A055211 -id:A055211 - cp's OEIS frontend

A058024 a(n) = A051451(n) - A058023(n).

3, 5, 7, 11, 11, 17, 19, 23, 17, 43, 59, 37, 29, 41, 53, 43, 37, 43, 47, 83, 71, 83, 61, 149, 73, 97, 89, 109, 113, 103, 113, 89, 137, 167, 157, 181, 239, 139, 241, 139, 179, 233, 193, 163, 241, 173, 283, 167, 271, 193, 277, 181, 179, 199, 269, 193, 223, 239

Offset: 3

Labos Elemer, Nov 15 2000

nonn

			So far, all terms are primes. The analogy with fortunate numbers (A005235) is clear.

Sean A. Irvine, Table of n, a(n) for n = 3..256

Cf. A000961, A003418, A005235, A051451, A052017, A055211, A057034.

Edited by N. J. A. Sloane, Aug 20 2021

Name corrected by Sean A. Irvine, Jul 18 2022

A082432 a(n) = p - A072181(n), where p is the least prime > A072181(n) + 1.

Original entry on oeis.org

2, 3, 5, 5, 7, 7, 11, 13, 13, 13, 13, 13, 17, 17, 17, 23, 59, 47, 41, 23, 23, 23, 83, 293, 383, 383, 103, 563, 107, 107, 71, 1399, 1399, 1399, 1399, 2803, 983, 983, 983, 10589, 5693, 5693, 19553, 827, 31699, 31699, 33001, 12193

Offset: 1

Naohiro Nomoto, Apr 25 2003

Is a(n) always prime?

			a(4) = 17 - A072181(4) = 17 - 12 = 5.

Cf. A005235, A055211, A037153, A037155, A072181.

a(36)-a(47) from Iain Fox, Nov 23 2017

a(48) from Iain Fox, Nov 29 2017

A098168 Prime index j such that prime(j) = the n-th fortunate number, A005235(n).

Original entry on oeis.org

2, 3, 4, 6, 9, 7, 8, 9, 12, 18, 19, 18, 20, 15, 28, 17, 18, 29, 24, 27, 22, 36, 45, 26, 27, 51, 48, 31, 48, 43, 38, 50, 117, 52, 37, 39, 85, 52, 46, 43, 46, 76, 51, 133, 65, 137, 111, 65, 76, 62, 86, 67, 61, 59, 58, 79, 63, 67, 75, 94, 67, 64, 78, 67, 71, 81, 82, 153, 101, 221

Offset: 1

Pierre CAMI, Aug 30 2004

nonn

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. A005235, A055211, A098166, A098167.

NextPrime[n_Integer] := Block[{k}, k = n + 1; While[ !PrimeQ[k], k++ ]; k]; Fortunate[n_Integer] := Block[{p = Product[Prime[i], {i, 1, n}] + 1, q}, q = NextPrime[p]; q - p + 1]; Table[ PrimePi[ Fortunate[n]], {n, 70}] (* Robert G. Wilson v, Sep 04 2004 *)

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

Better definition from R. J. Mathar, Oct 28 2007

A268608 a(n) is the least m > 1 such that (prime(n)#)^n - m is prime.

Original entry on oeis.org

5, 7, 19, 23, 41, 163, 67, 257, 83, 109, 43, 359, 293, 647, 277, 1567, 983, 419, 1723, 83, 103, 3089, 719, 733, 1723, 457, 331, 2729, 3389, 1123, 863, 1123, 1871, 6211, 19717, 5323, 5749, 419, 887, 811, 617, 2851, 2531, 5023, 6883, 6661, 2879, 16433, 19471

Offset: 2

Alexei Kourbatov, Feb 08 2016

nonn

Here prime(n)# denotes the primorial A002110(n), i.e., the product of the first n primes. a(1) is not defined (there are no primes less than 2).
The definition is similar to Lesser Fortunate numbers (A055211) - but here primorials A002110(n) are raised to the n-th power.
Similar to Fortunate numbers (A005235) and Lesser Fortunate numbers (A055211), the first fifty terms are all prime. (Cf. A263925 where the 6th term is composite.)

			a(2)=5 because m=5 is the least m > 1 such that A002110(2)^2 - m is prime.

Cf. A002110, A005235, A055211, A264050, A263925, A268607.

a(n)=my(s=prod(i=1, n, prime(i))^n); s-precprime(s-2)

A058020 Difference between lcm(1,..,n) and the smallest prime > lcm(1,...,n) + 1, where n runs over A000961, lcm(n) runs through A051451.

Original entry on oeis.org

3, 5, 5, 7, 11, 13, 11, 13, 31, 23, 19, 37, 41, 29, 31, 43, 53, 41, 53, 79, 59, 97, 59, 61, 113, 97, 179, 73, 73, 97, 103, 101, 109, 101, 229, 109, 139, 113, 227, 131, 191, 163, 139, 199, 151, 139, 181, 223, 229, 367, 239, 499, 251, 509, 251, 227, 373, 281, 233

Offset: 1

Labos Elemer, Nov 14 2000

nonn

Analogous to Fortunate numbers and like them so far proved to be primes. This holds for x<=421: if Q is the first follower prime, then Q(421)-lcm(1,...421) = 557. For first some cases when 1+LCM is also a prime, the 2nd primes give 3,5,5,7,11,11,.. deviations, i.e. give primes.

