A005235 -id:A005235 - cp's OEIS frontend

A376236 Ludic Fortunate numbers: a(n) = N(P(n)+1) - P(n), where N(x) = min {L in A003309 | L > x} is the next larger ludic number and P(n) = Prod_{k=1..n} A003309[n].

2, 3, 5, 7, 11, 23, 17, 37, 61

Offset: 1

M. F. Hasler, Nov 02 2024

Generalization of Fortunate numbers A005235 to ludic numbers A003309 instead of primes.

Are all terms ludic numbers? Will all ludic numbers > 1 appear in this sequence?

			The first ludic numbers are A003309 = 1, 2, 3, 5, 7, 11, 13, 17, 23, 25, 29, 37, ...
Their cumulative products are P = 1, 2, 6, 30, 210, 2310, 30030, 510510, 11741730, ...
Up to 510510 they are the same as primorials A002110 because ludic numbers > 1 coincide with the primes up to 17.
The first term of this sequence is a(1) = N(1 + P(1)) - P(1) = N(2) - 1 = 3 - 1 = 2, where we write N(x) for the least A003309(k) > x.
The second term is a(2) = N(1 + P(2)) - P(2) = N(3) - 2 = 5 - 2 = 3.
Then a(3) = N(1 + P(3)) - P(3) = N(7) - 6 = 11 - 6 = 5.
Then a(4) = N(1 + P(4)) - P(4) = N(31) - 30 = 37 - 30 = 7, still as in A005235 (because that sequence also uses the least strictly larger prime).
Then a(5) = N(1 + P(5)) - P(5) = N(211) - 210 = 221 - 210 = 11 (while A005235 has nextprime(211) - 210 = 223 - 210 = 13, where again it does not matter that 211 is a prime).

Cf. A003309 (ludic numbers), A376237 (ludic factorials), A005235 (Fortunate numbers: same idea with primes).

A376236(n)=next_A003309(1+n=A376237(n))-n

a(9) from Pontus von Brömssen, Nov 03 2024

A038771 a(n) is the smallest composite number c such that A002110(n) + c is prime.

Original entry on oeis.org

4, 9, 25, 49, 121, 221, 289, 529, 667, 899, 1147, 1591, 2021, 1849, 2773, 3551, 4087, 4819, 4757, 5041, 7519, 7663, 8549, 9991, 10379, 13231, 11227, 14659, 11881, 21877, 25283, 18209, 22331, 20989, 22499, 25591, 27221, 29503, 31313, 34547

Offset: 0

Labos Elemer, May 04 2000

nonn

The lower "envelope" of the sequence is prime(n+1)^2. See also Fortune-conjecture (A005235).
For some n, c=prime(n+1)^2; for others, it is larger, even not necessarily divisible by prime(n+1). E.g., at n=11, prime(11)=31 and a(11) = 1591 = 37*43 = prime(12)*prime(14), while for n=59, a(59) = 97969 = 313^2 = prime(65)^2, etc. Adding these to the suitable primorial numbers, primes are obtained.
Conjecture: lim inf_{n->oo} a(n)/prime(n+1)^2 = 1 < lim sup_{n->oo} a(n)/prime(n+1)^2 = 2. - Charles R Greathouse IV and Thomas Ordowski, Apr 24 2015
Conjecture: all the terms in this sequence have exactly two prime factors. This conjecture is true for the first 133 terms. - Dmitry Kamenetsky, Jan 06 2019

Dmitry Kamenetsky, Table of n, a(n) for n = 0..133

Cf. A002110, A054757, A054758, A005235.

a(n) = {my(q = prod(i=1, n, prime(i))); forcomposite(c = 1,, if (isprime(q+c), return(c);););} \\ Michel Marcus, May 24 2015

Name edited by Tom Edgar, Jun 08 2015

a(0) prepended by Dmitry Kamenetsky, Jan 06 2019

A038773 a(n) is the smallest prime of the form Q + c, where Q is the n-th primorial and c is a composite >= prime(n+1)^2.

