

A005235


Fortunate numbers: least m > 1 such that m + prime(n)# is prime, where p# denotes the product of all primes <= p.
(Formerly M2418)


54



3, 5, 7, 13, 23, 17, 19, 23, 37, 61, 67, 61, 71, 47, 107, 59, 61, 109, 89, 103, 79, 151, 197, 101, 103, 233, 223, 127, 223, 191, 163, 229, 643, 239, 157, 167, 439, 239, 199, 191, 199, 383, 233, 751, 313, 773, 607, 313, 383, 293, 443, 331, 283, 277, 271, 401, 307, 331
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OFFSET

1,1


COMMENTS

Reo F. Fortune conjectured that a(n) is always prime.
You might be searching for Fortunate Primes, which is an alternative name for this sequence. It is not the official name yet, because it is possible, although unlikely, that not all the terms are primes.  N. J. A. Sloane, Sep 30 2020
The strong form of Cramér's conjecture implies that a(n) is a prime for n > 1618, as previously noted by Golomb.  Charles R Greathouse IV, Jul 05 2011
a(n) is the smallest m such that m > 1 and A002110(n) + m is prime. For every n, a(n) must be greater than prime(n+1)  1.  Farideh Firoozbakht, Aug 20 2003
If a(n) < prime(n+1)^2 then a(n) is prime. According to Cramér's conjecture a(n) = O(prime(n)^2).  Thomas Ordowski, Apr 09 2013
If all terms are prime, then lim_{N>oo} (Sum_{n=1..N} primepi(a(n))) / (Sum_{n=1..N} n) = 3/2, and primepi(a(n))/n < 6 for all n.
Limit_{N>oo} (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} prime(n)) = Pi/2.
a(n)/prime(n) < 8 for all n. (End)
Conjecture: Limit_{N>oo} (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} prime(n)) = 3/2.  Alain Rocchelli, Dec 24 2022
The name "Fortunate numbers" was coined by Golomb (1981) after the New Zealand social anthropologist Reo Franklin Fortune (1903  1979). According to Golomb, Fortune's conjecture first appeared in print in Martin Gardner's Mathematical Games column in 1980.  Amiram Eldar, Aug 25 2020


REFERENCES

Martin Gardner, The Last Recreations (1997), pp. 194195.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 1994, Section A2, p. 11.
Stephen P. Richards, A Number For Your Thoughts, 1982, p. 200.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, Prime Numbers: The Most Mysterious Figures In Math, Hoboken, New Jersey: John Wiley & Sons (2005), pp. 108109.


LINKS



FORMULA

If x(n) = 1 + Product_{i=1..n} prime(i), q(n) = least prime > x(n), then a(n) = q(n)  x(n) + 1.
a(n) = 1 + the difference between the nth primorial plus one and the next prime.


EXAMPLE

a(4) = 13 because P_4# = 2*3*5*7 = 210, plus one is 211, the next prime is 223 and the difference between 210 and 223 is 13.


MAPLE

Primorial:= 2:
p:= 2:
A[1]:= 3:
for n from 2 to 100 do
p:= nextprime(p);
Primorial:= Primorial * p;
A[n]:= nextprime(Primorial+p+1)Primorial;
od:


MATHEMATICA

NPrime[n_Integer] := Module[{k}, k = n + 1; While[! PrimeQ[k], k++]; k]; Fortunate[n_Integer] := Module[{p, q}, p = Product[Prime[i], {i, 1, n}] + 1; q = NPrime[p]; q  p + 1]; Table[Fortunate[n], {n, 60}]
r[n_] := (For[m = (Prime[n + 1] + 1)/2, ! PrimeQ[Product[Prime[k], {k, n}] + 2 m  1], m++]; 2 m  1); Table[r[n], {n, 60}]
FN[n_] := Times @@ Prime[Range[n]]; Table[NextPrime[FN[k] + 1]  FN[k], {k, 60}] (* Jayanta Basu, Apr 24 2013 *)
NextPrime[#]#+1&/@(Rest[FoldList[Times, 1, Prime[Range[60]]]]+1) (* Harvey P. Dale, Dec 15 2013 *)


PROG

(Haskell)
a005235 n = head [m  m < [3, 5 ..], a010051'' (a002110 n + m) == 1]
(Sage)
def P(n): return prod(nth_prime(k) for k in range(1, n + 1))
it = (P(n) for n in range(1, 31))
print([next_prime(Pn + 2)  Pn for Pn in it]) # F. Chapoton, Apr 28 2020
(Python)
from sympy import nextprime, primorial
def a(n): psharp = primorial(n); return nextprime(psharp+1)  psharp


CROSSREFS

Cf. A046066, A002110, A006862, A035345, A035346, A055211, A129912, A010051, A005408, A038771, A038711.


KEYWORD

nonn,nice


AUTHOR



STATUS

approved



