A055211 -id:A055211 - cp's OEIS frontend

A098166 a(n) is the prime index of the Lesser Fortunate number A055211(n), or 0 if A055211(n) is not a prime.

2, 4, 5, 6, 7, 10, 9, 14, 13, 21, 17, 15, 24, 19, 21, 28, 24, 26, 31, 25, 23, 24, 25, 54, 32, 30, 36, 56, 54, 48, 41, 77, 60, 36, 45, 40, 52, 51, 43, 48, 48, 62, 108, 62, 88, 49, 64, 74, 55, 63, 65, 111, 69, 66, 63, 123, 91, 66, 83, 92, 97, 110, 67, 75, 99, 87

Offset: 2

Pierre CAMI, Aug 30 2004

nonn

Jinyuan Wang, Table of n, a(n) for n = 2..2000
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.

Cf. A000720, A055211, A098167.

a(n) = {my(k = prod(j=1, n, prime(j))); p = k-precprime(k-2); if(isprime(p), primepi(p), 0); } \\ Jinyuan Wang, Oct 02 2019

a(n) = A000720(A055211(n)) iff A055211(n) is a prime. - Michel Marcus, Oct 05 2019

Name edited by and more terms from Jinyuan Wang, Oct 02 2019

A087201 a(n) is the smallest m such that m > A055211(n) and A002110(n)-m is prime.

Original entry on oeis.org

11, 13, 17, 19, 47, 37, 61, 67, 79, 107, 53, 149, 97, 89, 109, 223, 107, 179, 181, 101, 197, 101, 257, 139, 137, 197, 313, 257, 257, 223, 449, 373, 233, 463, 479, 409, 257, 409, 383, 317, 587, 607, 401, 463, 347, 313, 751, 313, 443, 349, 809, 661, 587, 367

Offset: 3

Farideh Firoozbakht, Aug 27 2003

a(1) and a(2) are not defined. a(n) is the second m (first m is A055211(n)) such that m > 1 and A002110(n)-m is prime. I guess every term of this sequence (compare the conjecture about A055211) is prime. I checked this conjecture for n < 418.

Cf. A055211, A002110, A087200.

A055211[n_] := (For[m=2, !PrimeQ[Product[Prime[k], {k, n}]-m], m++ ]; m); a[n_] := (For[m=A055211[n]+1, !PrimeQ[Product[Prime[k], {k, n}]-m], m++ ]; m); Table[a[n], {n, 3, 62}]

A055211[n_] := (For[m=2, !PrimeQ[Product[Prime[k], {k, n}]-m], m++ ]; m); a[n_] := (For[m=A055211[n]+1, !PrimeQ[Product[Prime[k], {k, n}]-m], m++ ]; m);

A097588 a(n)=floor(P(k)/P(n)) with P(k)=lesser Fortunate numbers for P(n)# (A055211).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2

Offset: 2

Pierre CAMI, Aug 29 2004

Conjecture : a(n) is always < 8.

			a(11) = floor(73/31) = 2, A055211(11) = 73 prime(11) = 31,
2*3*5*7*11*13*17*19*23*29*31-73 is prime and 73 is the smallest odd number k such that p(11)# - k is prime.

Pierre CAMI, Table of n, a(n) for n = 2..2000

Cf. A055211.

A097590 a(n) = Sum_{i=2..n} A055211(i).

Original entry on oeis.org

3, 10, 21, 34, 51, 80, 103, 146, 187, 260, 319, 366, 455, 522, 595, 702, 791, 892, 1019, 1116, 1199, 1288, 1385, 1636, 1767, 1880, 2031, 2294, 2545, 2768, 2947, 3336, 3617, 3768, 3965, 4138, 4377, 4610, 4801, 5024, 5247, 5540, 6133, 6426, 6883, 7110, 7421

Offset: 2

Pierre CAMI, Aug 29 2004

nonn

Cf. A007504, A055211.

PrevPrime[ n_Integer] := Block[ {k = n - 1}, While[ ! PrimeQ[ k ], k-- ]; k ]; LF[ n_Integer ] := Block[ { p = Product[ Prime[ i ], {i, 1, n} ] - 1, q}, q = PrevPrime[ p ]; p - q + 1 ]; t = Table[LF[ n ], {n, 2, 48}]; Table[Plus @@ Take[t, n], {n, 47}] (* Robert G. Wilson v, Sep 04 2004 *)

Let F(n) := a(n)/A007504(n). Conjecture: as n tends to infinity F(n) tends to Pi/2 with Pi=3.14159......

