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A120776
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Composite numbers k such that k+d+1 is prime for all divisors d of k greater than 1.
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4
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8, 9, 35, 39, 65, 119, 125, 219, 341, 515, 749, 755, 905, 935, 989, 1043, 1119, 1343, 1355, 1469, 1649, 1829, 1859, 2519, 3005, 3161, 3563, 3953, 4193, 4269, 4359, 4613, 4685, 4769, 4859, 5123, 5165, 5249, 5585, 5699, 5723, 6005, 6059, 6239, 6629, 6879
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OFFSET
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1,1
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COMMENTS
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The sequence could begin with 1 by convention. The sequence in which d can be 1 is a subsequence. The elements are assumed composite so as to exclude the Sophie Germain primes (A005384) and (A045536). All terms except 8 and 9 are odd numbers in squarefree semiprimes (A006881) or 3-almost-primes (A014612). The only square is 9, the first few cubes are 8, 125, 357911=71^3, 6967871=191^3 and the first few 3-almost primes are 935=5*11*17, 1859=11*13^2, 11123=7^2*227, 305015=5*53*1151. The first 3-almost-prime divisible by 9 is 149049=3^2*16561. All elements not divisible by 3 are 5 or 11 mod 12. I have been unable to find an element with more than 3 prime factors. If one exists, it must be very large. One reason is that the number of divisors grows rapidly with the number of factors. For example, if n is squarefree with k factors, then tau(n)=2^k. The condition that the 2^k-1 numbers n+d+1 be prime is then quite strong. Another reason is that one or more of the numbers n+d+1 may always be composite. For example, if n=p^5, p prime, then both p^5+p^4+1 and p^5+p+1 are composite.
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LINKS
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EXAMPLE
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a(9)=935=5*11*17 since the divisors d greater than one are {5,11,17,55,85,187,935} and all elements in the set of n+d+1, {941,947,953,991,1021,1123,1871}, are primes.
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MAPLE
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with(numtheory); P:=[]: for w to 1 do for k from 2 do #start at 1, get first element 1 if not isprime(k) and isprime(2*k+1) then S:=divisors(k) minus {1, k}; Q:=map(z-> z+k+1, S); if andmap(isprime, Q) then P:=[op(P), k]; print(nops(P), k, ifactor(k)) fi; fd:=fopen("C:/temp/n+d+1=prime-1st-1000.txt", APPEND); fprintf(fd, "%d ", x); fclose(fd); if nops(P)=1000 then break fi; fi; od od;
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MATHEMATICA
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Select[Range[7000], CompositeQ[#]&&AllTrue[#+1+Rest[Divisors[#]], PrimeQ]&] (* Harvey P. Dale, Mar 14 2023 *)
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PROG
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(PARI) is(n)=if(isprime(n)||n<8, return(0)); fordiv(n, d, if(!isprime(n+d+1), return(0))); 1 \\ Charles R Greathouse IV, Feb 05 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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