A293756 -id:A293756 - cp's OEIS frontend

A294925 a(n) is the smallest number k with n prime factors such that p + k/p is prime for every prime p | k.

2, 6, 30, 210, 15810, 292110, 16893030, 984016110, 17088913842, 2446241358990, 1098013758964122

Offset: 1

Thomas Ordowski, Nov 11 2017

Such k is an even squarefree number.

Conjecture: the sequence is infinite.

			a(2) = 6 because k = 2*3 = 6 is the smallest number with 2 prime factors such that 2 + 6/2 = 3 + 6/3 = 5 is prime.
From _Michael De Vlieger_, Nov 13 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            15810   1, 1, 1, 4, 4
   6           292110   1, 1, 1, 1, 2, 22
   7         16893030   1, 1, 1, 1, 1, 15, 7
   8        984016110   1, 1, 1, 1, 1,  5, 2, 66
   9      17088913842   1, 1, 2, 1, 1,  1, 1,  1, 67
  10    2446241358990   1, 1, 1, 2, 1,  2, 2,  3,  1, 93
  11 1098013758964122   1, 1, 2, 1, 1,  3, 2,  8,  3, 22, 10
(End)

Cf. A293756.

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

a(5)-a(7) from Michel Marcus, Nov 11 2017
a(8) from Michel Marcus, Nov 12 2017
a(9)-a(10) from Michael De Vlieger, Nov 13 2017
a(11) (and update of table in Example section) from Jon E. Schoenfield, Nov 19 2017

A295124 a(n) = smallest number k with n prime factors such that 2d + k/d is prime for every d | k.

Original entry on oeis.org

1, 3, 15, 105, 93081

Offset: 0

Thomas Ordowski, Nov 15 2017

Such k must be an odd squarefree number.
a(n) has 2^n divisors and each gives another prime.
Conjecture: the sequence is infinite. It is hard to believe!
a(n) is the smallest k such that A088627(k) = A000005(k) = 2^n.

Cf. A088627, A293756.

Subsequence of A244520 (2d + k/d is prime for every d|k).

a(n) = A293756(n+1)/2.

a(4) from Michel Marcus, Nov 15 2017

A295741 a(n) is smallest divisor d of the n-th primorial such that d + prime(n)#/d is prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 5, 6, 10, 2, 1, 2, 11, 7, 2, 41, 2, 5, 5, 5, 3, 51, 10, 3, 14, 37, 10, 7, 2, 17, 17, 62, 21, 3, 38, 17, 47, 13, 17, 13, 7, 21, 83, 10, 138, 10, 26, 2, 7, 19, 31, 15, 13, 5, 23, 103, 19, 5, 95, 61, 6, 15, 2, 35, 13, 19, 65, 39, 7, 43, 73, 13, 26, 97, 1, 91, 2, 78, 19, 39, 34, 167, 82, 10, 38, 66, 58, 2, 46, 71

Offset: 0

Thomas Ordowski and Robert G. Wilson v, Nov 26 2017

nonn

Conjecture: a(n) < prime(n)^2 for n > 0.
a(n) = 1 iff n belongs to A014545.
Inspired by A293756.

			a(3) = 1 because 1 + prime(3)#/1 = 1 + 2*3*5 = 31 and 31 is a prime;
a(6) = 2 because 2 + prime(6)#/2 = 2 + 2*3*5*7*11*13/2 = 15017 and that is a prime;
a(7) = 5 because 5 + prime(7)#/5 = 5 + 2*3*5*7*11*13*17/5 = 102107 which is a prime; etc.

Robert G. Wilson v, Table of n, a(n) for n = 0..1000

Cf. A002110, A005235, A014545.

f[n_] := Block[{d = 1, p = Fold[Times, 1, Prime@ Range@ n]}, While[ !PrimeQ[d + p/d], d++]; d]; Array[f, 90]

a(n)=my(P=vecprod(primes(n))); for(d=1,P, if(P%d==0 && ispseudoprime(d+P/d), return(d))) \\ Charles R Greathouse IV, Nov 27 2017

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A294925 a(n) is the smallest number k with n prime factors such that p + k/p is prime for every prime p | k.

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A295124 a(n) = smallest number k with n prime factors such that 2d + k/d is prime for every d | k.

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A295741 a(n) is smallest divisor d of the n-th primorial such that d + prime(n)#/d is prime.

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