Cf. A000961, A003418, A051451, A057019, A037153, A035346, A005235, A054272, A055211, A037155, A045493, A038710 etc

N=1; for(n=2,1e3, if(isprimepower(n,&p), N*=p; print1(nextprime(N+2)-N", "))) \\ Charles R Greathouse IV, Nov 18 2015

Name corrected by Charles R Greathouse IV, Nov 18 2015

A098169 a(n) = Sum_{i=1..n} A098168(i).

Original entry on oeis.org

2, 5, 9, 15, 24, 31, 39, 48, 60, 78, 97, 115, 135, 150, 178, 195, 213, 242, 266, 293, 315, 351, 396, 422, 449, 500, 548, 579, 627, 670, 708, 758, 875, 927, 964, 1003, 1088, 1140, 1186, 1229, 1275, 1351, 1402, 1535, 1600, 1737, 1848, 1913, 1989, 2051, 2137, 2204

Offset: 1

Pierre CAMI, Aug 30 2004

nonn

Cf. A005235, A055211, A098166, A098167, A098168.

NextPrime[n_Integer] := Block[{k}, k = n + 1; While[ !PrimeQ[k], k++ ]; k]; Fortunate[n_Integer] := Block[{p = Product[Prime[i], {i, 1, n}] + 1, q}, q = NextPrime[p]; q - p + 1]; t = Table[ PrimePi[ Fortunate[n]], {n, 70}]; Table[Plus @@ Take[t, n], {n, 52}] (* Robert G. Wilson v, Sep 04 2004 *)

Conjecture: a(n)/triangular(n) -> Log(e*Pi/2).

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

A305099 Least prime m such that either prime(n)# - m is prime or prime(n)# + m is prime, where p# denotes the product of all primes <= p.

Original entry on oeis.org

3, 3, 7, 11, 13, 17, 19, 23, 37, 41, 67, 59, 47, 47, 67, 59, 61, 89, 89, 103, 79, 83, 89, 97, 103, 131, 113, 127, 223, 191, 163, 179, 389, 239, 151, 167, 173, 239, 199, 191, 199, 223, 233, 593, 293, 457, 227, 311, 373, 257, 307, 313, 283, 277, 271, 307, 307

Offset: 1

David Nicolas Lopez, May 22 2018

nonn

Since it is known that the first 2000 terms of A005235 are primes, and the first 1000 terms of A055211 are primes, then the first 1000 terms of this sequence are also the least m > 1 such that prime(n)# - m is prime or prime(n)# + m is prime. - Amiram Eldar, Nov 02 2018

			For n = 6, the sixth primorial is 30030. The nearest prime such that p(6)# plus or minus prime equals its 30030's closest prime is equal to 17 because 30030+17=30047 which is prime or 30030 - 17 = 30013 which is also prime. Given that we only care about the smallest prime distance to the closest prime to the primorial, then we return 17.
For n = 7, the seventh primorial is 510510. The closest prime to the primorial is 510529 which is 510510 + 19; therefore 19 is in the sequence.

Martin Gardner, The Last Recreations (1997), pp. 194-195.
R. K. Guy, Unsolved Problems in Number Theory, Section A2
Stephen P. Richards, A Number For Your Thoughts, 1982, p. 200.

S. W. Golomb, Evidence of Fortunes conjecture, Mathematics Magazine, Vol. 54, No. 4 (Sep., 1981), pp. 209-210.
The Prime Glossary, Fortunate and lesser fortunate numbers

Cf. A005235, A055211.

primorial[n_] := Product[Prime[i], {i, 1, n}]; a[n_] := Module[{k = 2, pr = primorial[n]}, While[! PrimeQ[pr - k] && ! PrimeQ[pr + k], k = NextPrime[k]]; k]; Array[a, 57] (* Amiram Eldar, Oct 31 2018 *)

a(n) = { my(pr = prod(k=1, n, prime(k)), m=2); while (!isprime(pr-m) && !isprime(pr+m), m = nextprime(m+1)); m;} \\ Michel Marcus, Nov 02 2018

# returns quasi-fortunate-numbers up to n
def generateQFN(n):
    quasi_fortunate_numbers = []
    primorialArray = []
    prime = Primes()
    num_length = n+1
    primorial = 1
    for i in range(num_length):
        primorial *= prime[i]
        primorialArray.append(primorial)
    for primorials in primorialArray:
        num = 0
        while num < num_length:
            if is_prime(primorials+prime[num]):
                quasi_fortunate_numbers.append(prime[num])
                break
            elif is_prime(primorials-prime[num]):
                quasi_fortunate_numbers.append(prime[num])
                break
            num += 1
    return quasi_fortunate_numbers
generateQFN(7)

a(n) = min(A005235(n), A055211(n)), for n > 1.

More terms from Amiram Eldar, Oct 31 2018

cp's OEIS Frontend

A058024 a(n) = A051451(n) - A058023(n).

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A082432 a(n) = p - A072181(n), where p is the least prime > A072181(n) + 1.

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A098168 Prime index j such that prime(j) = the n-th fortunate number, A005235(n).

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A268608 a(n) is the least m > 1 such that (prime(n)#)^n - m is prime.

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A058020 Difference between lcm(1,..,n) and the smallest prime > lcm(1,...,n) + 1, where n runs over A000961, lcm(n) runs through A051451.

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A098169 a(n) = Sum_{i=1..n} A098168(i).

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A305099 Least prime m such that either prime(n)# - m is prime or prime(n)# + m is prime, where p# denotes the product of all primes <= p.

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