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%I #32 Apr 28 2021 06:30:52
%S 2,7,9,23,47,79,81,167,359,727,839,1367,1847,2207,2399,3719,5039,7919,
%T 10607,11447,14639,16127,17159,19319,28559,29927,36479,44519,49727,
%U 54287,57119,59049,66047,85847,97967,113567,128879,177239,196247,201599,218087,241079,273527,292679
%N Numbers k such that tau(k) and tau(k+2) are both prime, where tau is the number of divisors function (the lesser of twin prime pairs are excluded).
%C 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.
%C 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.
%C 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?
%C Not less than 10^15. - _David A. Corneth_, Apr 18 2021
%H David A. Corneth, <a href="/A343495/b343495.txt">Table of n, a(n) for n = 1..10000</a>
%e 9 is a term because tau(9) = 3 and tau(11) = 2 and 3 and 2 are prime.
%e 47 is a term because tau(47) = 2 and tau(49) = 3 and 2 and 3 are prime.
%e 48 is not a term because tau(48) = 10 and tau(50) = 6 and 10 and 6 are not prime.
%t Select[Range@100000,And@@PrimeQ@DivisorSigma[0,{#,#+2}]&&Nand@@PrimeQ[{#,#+2}]&] (* _Giorgos Kalogeropoulos_, Apr 27 2021 *)
%o (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", ")))
%o (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
%Y Cf. A000005, A000040.
%K nonn
%O 1,1
%A _Claude H. R. Dequatre_, Apr 17 2021
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