A018239 -id:A018239 - cp's OEIS frontend

A038711 a(n) is the smallest m such that A002110(n) + m is prime.

1, 1, 1, 1, 1, 1, 17, 19, 23, 37, 61, 1, 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

Offset: 0

Labos Elemer, May 02 2000

nonn

Any composite a(n) would disprove Fortune's conjecture, see A005235. - Jeppe Stig Nielsen, Oct 31 2003

			For n=11, 1 + A002110(11) = 200560490131 < 200560490197 = 67 + A002110(11); therefore, a(11)=1 but A005235(11)=67.

Robert G. Wilson v, Table of n, a(n) for n = 0..1000

Cf. A002110, A005235, A035345, A018239, A038710, A060270.

p:= proc(n) option remember; `if`(n<1, 1, p(n-1)*ithprime(n)) end:
a:= n-> nextprime(p(n))-p(n):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 16 2020

nmax=2^16384; npd=1;n=1;npd=npd*Prime[n]; While[npdLei Zhou, Feb 15 2005 *)

a(n) = my(P=vecprod(primes(n))); nextprime(P+1) - P; \\ Michel Marcus, Dec 12 2023

a(n) = Min(1, A005235(n)); a(n)=1 for n=1, 2, 3, 4, 5, 11, 75, ...
a(n) = 1 for n=0, 1, 2, 3, 4, 5, 11, 75, ... (A014545); a(n) = A005235(n) otherwise. - Jeppe Stig Nielsen, Oct 31 2003
a(n) = A038710(n) - A002110(n). - Alois P. Heinz, Mar 16 2020

a(0)=1 prepended by Alois P. Heinz, Mar 16 2020

A038710 a(n) is the smallest prime > product of the first n primes (A002110(n)).

Original entry on oeis.org

2, 3, 7, 31, 211, 2311, 30047, 510529, 9699713, 223092907, 6469693291, 200560490131, 7420738134871, 304250263527281, 13082761331670077, 614889782588491517, 32589158477190044789, 1922760350154212639131, 117288381359406970983379, 7858321551080267055879179

Offset: 0

Labos Elemer, May 02 2000

nonn

			for n=1,2,3,4,5,11,75, A002110(n)+1 gives smaller primes than A002110(n)+p, where p is a fortunate number (prime). At n=5, both 2311 and 2333 are primes but the first is smaller.

Alois P. Heinz, Table of n, a(n) for n = 0..350

Cf. A002110, A005235, A007014, A018239, A035345, A038711.

p:= proc(n) option remember; `if`(n<1, 1, p(n-1)*ithprime(n)) end:
a:= n-> nextprime(p(n)):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 16 2020

nmax = 2^16384; npd = 1; n = 1; npd = npd*Prime[n]; While[npd < nmax, cp = npd + 1; While[ ! (PrimeQ[cp]), cp = cp + 2]; Print[cp]; n = n + 1; npd = npd*Prime[n]] (* Lei Zhou, Feb 15 2005 *)
NextPrime/@FoldList[Times,1,Prime[Range[25]]] (* Harvey P. Dale, Dec 17 2010 *)

a(n) = nextprime(1+factorback(primes(n))); \\ Michel Marcus, Sep 25 2016; Dec 24 2022

from sympy import nextprime, primorial
def a(n): return nextprime(primorial(n) if n else 1)
print([a(n) for n in range(20)]) # Michael S. Branicky, Dec 24 2022

a(n) = A002110(n) + A038711(n). - Alois P. Heinz, Mar 16 2020

Offset corrected, incorrect comment and formula removed, and more terms added by Jinyuan Wang, Mar 16 2020

A104372 Primes of the form A104350(k) + 1.

Original entry on oeis.org

2, 3, 7, 13, 61, 181, 2521, 7561, 415801, 1247401, 1099944846001, 146100174169950001, 1156675078903494150001, 750321420485151941966263672363958662088980270355720625000001

Offset: 1

Reinhard Zumkeller, Mar 06 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..16

Intersection of A104365 and A000040.

Cf. A104364, A104373, A088332, A018239.

Select[FoldList[Times, Array[FactorInteger[#][[-1, 1]] &, 100]] + 1, PrimeQ] (* Amiram Eldar, Apr 08 2024 *)

gpf(n) = {my(p = factor(n)[, 1]); if(n == 1, 1, p[#p]);}
lista(nmax) = {my(r = 1); for(k = 1, nmax, r * = gpf(k); if(isprime(r+1), print1(r+1, ", ")));} \\ Amiram Eldar, Apr 08 2024

a(14) from Amiram Eldar, Apr 09 2024

A113165 Numbers that divide primorial numbers plus one (p#+1).

Original entry on oeis.org

2, 3, 7, 19, 31, 59, 61, 73, 97, 131, 139, 149, 167, 173, 181, 211, 223, 271, 277, 307, 313, 317, 331, 347, 463, 467, 509, 571, 601, 673, 809, 827, 877, 881, 953, 983, 997, 1031, 1033, 1039, 1051, 1063, 1069, 1109, 1259, 1279, 1283, 1291, 1297, 1361, 1381

Offset: 1

Franklin T. Adams-Watters, Jan 05 2006

The smallest composite member of the sequence is 1843 (19 * 97), which divides 17#+1 (19 * 97 * 277). Based on Euclid's proof that there are infinitely many primes.

