

A071348


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


1



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
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OFFSET

1,1


COMMENTS

5 is not a term. Sigma[5]=6, and both 61=5 and 6+1=7 are primes. Phi[5]=4, and both 41=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



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 twinpairs; 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, s1, p+1, p1}, 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



STATUS

approved



