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A120811 Positive integers n such that n+d+1 is prime for all proper divisors d of n. Generalization of twin prime to all integers.

%I A120811 #10 Aug 21 2024 16:41:19
%S A120811 3,5,9,11,17,27,29,35,39,41,59,65,71,101,107,125,137,149,179,191,197,
%T A120811 227,237,239,269,281,305,311,347,417,419,431,437,461,521,569,597,599,
%U A120811 617,641,659,671,749,755,809,821,827,857,881,905,935,989,1019,1031,1049
%N A120811 Positive integers n such that n+d+1 is prime for all proper divisors d of n. Generalization of twin prime to all integers.
%C A120811 This sequence (A120811) is a generalization of twin prime (A001359), the sequence A120776 is a generalization of Sophie Germain prime (A005384), while A120806 is the generalization of Sophie Germain twin prime (A045536). The same observations apply to A120811 as to A120806: the elements are (a) twin primes, (b) semiprimes pq, (c) 3-almost-primes, (d) 4-almost-primes. Moreover, the sequence includes all twin primes but in (b), (c) and (d) the containments are proper. The first occurrence of (d) is A120811(3980)=3^3*13147. Any others? A120811 CONJECTURE: These are all the elements, that is, no element of A120811 has more than 3 prime factors with no degree (sum of exponents) higher than 4.
%H A120811 T. D. Noe, <a href="/A120811/b120811.txt">Table of n, a(n) for n=1..1000</a>
%F A120811 a(n)=n-th number such that n+d+1 is prime for all proper divisors d of n.
%e A120811 a(6)=27 since proper divisors={1,3,3^2} and 27+d+1={29,31,37} are all prime.
%e A120811 a(3980)=3^3*13147 since proper divisors={1,3,3^2,3^3,13147,3*13147,3^2*13147} and a(3980)+d+1={354971,354973,354979,354997,368117,394411,473293} are all prime.
%p A120811 with(numtheory); L:=[]: for w to 1 do for k from 1 while nops(L)<=5000 do x:=2*k+1; if isprime(x+2) then S:=divisors(x) minus {x}; Q:=map(z-> x+z+1, S); if andmap(isprime,Q) then fd:=fopen("C:/temp/n+d+1=prime-lower.txt",APPEND); fprintf(fd,"%d",x); fclose(fd); L:=[op(L),x]; print(nops(L),ifactor(x)); fi; #Q fi; #x od od;
%t A120811 Select[Range[2,1100],AllTrue[#+Most[Divisors[#]]+1,PrimeQ]&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Oct 22 2020 *)
%Y A120811 Cf. A120806, A120776, A120806, A001359, A005384, A045536.
%K A120811 nonn
%O A120811 1,1
%A A120811 _Walter Kehowski_, Jul 07 2006