OFFSET
1,1
COMMENTS
The lesser of twin prime pairs are excluded from this sequence since they are trivial terms; indeed twin primes have only two divisors by definition of primes (1 and themselves) and 2 is prime.
Considering the first 10^8 positive integers, we note the following: 276 of them meet the definition and only three (9, 81 and 59049, respectively the 3rd, 7th and 32nd terms of the sequence) are composite numbers and moreover perfect squares (9 = 3^2, 81 = 9^2 and 59049 = 243^2). All the other 273 terms are prime. From the 8th, all terms end with 7 or 9. The 276th term of the sequence is 98982599.
Open questions: are there any other terms greater than 98982599 that are composite numbers and more specifically perfect squares? If not, are there any other terms greater than 98982599 also prime but not ending with 7 or 9?
Not less than 10^15. - David A. Corneth, Apr 18 2021
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10000
EXAMPLE
9 is a term because tau(9) = 3 and tau(11) = 2 and 3 and 2 are prime.
47 is a term because tau(47) = 2 and tau(49) = 3 and 2 and 3 are prime.
48 is not a term because tau(48) = 10 and tau(50) = 6 and 10 and 6 are not prime.
MATHEMATICA
Select[Range@100000, And@@PrimeQ@DivisorSigma[0, {#, #+2}]&&Nand@@PrimeQ[{#, #+2}]&] (* Giorgos Kalogeropoulos, Apr 27 2021 *)
PROG
(PARI) upto(n) = print1(2", "); forstep(k=1, n, 2, if(isprime(numdiv(k)) && isprime(numdiv(k+2)) && (numdiv(k)!=2 || numdiv(k+2)!=2), print1(k", ")))
(PARI) upto(n) = {my(res = List([2])); n+=2; forprime(e = 3, logint(n, 3), forprime(p = 3, sqrtnint(n, e-1), c = p^(e-1); if(isprime(numdiv(c-2)), listput(res, c-2) ); if(isprime(numdiv(c+20)), listput(res, c) ) ) ); Set(res) } \\ David A. Corneth, Apr 18 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Claude H. R. Dequatre, Apr 17 2021
STATUS
approved