More terms from Robert G. Wilson v, Sep 04 2004

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

Original entry on oeis.org

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

N. J. A. Sloane

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

			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.

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.

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

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

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

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

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 *)

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

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

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

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

A179975 Smallest k such that k*10^n is a sum of two successive primes.

Original entry on oeis.org

5, 3, 1, 6, 6, 6, 14, 6, 9, 19, 21, 21, 42, 93, 21, 6, 11, 2, 12, 111, 37, 39, 63, 38, 42, 24, 15, 15, 60, 6, 39, 82, 47, 58, 337, 49, 72, 25, 34, 21, 6, 107, 128, 96, 20, 2, 63, 231, 70, 7, 62, 144, 28, 151, 157, 33, 98, 55, 134, 162, 87, 201, 124, 303, 64, 106, 130, 13, 43

Offset: 0

Zak Seidov, Aug 04 2010

nonn

From Robert G. Wilson v, Aug 11 2010: (Start)
A179975 n's such that a(n)=1: 3, 335, ..., .
A179975 First occurrence of k: 3, 18, 2, ???, 1, 4, 50, 162, 9, 335, 17, 19, 68, 7, 27, ..., .
Records: 5, 6, 14, 19, 21, 42, 93, 111, 337, 449, 862, 1049, 1062, 1122, 1280, 2278, 3168, 4290, ..., . (End)

			a(0)=5 because 5=2+3
a(1)=3 because 30=13+17
a(2)=1 because 100=47+53
a(3)=6 because 6000=2999+3001.

Robert G. Wilson v, Table of n, a(n) for n = 0..400.
Dario Alejandro Alpern, Brilliant numbers

Cf. A064397, A071220, A074924, A074925.

Cf. A033873, A033874, A005235, A055211, A038804, A084475.

Join[{5,3},Reap[Do[Do[n=10^m k; If[n==PreviousPrime[n/2]+NextPrime[n/2],Sow[k];Break[]],{k,2000}],{m,2,50}]][[2,1]]]
f[n_] := Block[{k = 1, tn = 10^n}, While[h = k*tn/2; NextPrime[h, -1] + NextPrime@h != k*tn, k++ ]; k]; f[1] = 3; Array[f, 70, 0] (* Robert G. Wilson v, Aug 11 2010 *)

More terms from Robert G. Wilson v, Aug 11 2010

A060270 Distance of n-th primorial from previous prime.

Original entry on oeis.org

1, 1, 11, 1, 1, 29, 23, 43, 41, 73, 59, 1, 89, 67, 73, 107, 89, 101, 127, 97, 83, 89, 1, 251, 131, 113, 151, 263, 251, 223, 179, 389, 281, 151, 197, 173, 239, 233, 191, 223, 223, 293, 593, 293, 457, 227, 311, 373, 257, 307, 313, 607, 347, 317, 307, 677, 467, 317

Offset: 2

Labos Elemer, Mar 23 2001

nonn

			Before 7th primorial 510481 is the largest prime. Its distance from 510510 is a(7)=29.

Robert G. Wilson v, Table of n, a(n) for n = 2..1000

Cf. A002110, A038710, A007014, A058044, A038711, A055211.

[seq(product(ithprime(j),j=1..n)-prevprime(product(ithprime(j),j=1..n)), n=2..50)];

Map[# - NextPrime[#, -1] &, Rest@ FoldList[Times, Prime@ Range[59]]] (* Michael De Vlieger, Aug 10 2023 *)

a(n) = my(P=vecprod(primes(n))); P-precprime(P-1); \\ Michel Marcus, Aug 11 2023

a(n)=1 for n=2, 3, 5, 6, 13, 24, 66, 68, 167, ... (A057704); a(n)=A055211(n) otherwise. - Jeppe Stig Nielsen, Oct 31 2003

More terms from Jeppe Stig Nielsen, Oct 31 2003

A037155 a(n) = n!-p, where p is the largest prime < (n!-1).

