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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. 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 (list; graph; refs; listen; history; text; internal format)
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

Cf. A000010, A000203, A068017, A068019.

Sequence in context: A045039 A322972 A080059 * A055042 A044071 A186279

Adjacent sequences:  A071345 A071346 A071347 * A071349 A071350 A071351

KEYWORD

nonn

AUTHOR

Labos Elemer, May 21 2002

STATUS

approved

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Last modified December 2 08:31 EST 2021. Contains 349437 sequences. (Running on oeis4.)