, 8, 0, 18, 8, 8, 8, 8, 24, 4, 28, 2, 24, 0, 32
Offset: 1

Views

Author

Antti Karttunen, Nov 27 2017

Keywords

Links

Crossrefs

Cf. A000010, A010554, A051953, A070556, A083254, A295660.
Cf. A003401 (gives the positions of zeros after the two initial -1's).

Programs

  • PARI
    A293516(n) = (eulerphi(n) - 2*eulerphi(eulerphi(n)));

Formula

a(n) = A000010(n) - 2*A010554(n).
a(n) = A070556(n) - A010554(n).
a(n) = -A083254(A000010(n)).

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
Showing 1-7 of 7 results.

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