			59 is in the sequence because 13#+1 = 30031 = 59 * 509.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..1000

Cf. A002110 (primorials), A018239 (primorial primes), A000945 (Euclid-Mullin sequence), A006862 (primorials plus one).

n=0;for(i=2,1e5,p=Mod(1,i);forprime(q=2,factor(i)[1,1],if(p==-1,print(n++," ",i);break());p*=q)) \\ Jeppe Stig Nielsen, Mar 25 2017

A136351 Primorial numbers p# such that p# + 1 is a prime.

Original entry on oeis.org

1, 2, 6, 30, 210, 2310, 200560490130

Offset: 1

Enoch Haga, Dec 25 2007

This sequence is different from A121069 and A002110.

a(8) = A002110(75) has 154 digits and is too long to be listed. - R. J. Mathar, Jul 23 2008

			a(6)=2310 is followed by prime 2311 whereas 30030 is not followed by a prime.

James C. McMahon, Table of n, a(n) for n = 1..10 (a(n) for n = 2..10 by Miles Englezou).

Cf. A002110, A014545, A018239, A121069, A136349, A136350, A136352.

Select[FoldList[Times, 1, Prime[Range[18]]],PrimeQ[#+1]&] (* James C. McMahon, May 08 2025 *)

S=[];for(n=0, 80, k=vecprod(primes(n)); if(isprime(k+1), S=concat(S,k))); S \\ Miles Englezou, Oct 28 2024

{A002110(j): A002110(j)+1 in A000040}. - R. J. Mathar, Jul 23 2008
a(n) = A002110(A014545(n)). - Michel Marcus, Apr 05 2021
a(n) = A018239(n) - 1. - James C. McMahon, May 08 2025

Changed a(1) from 4 to 2 and edited by R. J. Mathar, Jul 23 2008

a(1)=1 inserted by James C. McMahon, May 08 2025

A177689 Sums of 2 distinct primorials.

Original entry on oeis.org

3, 7, 8, 31, 32, 36, 211, 212, 216, 240, 2311, 2312, 2316, 2340, 30031, 30032, 30036, 30060, 30240, 32340, 510511, 510512, 510516, 510540, 510720, 512820, 540540, 9699691, 9699692, 9699696, 9699720, 9699900, 9702000, 9729720, 10210200

Offset: 1

Jonathan Vos Post, May 11 2010

This is to numbers that are the sum of 2 different primes (A038609) as primorials (A002110) are to primes (A000040). The subsequence of primes among these sums of 2 distinct primorials is the sequence of primorial primes (A018239) which is the same as the subsequence of primes among the Euclid numbers (A006862).

Cf. A000040, A002110, A006862, A038609.

{a(n)} = {A002110(i) + A002110(j) for i =/= j}.

A103515 Primes of the form primorial P(k)*2^n-1 with minimal n, n>=0, k>=2.

Original entry on oeis.org

5, 29, 419, 2309, 30029, 1021019, 19399379, 892371479, 51757545839, 821495767572479, 14841476269619, 304250263527209, 54873078184468933509119, 2459559130353965639, 521426535635040715679, 15751252788463309939261439

Offset: 1

Lei Zhou, Feb 15 2005

nonn

Conjecture: sequence is defined for all k>=2

			P(2)*2^0-1=3*2-1=5 is prime, so a(2)=5;
P(4)*2^1-1=7*5*3*2*2-1=419 is prime, so a(4)=419;

Cf. A002110, A005234, A014545, A018239, A006794, A057704, A057705, A103153, A103154.

nmax = 2^2048; npd = 2; n = 2; npd = npd*Prime[n]; While[npd < nmax, tt = 1; cp = npd*tt - 1; While[ ! (PrimeQ[cp]), tt = tt*2; cp = npd*tt - 1]; Print[cp]; n = n + 1; npd = npd*Prime[n]]

A324550 Primes written in primorial base (A049345).

Original entry on oeis.org

10, 11, 21, 101, 121, 201, 221, 301, 321, 421, 1001, 1101, 1121, 1201, 1221, 1321, 1421, 2001, 2101, 2121, 2201, 2301, 2321, 2421, 3101, 3121, 3201, 3221, 3301, 3321, 4101, 4121, 4221, 4301, 4421, 5001, 5101, 5201, 5221, 5321, 5421, 6001, 6121, 6201, 6221, 6301, 10001, 10201, 10221, 10301, 10321, 10421, 11001, 11121, 11221

