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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. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

LINKS

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

FORMULA

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

EXAMPLE

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.

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 *)

CROSSREFS

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

Sequence in context: A302596 A230721 A094509 * A340287 A123069 A270836

Adjacent sequences:  A120808 A120809 A120810 * A120812 A120813 A120814

KEYWORD

nonn

AUTHOR

Walter Kehowski, Jul 07 2006

STATUS

approved

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Last modified December 9 06:07 EST 2021. Contains 349627 sequences. (Running on oeis4.)