Original entry on oeis.org

3, 5, 7, 11, 17, 31, 13, 11, 13, 13, 23, 17, 47, 53, 59, 41, 101, 31, 31, 73, 89, 73, 149, 37, 43, 101, 31, 79, 61, 163, 47, 193, 113, 127, 97, 79, 73, 83, 131, 79, 109, 109, 53, 89, 79, 103, 59, 97, 179, 67, 59, 127, 61, 461, 277, 109, 137, 139, 71, 71, 101, 359

Offset: 3

Jud McCranie

nonn

Analogous to the lesser Fortunate numbers and like them, all entries so far are primes.

			a(4) = 4!-19 = 24-19 = 5.

Charles R Greathouse IV, Table of n, a(n) for n = 3..1000
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.

Cf. A055211.

PrevPrime[ n_Integer ] := (k=n-1; While[ !PrimeQ[ k ], k-- ]; Return[ k ]); f[ n_Integer ] := (p = n! - 1; q = NextPrime[ p ]; Return[ p - q + 1 ]); Table[ f[ n ], {n, 3, 75} ]
f[n_]:=Module[{nf=n!},nf-NextPrime[nf-1,-1]];f/@Range[3,90]  (* Harvey P. Dale, Mar 20 2011 *)

a(n)=my(N=n!); N-precprime(N-3) \\ Charles R Greathouse IV, Jan 28 2018

from sympy import factorial, prevprime
def a(n): fn = factorial(n); return fn - prevprime(fn-1)
print([a(n) for n in range(3, 65)]) # Michael S. Branicky, May 22 2022

a(n) >= n. - Seiichi Manyama, Mar 21 2018

More terms from James Sellers, Jul 06 2000

A098167 Partial sums of A098166.

Original entry on oeis.org

2, 6, 11, 17, 24, 34, 43, 57, 70, 91, 108, 123, 147, 166, 187, 215, 239, 265, 296, 321, 344, 368, 393, 447, 479, 509, 545, 601, 655, 703, 744, 821, 881, 917, 962, 1002, 1054, 1105, 1148, 1196, 1244, 1306, 1414, 1476, 1564, 1613, 1677, 1751, 1806, 1869, 1934, 2045

Offset: 2

Pierre CAMI, Aug 30 2004

nonn

Cf. A000217, A055211, A098166.

More terms from Jinyuan Wang, Mar 05 2020

A268607 a(n) is the least m > 1 such that 2^n - m is prime.

Original entry on oeis.org

2, 3, 3, 3, 3, 15, 5, 3, 3, 9, 3, 13, 3, 19, 15, 9, 5, 19, 3, 9, 3, 15, 3, 39, 5, 39, 57, 3, 35, 19, 5, 9, 41, 31, 5, 25, 45, 7, 87, 21, 11, 57, 17, 55, 21, 115, 59, 81, 27, 129, 47, 111, 33, 55, 5, 13, 27, 55, 93, 31, 57, 25, 59, 49, 5, 19, 23, 19, 35, 231, 93

Offset: 2

Alexei Kourbatov, Feb 08 2016

nonn

a(1) is not defined (there are no primes less than 2).

The definition is similar to Lesser Fortunate numbers (A055211) but uses 2^n instead of primorials A002110(n).

			a(7)=15 because m=15 is the least m > 1 such that 2^7 - m is prime.

Paolo Xausa, Table of n, a(n) for n = 2..1000

Cf. A005235, A013603, A055211, A264050, A263925, A268608.

Map[# - NextPrime[#-1, -1] &, 2^Range[2, 100]] (* Paolo Xausa, Mar 10 2025 *)

PARI
```
a(n)=2^n-precprime(2^n-2)
```

a(n) = A013603(n), if A013603(n) > 1. - Jason Yuen, Mar 10 2025

cp's OEIS Frontend

A098166 a(n) is the prime index of the Lesser Fortunate number A055211(n), or 0 if A055211(n) is not a prime.

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A087201 a(n) is the smallest m such that m > A055211(n) and A002110(n)-m is prime.

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A097588 a(n)=floor(P(k)/P(n)) with P(k)=lesser Fortunate numbers for P(n)# (A055211).

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A097590 a(n) = Sum_{i=2..n} A055211(i).

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

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A179975 Smallest k such that k*10^n is a sum of two successive primes.

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A060270 Distance of n-th primorial from previous prime.

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A037155 a(n) = n!-p, where p is the largest prime < (n!-1).

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