

A005235


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


53



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

R. F. Fortune conjectured that a(n) is always prime.
The first 500 terms are primes.  Robert G. Wilson v [The first 2000 terms are prime.  Joerg Arndt, Apr 15 2013]
The strong form of CramÃ©r'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 Cramer's conjecture a(n) = O(prime(n)^2).  Thomas Ordowski, Apr 09 2013
From Pierre CAMI, Sep 08 2017: (Start)
a(n) = prime(i), lim_{N>inf} (Sum_{n=1..N} i) / (Sum_{n=1..N} n) = 3/2.
i/n is always < 6.
Lim_{N>inf} (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} prime(n)) = Pi/2.
a(n) / prime(n) is always < 8.
(End)


REFERENCES

Martin Gardner, The Last Recreations (1997), pp. 19495.
R. K. Guy, Unsolved Problems in Number Theory, Section A2
Richards, Stephen P., 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).


LINKS

Pierre CAMI, Table of n, a(n) for n = 1..3000 (first 2000 terms from T. D. Noe)
C. Banderier, Conjecture checked for n < 1000 [It has been reported that the data given here contains several errors]
S. W. Golomb, The evidence for Fortune's conjecture, Math. Mag. 54 (1981), 209210.
R. K. Guy, Letter to N. J. A. Sloane, 1987
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697712.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697712. [Annotated scanned copy]
Bill McEachen, McEachen Conjecture
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012.
Eric Weisstein's World of Mathematics, Fortunate Prime
R. G. Wilson, V, Letter to N. J. A. Sloane with attachment, Jan 1992


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


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:
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 *)


PROG

(PARI) a(n)=my(P=prod(k=1, n, prime(k))); nextprime(P+2)P \\ Charles R Greathouse IV, Jul 15 2011; corrected by JeanMarc Rebert, Jul 28 2015
(Haskell)
a005235 n = head [m  m < [3, 5 ..], a010051'' (a002110 n + m) == 1]
 Reinhard Zumkeller, Apr 02 2014
(Python)
from operator import mul
def x(n): return 1 + reduce(mul, [prime(i) for i in xrange(1, n + 1)])
print [nextprime(x(n))  x(n) + 1 for n in xrange(1, 101)] # Indranil Ghosh, Mar 23 2017


CROSSREFS

Cf. A046066, A002110, A006862, A035345, A035346, A055211, A129912, A010051, A005408, A038771, A038711.
Sequence in context: A051507 A173145 A060274 * A107664 A321671 A085013
Adjacent sequences: A005232 A005233 A005234 * A005236 A005237 A005238


KEYWORD

nonn,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from Jud McCranie
Guy lists 100 terms, as computed by Stan Wagon


STATUS

approved



