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Difference phi(p-1) - phi(p+1) for lesser of twin primes.
+10
4
-1, 0, 0, 2, 4, 4, 12, 0, 8, 16, 20, 32, 40, 8, 24, 40, 32, 60, 4, 24, 60, 84, 24, 56, 24, 136, 104, 36, 44, 116, 184, 48, 84, 184, 68, 252, 72, 280, 68, 144, 56, 292, 140, 192, 120, 338, 276, 120, 144, 262, 192, 376, 120, 268, 192, 236, 64, 168, 240, 492, 348, 388
MATHEMATICA
(EulerPhi[#-1] - EulerPhi[#+1]) &@ Select[Prime@ Range@ 310, PrimeQ[# + 2] &] (* Giovanni Resta, May 13 2017 *)
PROG
(PARI) lista(nn) = forprime(p=2, nn, if (isprime(p+2), print1(eulerphi(p-1)-eulerphi(p+1), ", ")));
Lesser of twin primes p for which phi(p-1) = phi(p+1), where phi(n) is the Euler totient function ( A000010).
+10
1
5, 11, 71, 2591, 208391, 16692551, 48502931, 92012201, 249206231, 419445251, 496978301, 1329067391, 1837151681, 2277479051, 2647600061, 4733566391, 6435087011, 10327948751, 14089345691, 14923624031, 22415286251, 27508270301, 39662281331, 59013882071, 70353395351
COMMENTS
The terms below 10^8 were taken from the paper by Garcia et al.
EXAMPLE
p = 5 is the lesser of the twin primes (5, 7), and phi(5-1) = phi(5+1) = 2.
MATHEMATICA
seq={}; Do[p = Prime[i]; If[PrimeQ[p+2] && EulerPhi[p-1] == EulerPhi[p+1], AppendTo[seq, p]], {i, 1, 1000000}]; seq
PROG
(PARI) isok(p) = isprime(p) && isprime(p+2) && (eulerphi(p-1) == eulerphi(p+1)); \\ Michel Marcus, Apr 26 2018
Lesser of twin primes p >= 5 for which phi(p+1)/phi(p-1) reaches record value, where phi(n) is the Euler totient function ( A000010).
+10
1
5, 2381, 3851, 20021, 50051, 52361, 424271, 470471, 602141, 2302301, 6806801, 16926911, 17497481, 69989921, 78278201, 183953771, 242662421, 468818351, 2156564411, 24912037151, 43874931101, 73769375681, 131104243271, 1360122864101, 1943064533411, 2635321709021, 3075260848661, 4078063299311
COMMENTS
Terms a(2)-a(23) were taken from the paper by Garcia et al.
Garcia et al. proved that assuming Dickson's conjecture, {phi(p+1)/phi(p-1) : p and p+2 are prime} is dense in [0, oo), and thus this sequence is infinite.
They give an example of a term p with 1099 digits with phi(p+1)/phi(p-1) = 3.11615...
What is the least value of lesser of twin primes p such that phi(p+1)/phi(p-1) > 2?
A candidate is p = 8183287190196092135163947564054981234789530779544672356881 for which the ratio is equal to 2.00047615... . - Giovanni Resta, Nov 01 2019
EXAMPLE
The values of phi(p+1)/phi(p-1) for the first terms are 1 < 1.031... < 1.06 < 1.118... < 1.12 < ...
MATHEMATICA
s = {}; rm = 0; p = 5; Do[q = NextPrime[p]; If[q - p != 2, p = q; Continue[]]; r = EulerPhi[p + 1]/EulerPhi[p - 1]; If[r > rm, rm = r; AppendTo[s, p]]; p = q, {10^6}]; s
CROSSREFS
Except for 5, subsequence of A286715.
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