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%I #12 Jul 14 2024 19:07:18
%S 10,26,38,135,206,209,216,278,371,398,416,545,560,650,698,792,866,924,
%T 1062,1125,1286,1364,1403,1482,1512,1946,2021,2151,2306,2432,2516,
%U 2920,3040,3239,3263,3338,3363,3398,3443,3537,3758,3815,4028,4041,4058,4131
%N Intersection of A068017 and A068019: numbers n such that both sigma(n) and phi(n) are middle terms between (different) twin prime pairs.
%C 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
%H Harvey P. Dale, <a href="/A071348/b071348.txt">Table of n, a(n) for n = 1..1000</a>
%e 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.
%t 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}]
%t 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 *)
%Y Cf. A000010, A000203, A068017, A068019.
%K nonn
%O 1,1
%A _Labos Elemer_, May 21 2002
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