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 1-6 of 6 results.

A121048 a(n) = n + phi(n), where phi is the Euler totient function.

Original entry on oeis.org

2, 3, 5, 6, 9, 8, 13, 12, 15, 14, 21, 16, 25, 20, 23, 24, 33, 24, 37, 28, 33, 32, 45, 32, 45, 38, 45, 40, 57, 38, 61, 48, 53, 50, 59, 48, 73, 56, 63, 56, 81, 54, 85, 64, 69, 68, 93, 64, 91, 70, 83, 76, 105, 72, 95, 80, 93, 86, 117, 76, 121, 92, 99, 96, 113, 86, 133, 100, 113
Offset: 1

Views

Author

Jonathan Vos Post, Aug 08 2006

Keywords

Links

Crossrefs

Cf. A000010, A000040, A050530, A051953, A059956, A068080.

Programs

  • Magma
    [n + EulerPhi(n): n in [1..70]]; // Vincenzo Librandi, Jul 13 2012
  • Mathematica
    Table[n+EulerPhi[n],{n,1,70}]
  • PARI
    a(n) = n+eulerphi(n); \\ Michel Marcus, Sep 19 2022

Formula

a(n) = n + phi(n) = n + A000010(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 + 1/zeta(2) = 1.607927... . - Amiram Eldar, Dec 16 2023

A068081 Numbers n such that n + phi(n) and n - phi(n) are prime.

Original entry on oeis.org

15, 33, 35, 51, 65, 77, 91, 95, 143, 161, 177, 209, 213, 215, 217, 247, 255, 303, 335, 341, 371, 411, 427, 435, 447, 455, 533, 545, 561, 573, 591, 611, 665, 707, 713, 717, 779, 803, 871, 917, 933, 965, 1001, 1041, 1067, 1105, 1115, 1133, 1157, 1159, 1211
Offset: 1

Views

Author

Amarnath Murthy, Feb 17 2002

Keywords

Links

Crossrefs

Cf. A050530, A068080.

Programs

  • GAP
    A068081:=[];; for n in [1,3..10^4+1] do if IsPrime(n+Phi(n)) and IsPrime(n-Phi(n)) then Add(A068081,n); fi; od; A068081;  # Muniru A Asiru, Aug 31 2017
  • Maple
    with(numtheory): for n from 1 by 2 to 10^4 do if [isprime(n+phi(n)),
isprime(n-phi(n))]=[true,true] then print(n); fi; od; # Muniru A Asiru, Aug 31 2017
  • Mathematica
    Select[ Range[1500], PrimeQ[ # + EulerPhi[ # ]] && PrimeQ[ # - EulerPhi[ # ]] & ]
epQ[n_]:=Module[{ep=EulerPhi[n]},AllTrue[n+{ep,-ep},PrimeQ]]; Select[ Range[ 1500],epQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 22 2016 *)
  • PARI
    is(n)=my(t=eulerphi(n)); isprime(n-t) && isprime(n+t) \\ Charles R Greathouse IV, Jan 25 2017

Extensions

Edited and extended by Robert G. Wilson v, Feb 18 2002

A116035 Numbers k such that k + phi(k) + sigma(k) is a prime.

Original entry on oeis.org

1, 4, 15, 33, 35, 36, 50, 55, 57, 64, 65, 75, 77, 85, 87, 93, 98, 115, 119, 129, 133, 143, 155, 159, 185, 187, 189, 205, 213, 215, 217, 219, 242, 243, 247, 253, 265, 287, 295, 303, 309, 323, 324, 327, 339, 345, 365, 385, 393, 395, 407, 425, 427, 453, 469, 493
Offset: 1

Views

Author

Giovanni Resta, Feb 13 2006

Keywords

Examples

			205 + phi(205) + sigma(205) = 617 (prime).

Links

Crossrefs

Cf. A078762, A068080.

Programs

  • Mathematica
    Select[Range[500],PrimeQ[#+EulerPhi[#]+DivisorSigma[1,#]]&] (* Harvey P. Dale, Nov 09 2014 *)

A116045 n+phi(n)+phi(phi(n)) is a prime.

