A068019 -id:A068019 - cp's OEIS frontend

A071348 Intersection of A068017 and A068019: numbers n such that both sigma(n) and phi(n) are middle terms between (different) twin prime pairs.

10, 26, 38, 135, 206, 209, 216, 278, 371, 398, 416, 545, 560, 650, 698, 792, 866, 924, 1062, 1125, 1286, 1364, 1403, 1482, 1512, 1946, 2021, 2151, 2306, 2432, 2516, 2920, 3040, 3239, 3263, 3338, 3363, 3398, 3443, 3537, 3758, 3815, 4028, 4041, 4058, 4131

Offset: 1

Labos Elemer, May 21 2002

nonn

5 is not a term. Sigma[5]=6, and both 6-1=5 and 6+1=7 are primes. Phi[5]=4, and both 4-1=3 and 4+1=5 are primes. But, even though (3,5) and (5,7) are in some sense "(different) twin prime pairs" (quoting the sequence's definition), because 5 is a member of both they are treated, for purposes of this sequence, as not being "different." - Harvey P. Dale, Jun 05 2019

			n=4440,6328,6808,7030: sigma[n]=13680 between 13679 and 13681 prime, while Phi[4440]=1152,Phi[6328]=2688,Phi[6808]=3168,Phi[7030]=2592 are middle terms between different twin-pairs; n=545,866,1482,1512: phi[n]=432 between 431 and 433; sigma[n]-s give middle terms between different twin prime pairs.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000010, A000203, A068017, A068019.

Do[s=-1+DivisorSigma[1, n]; s1=1+DivisorSigma[1, n]; z=-1+EulerPhi[n]; z1=1+EulerPhi[n]; If[PrimeQ[s]&&PrimeQ[s1]&& PrimeQ[z]&&PrimeQ[z1]&&!PrimeQ[n], Print[{n, s, s1, z, z1}]], {n, 1, 10000}]
spmtQ[n_]:=Module[{s=DivisorSigma[1,n],p=EulerPhi[n]},s!=p&&AllTrue[ {s+1, s-1,p+1,p-1},PrimeQ]]; Select[Range[6,4200],spmtQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jun 05 2019 *)

A072281 Numbers n such that phi(n) + 1 and phi(n) - 1 are twin primes.

Original entry on oeis.org

5, 7, 8, 9, 10, 12, 13, 14, 18, 19, 21, 26, 27, 28, 31, 36, 38, 42, 43, 49, 54, 61, 62, 73, 77, 86, 91, 93, 95, 98, 99, 103, 109, 111, 117, 122, 124, 133, 135, 139, 146, 148, 151, 152, 154, 171, 181, 182, 186, 189, 190, 193, 198, 199, 206, 209, 216, 217, 218, 221, 222

Offset: 1

Joseph L. Pe, Jul 10 2002

Phi(n) is middle term between twin primes (A014574). Union of A006512 and A068019; intersection of A039698 and A078892. - Ray Chandler, May 26 2008

The positions of isolated nonprimes in A000010. - Juri-Stepan Gerasimov, Nov 10 2009

			phi(14) + 1 = 7 and phi(14) - 1 = 5, so 14 is a term of the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A006512, A014574, A039698, A068019, A078892.

Select[Range[10^3], PrimeQ[EulerPhi[ # ] + 1] && PrimeQ[EulerPhi[ # ] - 1] &]
Select[Range[300],And@@PrimeQ[EulerPhi[#]+{1,-1}]&] (* Harvey P. Dale, Apr 07 2012 *)

isok(n) = my(p); isprime(p=eulerphi(n)-1) && isprime(p+2); \\ Michel Marcus, Sep 29 2019

Extended by Ray Chandler, May 26 2008

A078893 Composite numbers k such that phi(k) - 1 is prime, where phi is Euler's totient function (A000010).

Original entry on oeis.org

8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 26, 27, 28, 30, 33, 35, 36, 38, 39, 42, 44, 45, 49, 50, 51, 52, 54, 56, 62, 64, 65, 66, 68, 69, 70, 72, 77, 78, 80, 81, 84, 86, 90, 91, 92, 93, 95, 96, 98, 99, 102, 104, 105, 111, 112, 117, 120, 121, 122, 123, 124, 129, 130, 133

Offset: 1

Reinhard Zumkeller, Dec 12 2002

nonn

A078892 with the primes removed. - Ray Chandler, May 26 2008

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A008864, A109606, A078892, A066071, A068019.

Select[Range[150],CompositeQ[#]&&PrimeQ[EulerPhi[#]-1]&] (* Harvey P. Dale, Dec 28 2021 *)

is(n)=!isprime(n) && isprime(eulerphi(n)-1) \\ Charles R Greathouse IV, Feb 21 2013

cp's OEIS Frontend

A071348 Intersection of A068017 and A068019: numbers n such that both sigma(n) and phi(n) are middle terms between (different) twin prime pairs.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A072281 Numbers n such that phi(n) + 1 and phi(n) - 1 are twin primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A078893 Composite numbers k such that phi(k) - 1 is prime, where phi is Euler's totient function (A000010).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI