A294925 -id:A294925 - cp's OEIS frontend

A140775 Numbers k > 1 such that p + k/p is prime for every prime p that divides k.

2, 6, 10, 22, 30, 34, 42, 58, 70, 78, 82, 102, 118, 130, 142, 190, 202, 210, 214, 274, 298, 310, 322, 330, 358, 382, 394, 442, 454, 462, 478, 510, 538, 562, 582, 610, 622, 658, 694, 714, 730, 742, 790, 838, 862, 922, 930, 970, 1002, 1038, 1042, 1110, 1138, 1198

Offset: 1

Leroy Quet, May 29 2008

nonn

All terms of this sequence are even and squarefree.

The only term == 2 (mod 3) is 2. - Robert Israel, Jan 09 2024

			The primes dividing 70 are 2, 5, 7. Now, 2 + 70/2 = 37; 5 + 70/5 = 19; 7 + 70/7 = 17. Since 37, 19 and 17 are each prime, then 70 is included in this sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A140776, A140777, A294925.

filter:= t -> andmap(p -> isprime(p+t/p), numtheory:-factorset(t)):
select(filter, [seq(i,i=2..2000,4)]); # Robert Israel, Jan 09 2024

fQ[n_] := Block[{p = First@ Transpose@ FactorInteger@ n}, Union@ PrimeQ[p + n/p] == {True}]; Select[ Range[2, 1221], fQ@# &] (* Robert G. Wilson v, May 30 2008 *)
pnpQ[n_]:=AllTrue[#+n/#&/@Transpose[FactorInteger[n]][[1]],PrimeQ]; Select[ Range[2,1200],pnpQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 15 2016 *)

More terms from Robert G. Wilson v, May 30 2008

Definition edited by Robert Israel, Jan 09 2024

A293756 a(n) = smallest number k with n prime factors such that d + k/d is prime for every d | k.

Original entry on oeis.org

1, 2, 6, 30, 210, 186162

Offset: 0

Thomas Ordowski, Nov 11 2017

For n > 0, a(n) is even and squarefree.
For n > 0, a(n) gives 2^(n-1) distinct primes.
If the k-tuple conjecture is true, then this sequence is infinite. - Carl Pomerance, Nov 12 2017
a(n) is the least integer k with n prime divisors such that A282849(k) = A000005(k). - Michel Marcus, Nov 13 2017
a(n) is the smallest k with n prime factors such that A282849(k) = 2^n. - Thomas Ordowski, Nov 13 2017
a(6), if it exists, has a prime divisor greater than 10^3. - Arkadiusz Wesolowski, Nov 14 2017

			a(2) = 2*3 = 6 because k = 6 is the smallest number with 2 prime factors such that for d = {1, 2, 3, 6} we have 1 + 6/1 = 6 + 6/6 = 7 is prime and 2 + 6/2 = 3 + 6/3 = 5 is prime.
From _Michael De Vlieger_, Nov 14 2017: (Start)
First differences of prime indices of a(n):
n       a(n)   A287352(a(n))
-----------------------------
1         2    1
2         6    1, 1
3        30    1, 1, 1
4       210    1, 1, 1, 1
5    186162    1, 1, 6, 1, 11
(End)

Eric Weisstein's World of Mathematics, k-Tuple Conjecture

Subsequence of A080715 (d + k/d is prime for every d|k).

Cf. A103787, A282849, A294925, A295124, A295169.

with(numtheory): P:=proc(q) local a,b,j,k,n,ok; print(1);for n from 1 to q do for k from 2 to q do a:=ifactors(k)[2]; a:=add(a[j][2],j=1..nops(a)); if a=n then b:=divisors(k); ok:=1;
for j from 1 to nops(b) do if not isprime(b[j]+k/b[j]) then ok:=0; break; fi; od; if ok=1 then print(k); break; fi; fi; od; od; end: P(10^8); # Paolo P. Lava, Nov 16 2017

isok(k, n)=if (!issquarefree(k), return (0)); if (omega(k) != n, return (0)); fordiv(k, d, if (!isprime(d+k/d), return(0))); 1;
a(n) = {my(k=1); while( !isok(k, n), k++); k;} \\ Michel Marcus, Nov 11 2017

a(n) = 2*A295124(n-1) for n > 0. - Thomas Ordowski, Nov 15 2017

a(5) from Michel Marcus, Nov 11 2017

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A140775 Numbers k > 1 such that p + k/p is prime for every prime p that divides k.

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