Offset: 1

Antti Karttunen, Mar 11 2019

When the primorial base representation is expressed with decimal digits as here, the sequence stays unambiguous only up to the 317th prime, 2099, written as 96421, because after that primorial base digits larger than 9 would be needed.
By writing down terms from a(6) to a(46) (primes 13 .. 199):
  201, 221, 301, 321, 421, 1001, 1101, 1121, 1201, 1221, 1321, 1421, 2001, 2101, 2121, 2201, 2301, 2321, 2421, 3101, 3121, 3201, 3221, 3301, 3321, 4101, 4121, 4221, 4301, 4421, 5001, 5101, 5201, 5221, 5321, 5421, 6001, 6121, 6201, 6221, 6301,
and then from a(48) to a(80) (primes 223 .. 409):
  10201, 10221, 10301, 10321, 10421, 11001, 11121, 11221, 11321, 11421, 12001, 12101, 12121, 12201, 12321, 13101, 13121, 13201, 13221, 14001, 14101, 14221, 14301, 14321, 14421, 15101, 15201, 15301, 15321, 15421, 16101, 16121, 16301,
it is clearly seen that if n is a prime, then p+n is also likely to be prime, where p is the next higher primorial (A002110) > n. See also A324656.

Antti Karttunen, Table of n, a(n) for n = 1..317
Index entries for sequences related to primorial base.

Cf. A000040, A018239, A049345, A324642, A324656, A324657.

Cf. also A004676, A214617, A286941.

a[n_] := Module[{k = Prime[n], p = 2, s = {}, r}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, AppendTo[s, r]; p = NextPrime[p]]; FromDigits[Reverse[s]]]; Array[a, 100] (* Amiram Eldar, Mar 06 2024 *)

A324550(n) = A049345(prime(n)); \\ For A049345, see under that entry.

a(n) = A049345(A000040(n)).

A344384 Prime numbers p such that p-1 or p+1 is a number of least prime signature (A025487).

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 23, 29, 31, 37, 47, 59, 61, 71, 73, 97, 127, 179, 181, 191, 193, 211, 239, 241, 257, 359, 383, 419, 421, 431, 433, 479, 577, 719, 769, 839, 863, 1151, 1153, 1259, 1297, 1439, 1801, 2161, 2309, 2311, 2521, 2591, 2593, 2879, 3359, 3361

Offset: 1

Hal M. Switkay, May 16 2021

nonn

The corresponding numbers of least prime signature are A344385.
19 is the first prime not in this sequence.
This sequence unites many familiar sequences of primes, including Fermat primes (A019434), Mersenne primes (A000668), primorial primes (A018239 and A057705), factorial primes (A088054), A007505, and A039687.
Questions: 1) Is this sequence infinite? 2) Is log(a(n)) = O(log(n)^2)?

			17 is a term because 17 - 1 = 16 is a number of least prime signature.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..158 from Hal M. Switkay)

Cf. A025487, A344385.

Cf. A000668, A007505, A018239, A019434, A039687, A057705, A088054.

{2}~Join~Select[Prime@ Range[2, 900], AnyTrue[# + {-1, 1}, Times @@ MapIndexed[Prime[First[#2]]^#1 &, Sort[FactorInteger[#][[All, -1]], Greater] ] == # &] &] (* Michael De Vlieger, May 16 2021 *)

A103782 a(n) = minimal m >= 0 that makes primorial P(n)*2^m-1 prime.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 1, 2, 3, 12, 1, 0, 22, 2, 4, 13, 12, 6, 1, 4, 1, 4, 0, 2, 9, 5, 6, 2, 1, 9, 17, 22, 7, 19, 73, 23, 12, 5, 27, 33, 64, 33, 5, 7, 41, 44, 35, 29, 3, 19, 6, 26, 5, 11, 9, 33, 34, 16, 63, 46, 8, 4, 24, 48, 0, 11, 0, 26, 6, 25, 17, 31, 6, 46, 33, 46, 17, 8, 61, 12, 23, 76, 20, 17

Offset: 2

Lei Zhou, Feb 15 2005

nonn

The values of n in A103515

			P(2)*2^0-1=5 is prime, so a(2)=0; P(9)*2^2-1=892371479 is prime, so a(9)=2;

Cf. A002110, A005234, A014545, A018239, A006794, A057704, A057705, A103513, A103515.

nmax = 2^2048; npd = 2; n = 2; npd = npd*Prime[n]; While[npd < nmax, tn = 0; tt = 1; cp = npd*tt - 1; While[(cp > 1) && (! (PrimeQ[cp])), tn = tn + 1; tt = tt*2; cp = npd*tt - 1]; Print[tn]; n = n + 1; npd = npd*Prime[n]]

cp's OEIS Frontend

A038711 a(n) is the smallest m such that A002110(n) + m is prime.

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A038710 a(n) is the smallest prime > product of the first n primes (A002110(n)).

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A104372 Primes of the form A104350(k) + 1.

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A113165 Numbers that divide primorial numbers plus one (p#+1).

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A136351 Primorial numbers p# such that p# + 1 is a prime.

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A177689 Sums of 2 distinct primorials.

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A103515 Primes of the form primorial P(k)*2^n-1 with minimal n, n>=0, k>=2.

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A324550 Primes written in primorial base (A049345).

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