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.

3, 5, 9, 11, 17, 27, 29, 35, 39, 41, 59, 65, 71, 101, 107, 125, 137, 149, 179, 191, 197, 227, 237, 239, 269, 281, 305, 311, 347, 417, 419, 431, 437, 461, 521, 569, 597, 599, 617, 641, 659, 671, 749, 755, 809, 821, 827, 857, 881, 905, 935, 989, 1019, 1031, 1049

Offset: 1

Author

Walter Kehowski, Jul 07 2006

Keywords

• nonn

Comments

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.

Examples

			a(6)=27 since proper divisors={1,3,3^2} and 27+d+1={29,31,37} are all prime.
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.
		

Links

• T. D. Noe, Table of n, a(n) for n=1..1000

Crossrefs

Cf. A120806, A120776, A120806, A001359, A005384, A045536.

Programs

Maple

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;

Mathematica

Select[Range[2,1100],AllTrue[#+Most[Divisors[#]]+1,PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 22 2020 *)

Formula

a(n)=n-th number such that n+d+1 is prime for all proper divisors d of n.