Original entry on oeis.org

11, 31, 79, 331, 2531, 30319, 511039, 9700357, 223093769, 6469694377, 200560491721, 7420738136831, 304250263529059, 13082761331672803, 614889782588494961, 32589158477190048817, 1922760350154212643889, 117288381359406970988027, 7858321551080267055884131, 557940830126698960967422909

Offset: 1

Labos Elemer, May 04 2000

nonn

Between 2310 and 2531 there are 26 primes (2311, ..., 2521), all of which are of the form (primorial + prime). (2311 = 2 + 2309 (prime) = 2*3*5 + 2281 (prime); each of the other 25 primes is of the form 2*3*5*7*11 + prime.)

Observe that a(2) = 31 = 2*3 + 5^2 = 2*3*5 + 1, so it has two "primorial forms".

			At n=5, the 5th primorial is A002110(5)=2310 and 2310 + 13*17 = 2310 + 221 = 2531 is the prime that meets the criteria of the definition.

Michael De Vlieger, Table of n, a(n) for n = 1..100

Cf. A002110, A054757, A054758, A005235.

Array[Block[{Q = Product[Prime@ i, {i, #}], c = Prime[# + 1]^2}, While[Nand[PrimeQ[Q + c], CompositeQ@ c], c++]; Q + c] &, 17] (* Michael De Vlieger, May 22 2018 *)

a(n) = {my(pr = prod(k=1, n, prime(k)), c = prime(n+1)^2); while (isprime(c) || !isprime(pr + c), c++); pr + c;} \\ Michel Marcus, May 26 2018

Edited by Jon E. Schoenfield, May 22 2018

More terms from Michael De Vlieger, May 22 2018

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

A045493 Unfortunate primes - primes that are not Fortunate numbers.

Original entry on oeis.org

2, 11, 29, 31, 41, 43, 53, 73, 83, 97, 113, 131, 137, 139, 149, 173, 179, 181, 193, 211, 227, 241, 251, 257, 263, 269, 281, 317, 337, 347, 349, 359, 367, 389, 431, 433, 449, 463, 467, 479, 487, 503, 521, 541, 557, 563, 569, 571, 577, 587, 593, 599, 647, 659

Offset: 1

Jud McCranie

nonn

Complement of A046066 in the primes. - Charles R Greathouse IV, Sep 06 2012

Martin Gardner, "The Last Recreations", chapter 12 discusses Fortunate numbers.

Eric Weisstein's World of Mathematics, Fortunate Prime

Cf. A005235, A046066.

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

Original entry on oeis.org

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

A160433 a(n) is the least number k such that (k-th prime after n!+1)-n! is not a prime.

Original entry on oeis.org

2, 2, 3, 7, 8, 15, 8, 18, 16, 19, 12, 20, 11, 8, 11, 6, 12, 23, 24, 15, 31, 21, 27, 15, 16, 26, 25, 17, 17, 29, 20, 27, 27, 30, 23, 16, 28, 23, 25, 29, 15, 24, 19, 36, 36, 39, 15, 36, 24, 44, 35, 29, 27, 25, 36, 22, 37, 31, 32, 41, 29, 55, 27, 45, 29, 59, 34, 37, 24, 49, 25, 40

Offset: 0

Frederick Magata (frederick.magata(AT)web.de), May 13 2009

nonn

The conjectures from A037153 and A087202 can be rephrased using a(n):
Is a(n)>=2 for all n>=0 and a(n)>=3 for all n>=2?
Also compare this with the conjecture on the fortunate numbers A005235.
Is the following true: for every m there is an N such that for all n>N a(n)>m?
There even seems to be the estimate a(n)>log(n+1)*sqrt(n+1)/2.

			a(3)=7: The seven primes following 3!+1=7 are 11,13,17,19,23,29 and 31.
Subtracting 3!=6 from each of them gives 5,7,11,13,17,23 and 25.
The first six values are prime, while the seventh 25=5^2 is not.

