A068017 -id:A068017 - 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 *)

A068019 Composite n such that both 1 + phi(n) and -1 + phi(n) are primes, i.e., phi(n) is the middle term between twin primes (A014574).

Original entry on oeis.org

8, 9, 10, 12, 14, 18, 21, 26, 27, 28, 36, 38, 42, 49, 54, 62, 77, 86, 91, 93, 95, 98, 99, 111, 117, 122, 124, 133, 135, 146, 148, 152, 154, 171, 182, 186, 189, 190, 198, 206, 209, 216, 217, 218, 221, 222, 228, 234, 252, 266, 270, 278, 279, 287, 291, 297, 302

Offset: 1

Labos Elemer, Feb 08 2002

nonn

A072281 with the primes removed; intersection of A066071 and A078893. - Ray Chandler, May 26 2008

			n = 21, 26, 28, 36, 42 give phi(n)=12; the corresponding twin primes are {11,13}.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A014574, A066071, A072281, A078893, A068017.

Filtered([1..310],n->not IsPrime(n) and IsPrime(1+Phi(n)) and IsPrime(-1+Phi(n))); # Muniru A Asiru, Dec 08 2018

Do[s=-1+EulerPhi[n]; s1=1+EulerPhi[n]; If[PrimeQ[s]&&PrimeQ[s1]&&!PrimeQ[n], Print[n]], {n, 1, 2000}]
(* Second program: *)
Select[Range[4, 302], And[CompositeQ@ #, AllTrue[EulerPhi@ # + {-1, 1}, PrimeQ]] &] (* Michael De Vlieger, Dec 08 2018 *)

isok(n) = !isprime(n) && isprime(eulerphi(n)+1) && isprime(eulerphi(n)-1); \\ Michel Marcus, Dec 08 2018

A072282 Numbers n such that sigma(n) + 1 and sigma(n) - 1 are twin primes.

Original entry on oeis.org

3, 5, 6, 10, 11, 17, 20, 24, 26, 29, 30, 38, 41, 46, 51, 55, 59, 71, 85, 88, 101, 105, 107, 114, 118, 126, 135, 136, 137, 141, 145, 147, 149, 155, 158, 161, 177, 178, 179, 185, 191, 197, 203, 206, 207, 209, 216, 227, 230, 236, 238, 239, 255, 269, 278, 281, 296

Offset: 1

Joseph L. Pe, Jul 10 2002

			sigma(20) + 1 = 43 and sigma(20) - 1 = 41, so 20 is a term of the sequence.

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

Cf. A000203, A068017.

Select[Range[10^3], PrimeQ[DivisorSigma[1, # ] + 1] && PrimeQ[DivisorSigma[1, # ] - 1] &]
Select[Range[300],AllTrue[DivisorSigma[1,#]+{1,-1},PrimeQ]&] (* Harvey P. Dale, Apr 06 2023 *)

isok(n) = my(p=sigma(n)-1); isprime(p) && isprime(p+2); \\ Michel Marcus, Sep 30 2019

More terms from Amiram Eldar, Sep 30 2019

A349981 Midpoints k of a pair of twin primes such that sigma(k) is also the midpoint of a pair of twin primes.

Original entry on oeis.org

6, 30, 462, 1062, 1290, 1482, 1878, 2088, 2790, 3558, 4272, 4338, 6552, 6660, 7308, 8010, 8598, 8820, 10038, 10428, 10530, 10890, 11940, 12042, 12918, 13338, 13758, 16980, 17418, 17580, 18252, 19992, 21588, 22038, 22740, 23742, 25848, 26862, 27738, 32028, 33288, 35730, 37548, 37782, 42180, 42570

Offset: 1

J. M. Bergot and Robert Israel, Feb 06 2022

nonn

			a(3) = 462 is a term because sigma(462) = 1152 and 461, 463, 1151 and 1153 are primes.

Robert Israel, Table of n, a(n) for n = 1..10000

Intersection of A014574 and A068017.

Cf. A000203, A202607.

filter:= proc(x) local t;
  if not (isprime(x-1) and isprime(x+1)) then return false fi;
  t:= numtheory:-sigma(x);
  t mod 6 = 0 and isprime(t-1) and isprime(t+1)
end proc:
select(filter, [seq(i,i=6..100000,6)]);

a(n) = 6*A202607(n). - Ivan N. Ianakiev, Feb 07 2022

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

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A068019 Composite n such that both 1 + phi(n) and -1 + phi(n) are primes, i.e., phi(n) is the middle term between twin primes (A014574).

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A072282 Numbers n such that sigma(n) + 1 and sigma(n) - 1 are twin primes.

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A349981 Midpoints k of a pair of twin primes such that sigma(k) is also the midpoint of a pair of twin primes.

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