OFFSET
1,1
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
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..1000
EXAMPLE
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.
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Labos Elemer, May 21 2002
STATUS
approved