A038710 -id:A038710 - 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

A007014 Largest prime <= Product prime(k).

Original entry on oeis.org

2, 5, 29, 199, 2309, 30029, 510481, 9699667, 223092827, 6469693189, 200560490057, 7420738134751, 304250263527209, 13082761331669941, 614889782588491343, 32589158477190044657, 1922760350154212638963, 117288381359406970983181, 7858321551080267055878989

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

A057705 contains terms of a(n) such that A002110(n) - a(n) = 1. -Michael De Vlieger, May 15 2017

			From _Michael De Vlieger_, May 15 2017: (Start)
a(1) = 2 since A002110(1) = 2. 2 is prime thus the largest prime <= 2 = 2.
a(2) = 5 since A002110(2) = 6. 5 is the largest prime <= 6. (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Michael De Vlieger, Table of n, a(n) for n = 1..350
R. G. Wilson, V, Letter to N. J. A. Sloane, Jan. 1994

Cf. A000849, A002110, A038710, A057705, A151799.

Array[Abs@ NextPrime[Product[Prime@ i, {i, #}], -1] &, 14] (* Michael De Vlieger, May 15 2017 *)

lista(n) = {prd = 1; for (i=1, n, prd *= prime(i); print1(precprime(prd), ", "););} \\ Michel Marcus, Jun 17 2013

a(n)=precprime(prod(i=1,n,prime(i))) \\ Charles R Greathouse IV, Jun 17 2013

From Michael De Vlieger, May 15 2017: (Start)
a(n) = prime(A000849(n)).
a(n) = A151799(A002110(n)). (End)

Corrected by Jud McCranie, Jan 03 2001

More terms from Michael De Vlieger, May 15 2017

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

A058044 Difference between the smallest prime following and largest prime preceding n-th primorial number.

Original entry on oeis.org

2, 2, 12, 2, 18, 48, 46, 80, 102, 74, 120, 72, 136, 174, 132, 168, 198, 190, 230, 176, 234, 286, 102, 354, 364, 336, 278, 486, 442, 386, 408, 1032, 520, 308, 364, 612, 478, 432, 382, 422, 606, 526, 1344, 606, 1230, 834, 624, 756, 550

Offset: 2

Labos Elemer, Nov 17 2000

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..100

Cf. A002110, A007014, A038710.

[seq(nextprime(product(ithprime(k), k=1..w))-prevprime (product(ithprime(k), k=1..w)), w=2..50)];

Rest[NextPrime[#]-NextPrime[#,-1]&/@Rest[FoldList[Times,1,Prime[Range[ 50]]]]] (* Harvey P. Dale, Mar 24 2013 *)

a(n) = A038710(n)-A007014(n).

Offset corrected by Alois P. Heinz, Jun 08 2014

A035345 Smallest prime > prime(1)prime(2)...*prime(n)+1.

Original entry on oeis.org

3, 5, 11, 37, 223, 2333, 30047, 510529, 9699713, 223092907, 6469693291, 200560490197, 7420738134871, 304250263527281, 13082761331670077, 614889782588491517, 32589158477190044789, 1922760350154212639131

Offset: 0

N. J. A. Sloane

nonn

			Next prime after 2*3*5 + 1 = 31 is 37, so a(3)=37.

S. W. Golomb, The evidence for Fortune's conjecture, Math. Mag. 54 (1981), 209-210.
Eric Weisstein's World of Mathematics, Fortunate Prime

Cf. A002110, A005235, A006862, A038710.

Table[NextPrime[Product[Prime@ k, {k, n}] + 1], {n, 0, 17}] (* Michael De Vlieger, Dec 02 2015 *)

a(n) = nextprime(2+factorback(primes(n))); \\ Michel Marcus, Dec 24 2022

from sympy import nextprime, primorial
def a(n): return nextprime(1 + (primorial(n) if n else 1))
print([a(n) for n in range(18)]) # Michael S. Branicky, Dec 24 2022

a(n) = A002110(n) + A005235(n) for n > 0. - Jonathan Sondow, Dec 02 2015

A060269 Distance of n-th primorial from closest prime.

Original entry on oeis.org

1, 1, 1, 1, 19, 23, 37, 41, 1, 59, 1, 47, 67, 59, 61, 89, 89, 103, 79, 83, 89, 1, 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

Offset: 3

Labos Elemer, Mar 23 2001

nonn

			7th primorial is surrounded by {510529,510481} primes in {19,71} distances of which the smaller is 19=a(7).

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

with(numtheory): [seq(min(nextprime(product(ithprime(j),j=1..n))-product(ithprime(j),j= >1..n),product(ithprime(j),j=1..n)-prevprime(product(ithprime(j),j=1..n >))), n=3..50)];

dnp[n_]:=Module[{a=NextPrime[n,-1],b=NextPrime[n]},Min[n-a,b-n]]; dnp/@ FoldList[Times,Prime[Range[50]]] (* Harvey P. Dale, Jul 11 2017 *)

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

A277005 Least prime greater than n-th compositorial.

Original entry on oeis.org

2, 5, 29, 193, 1733, 17291, 207367, 2903041, 43545611, 696729629, 12541132817, 250822656001, 5267275776047, 115880067072017, 2781121609728037, 69528040243200079, 1807729046323200001, 48808684250726400031, 1366643159020339200397

Offset: 0

Walter Carlini, Sep 25 2016

			a(0) = A151800(A036691(0)) = A151800(1) = 2; where the zeroth compositorial, A036691(0), is the empty product = 1.
a(3) = 193, which is the least prime number greater than the third compositorial number, 192 = 4 * 6 * 8.

Cf. A036691, A038710, A151800.

findComp[n_] := FixedPoint[n + PrimePi@ # + 1 &, n + PrimePi@ n + 1] ; Table[NextPrime@ Product[findComp@ k, {k, n}], {n, 0, 18}] (* Michael De Vlieger, Sep 25 2016, after Robert G. Wilson v at A036691 *)

a(n) = A151800(A036691(n)). - Michel Marcus, Sep 25 2016

a(18) corrected by Sean A. Irvine, Sep 26 2023

cp's OEIS Frontend

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

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A007014 Largest prime <= Product prime(k).

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A060270 Distance of n-th primorial from previous prime.

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A058044 Difference between the smallest prime following and largest prime preceding n-th primorial number.

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A035345 Smallest prime > prime(1)*prime(2)*...*prime(n)+1.

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A060269 Distance of n-th primorial from closest prime.

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Maple

Mathematica

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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A035345 Smallest prime > prime(1)prime(2)...*prime(n)+1.