Cf. A005235, A037153, A087202.

a:=proc(n) option remember; local k:
for k from 1 while isprime((nextprime@@k)(n!+1)-n!) do od:
k; end;

A174023 The number of primes between prime(n)# and prime(n)# + prime(n)^2.

Original entry on oeis.org

2, 3, 6, 9, 17, 18, 20, 28, 25, 30, 41, 46, 41, 53, 56, 73, 62, 66, 81, 93, 85, 84, 89, 97, 101, 127, 121, 122, 119, 128, 150, 141, 144, 152, 150, 143, 174, 203, 197, 195, 196, 194, 213, 213, 218, 223, 230, 235, 249, 258, 256, 244, 264, 262, 274, 275, 278, 295

Offset: 1

T. D. Noe, Mar 12 2010

nonn

Here prime(n)# denotes the product of the first n primes, A002110(n). This sequence provides numerical evidence that the smallest prime p greater than prime(n)#+1 is a prime distance from prime(n)#; that is, p-prime(n)# is a prime number, as shown in the sequence of Fortunate numbers, A005235. For p-prime(n)# to be a composite number, p would have to be greater than prime(n)#+prime(n)^2, which would imply that a(n)=0.

			For 3, the second prime, 3# is 6 and 3#+3^2 is 15. There are 3 primes between 6 and 15: 7, 11, and 13. Hence a(2)=3.

T. D. Noe, Table of n, a(n) for n = 1..100

Table[p=Prime[n]; prod=prod*p; Length[Select[Range[prod+1,prod+p^2-1], PrimeQ]], {n,50}]

Limit_{N->infinity} (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} prime(n)) = 1. - Alain Rocchelli, Nov 03 2022

A228891 Let A = A050376. Let Q be the smallest term of A more than 1 + Product_{i=1..n} A(i). a(n) = Q - Product_{i=1..n} A(i).

Original entry on oeis.org

2, 3, 5, 7, 13, 13, 17, 17, 17, 41, 59, 29, 41, 53, 37, 67, 79, 61, 89, 101, 139, 71, 67, 83, 151, 101, 89, 127, 163, 137, 101, 103, 131, 181, 139, 139, 181, 181, 139, 317, 191, 313, 163, 197, 199, 389, 191, 233, 229, 337, 239, 229, 347, 881, 239, 283, 487

Offset: 1

Vladimir Shevelev, Sep 07 2013

nonn

This sequence is a Fermi-Dirac analog of the Fortunate numbers (A005235).

			a(1) = 2, since 1 + Product_{i=1} A(i) = 1 + 2 = 3, the smallest term Q of A050376 more than 3 is 4 and a(1) = 4-2 = 2; let n=4, then 1 + Product_{i=1..4} A(i) = 1 + 2*3*4*5 = 121 and the smallest term Q of A050376 more than 121 is 127. So a(4) = 127 - 120 = 7.

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A005235, A050376.

More terms from Peter J. C. Moses, Sep 10 2013

cp's OEIS Frontend

A376236 Ludic Fortunate numbers: a(n) = N(P(n)+1) - P(n), where N(x) = min {L in A003309 | L > x} is the next larger ludic number and P(n) = Prod_{k=1..n} A003309[n].

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A038771 a(n) is the smallest composite number c such that A002110(n) + c is prime.

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A038773 a(n) is the smallest prime of the form Q + c, where Q is the n-th primorial and c is a composite >= prime(n+1)^2.

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

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Mathematica

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A045493 Unfortunate primes - primes that are not Fortunate numbers.

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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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A160433 a(n) is the least number k such that (k-th prime after n!+1)-n! is not a prime.

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Maple

A174023 The number of primes between prime(n)# and prime(n)# + prime(n)^2.

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