A005235 Fortunate numbers: least m > 1 such that m + prime(n)# is prime, where p# denotes the product of all primes <= p.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

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 first 500 terms are primes. - Robert G. Wilson v. The first 2000 terms are primes. - Joerg Arndt, Apr 15 2013

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

Conjectures from Pierre CAMI, Sep 08 2017: (Start)

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

Examples

			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.
		

References

• Martin Gardner, The Last Recreations, Chapter 12: Strong Laws of Small Primes, Springer-Verlag, 1997, pp. 191-205, especially pp. 194-195.
• 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).
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 114-115.
• David Wells, Prime Numbers: The Most Mysterious Figures In Math, Hoboken, New Jersey: John Wiley & Sons (2005), pp. 108-109.

Links

• Pierre CAMI, Table of n, a(n) for n = 1..3000 (first 2000 terms from T. D. Noe)
• Ray Abrahams and Huon Wardle, Fortune's 'Last Theorem', Cambridge Anthropology, Vol. 23, No. 1 (2002), pp. 60-62.
• Cyril Banderier, Conjecture checked for n < 1000 [It has been reported that the data given here contains several errors]
• C. K. Caldwell, Fortunate number, The Prime Glossary.
• Antonín Čejchan, Michal Křížek, and Lawrence Somer, On Remarkable Properties of Primes Near Factorials and Primorials, Journal of Integer Sequences, Vol. 25 (2022), Article 22.1.4.
• Martin Gardner, Patterns in primes are a clue to the strong law of sma11 numbers, Mathematical Games, Scientific American, Vol. 243, No. 6 (December, 1980), pp. 18-28.
• Solomon W. Golomb, The evidence for Fortune's conjecture, Mathematics Magazine, Vol. 54, No. 4 (1981), pp. 209-210.
• Richard K. Guy, Letter to N. J. A. Sloane, 1987
• Richard K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
• Richard K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
• Bill McEachen, McEachen Conjecture
• Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012.
• Eric Weisstein's World of Mathematics, Fortunate Prime
• Robert G. Wilson v, Letter to N. J. A. Sloane with attachment, Jan 1992

Crossrefs

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

Programs

Haskell

a005235 n = head [m | m <- [3, 5 ..], a010051'' (a002110 n + m) == 1]
-- Reinhard Zumkeller, Apr 02 2014

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:
seq(A[n],n=1..100); # Robert Israel, Dec 02 2015

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 *)

PARI

a(n)=my(P=prod(k=1,n,prime(k)));nextprime(P+2)-P \\ Charles R Greathouse IV, Jul 15 2011; corrected by Jean-Marc Rebert, Jul 28 2015

Python

from sympy import nextprime, primorial
def a(n): psharp = primorial(n); return nextprime(psharp+1) - psharp
print([a(n) for n in range(1, 59)]) # Michael S. Branicky, Jan 15 2022

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

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 n-th primorial plus one and the next prime.

a(n) = A035345(n) - A002110(n). - Jonathan Sondow, Dec 02 2015