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

Author

Labos Elemer, May 21 2002

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A000010, A000203, A068017, A068019.

Programs

Mathematica

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 *)