Original entry on oeis.org

1, 4, 5, 9, 13, 17, 19, 21, 25, 33, 35, 39, 41, 43, 49, 61, 75, 79, 85, 87, 95, 101, 121, 133, 137, 141, 153, 155, 159, 163, 165, 169, 177, 181, 193, 199, 201, 203, 207, 213, 215, 231, 233, 245, 251, 257, 259, 265, 267, 271, 277, 291, 299, 307, 309, 337, 339
Offset: 1

Views

Author

Giovanni Resta, Feb 13 2006

Keywords

Examples

			13+phi(13)+phi(phi(13)) = 29.

Crossrefs

Cf. A068080.

Programs

  • Mathematica
    epQ[n_]:=Module[{ep=EulerPhi[n]},PrimeQ[n+ep+EulerPhi[ep]]]
Select[Range[400],epQ]  (* Harvey P. Dale, Mar 06 2011 *)

A158458 Numbers k such that k + bigomega(k) is prime.

Original entry on oeis.org

2, 8, 9, 15, 20, 21, 28, 32, 35, 39, 44, 48, 50, 51, 57, 65, 68, 69, 70, 76, 77, 87, 95, 98, 108, 110, 111, 124, 129, 135, 148, 154, 155, 161, 162, 164, 168, 170, 176, 177, 188, 189, 190, 192, 209, 221, 225, 230, 236, 237, 238, 249, 252, 264, 266, 267, 268, 272, 290
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Mar 19 2009

Keywords

Comments

2 is the only prime number in the sequence. - Michel Lagneau, May 17 2010

Examples

			a(2) = 8 = 2*2*2; 8+3 = 11 is prime.
a(3) = 9 = 3*3; 9+2 = 11 is prime.
a(4) = 15 = 3*5; 15+2 = 17 is prime.

Links

Crossrefs

Cf. A000040, A001222.
Cf. A067532, A078762, A068080. - Michel Lagneau, May 17 2010

Programs

  • Maple
    for k from 2 to 400 do if isprime(k+numtheory[bigomega](k)) then printf("%d,",k) ; fi; od: # R. J. Mathar, May 19 2009, May 23 2010
  • Mathematica
    Select[Range[10^3], PrimeQ[ # + Plus @@ Last /@ FactorInteger[ # ]] &] (* Michel Lagneau, May 17 2010 *)
Select[Range[300],PrimeQ[#+PrimeOmega[#]]&] (* Harvey P. Dale, Dec 20 2021 *)
  • PARI
    is(n)=isprime(n+bigomega(n)) \\ Eric Chen, Jun 13 2018

Formula

{k: k+A001222(k) in A000040}.

Extensions

191 replaced with 192 and extended by R. J. Mathar, May 19 2009
Generalized (by inserting a(1)=2) by Michel Lagneau, May 17 2010

A363583 Numbers k such that 2*phi(k)+k is a prime, where phi is A000010.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 23, 33, 35, 37, 43, 47, 53, 61, 67, 69, 71, 77, 87, 95, 103, 113, 119, 123, 127, 133, 137, 143, 159, 163, 167, 177, 181, 191, 193, 209, 211, 217, 249, 251, 257, 259, 263, 267, 271, 277, 293, 299, 307, 313, 329, 331, 335, 337, 339
Offset: 1

Views

Author

Darío Clavijo, Aug 17 2023

Keywords

Comments

All the terms are odd squarefree numbers.

Crossrefs

Cf. A000010, A068080.
Subsequence of A056911.
Subsequence: A088878 (the prime terms).

Programs

  • Mathematica
    Select[Range[1, 350, 2], PrimeQ[2*EulerPhi[#] + #] &] (* Amiram Eldar, Aug 17 2023 *)
  • PARI
    isok(k) = isprime(k+2*eulerphi(k)); \\ Michel Marcus, Aug 20 2023
  • Python
    from sympy import totient, isprime
print([k for k in range(1, 340) if isprime(2*totient(k) + k)])
Showing 1-6 of 6 results.

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