A038711 -id:A038711 - cp's OEIS frontend

A340007 Number of times the n-th prime (=A000040(n)) occurs in A038711.

0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 3, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 2, 0, 1, 2, 0, 2, 0, 1, 2, 2, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 2, 0, 4, 0, 0, 0, 1, 0, 0, 1, 1, 2, 0, 1, 1, 0, 1, 2, 0, 0, 2, 1

Offset: 1

A.H.M. Smeets, Dec 26 2020

nonn

Each term in A038711 is either 1 or a prime number. Moreover it is known that each prime occurs only a finite number of times in A038711.

By excluding the terms that equal one from A038711, we observe the smallest value of A038711(n)/log(A002110(n)) in the range n = 2..1000 to be ~1.017. From this it is believed that the primes less than 0.9*log(A002110(1001))*1.017 (~ 7157) will not occur anymore in the sequence A038711 for n > 1000; the applied factor 0.9 is a safety factor to be more or less sure that the prime numbers up to about 7157 will no longer occur in A038711.

			The prime number 17 occurs 1 time in A038711, and A000040(7) = 17, so a(7) = 1.
The prime number 5 does not occur in A038711, and A000040(3) = 5, so a(3) = 0.

A.H.M. Smeets, Table of n, a(n) for n = 1..916
A.H.M. Smeets, Sum_{k = 1..n} a(k) versus A000040(n)

Cf. A000040, A002110, A038711, A060270, A340006.

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

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

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

A340006 Number of times the n-th prime (=A000040(n)) occurs in A060270.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 2, 0, 1, 3, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 3, 1, 0, 1, 1, 0, 2, 1, 1, 0, 0, 0, 1, 0, 2, 2, 1, 1, 2, 1, 0, 1, 0, 0, 0, 0, 2, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 2

Offset: 1

A.H.M. Smeets, Dec 26 2020

nonn

Each term in A060270 is either 1 or a prime number. Moreover it is known that each prime occurs only a finite number of times in A060270.

By excluding the terms that equal one from A060270, we observe the smallest value of A060270(n)/log(A002110(n)) in the range n = 2..1000 to be ~1.014. From this it is believed that the primes less than 0.9*log(A002110(1001))*1.014 (~ 7138) will not occur anymore in the sequence A060270 for n > 1000; the applied factor 0.9 is a safety factor to be more or less sure that the prime numbers up to about 7138 will no longer occur in A060270.

			The prime number 7 does not occur in A060270, and A000040(4) = 7, so a(4) = 0.
The prime number 11 occurs 1 time in A060270, and A000040(5) = 11, so a(5) = 1.

A.H.M. Smeets, Table of n, a(n) for n = 1..916
A.H.M. Smeets, Sum_{k = 1..n} a(k) versus A000040(n)

Cf. A000040, A002110, A038711, A060270, A340007.

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 *)

A350460 Positive integers k such that if p is the next prime > k then p - k is prime.

Original entry on oeis.org

3, 5, 8, 9, 11, 14, 15, 17, 20, 21, 24, 26, 27, 29, 32, 34, 35, 38, 39, 41, 44, 45, 48, 50, 51, 54, 56, 57, 59, 62, 64, 65, 68, 69, 71, 74, 76, 77, 80, 81, 84, 86, 87, 90, 92, 94, 95, 98, 99, 101, 104, 105, 107, 110, 111, 114, 116, 120, 122, 124, 125, 128, 129

Offset: 1

Ryan Bresler, Jan 01 2022

nonn

a(n) is only prime when n is the lesser of a twin prime pair (A001359). All other terms are composite.

			3 is a term because the next prime > 3 is 5, and 5 - 3 = 2, which is prime.
14 is a term because the next prime > 14 is 17, and 17 - 14 = 3, which is prime.

Cf. A038711, A001359, A068780.

Select[Range[130], PrimeQ[NextPrime[#] - #] &] (* Amiram Eldar, Jan 01 2022 *)

isok(k) = my(p=nextprime(k+1)); isprime(p-k); \\ Michel Marcus, Jan 01 2022

from sympy import isprime, nextprime
def ok(n): return n > 0 and isprime(nextprime(n) - n)
print([k for k in range(130) if ok(k)]) # Michael S. Branicky, Jan 01 2022

A268480 Integers k such that A002110(k) is the average of two consecutive primes.

Original entry on oeis.org

2, 3, 5, 8, 38, 40, 64, 73, 89, 236, 480, 486

Offset: 1

Altug Alkan, Mar 21 2016

In other words, the primorial numbers that are considered are those of the form (p + q)/2 where p and q are consecutive primes. Note that the initial values of (p - q)/2 are 1, 1, 1, 23, 239, 191, 331, 373, 1021.
A088256 is a subsequence of these primorials, which in turn are a subsequence of A024675.
Numbers k such that A038711(k) = A060270(k). - Amiram Eldar, May 19 2024

			5 is a term because 2*3*5*7*11 = 2310 = (2309 + 2311)/2.
8 is a term because 2*3*5*7*11*13*17*19 = 9699690 = (9699667 + 9699713)/2.

Index entries for sequences related to primorial numbers.

Cf. A002110, A024675, A038711, A060270, A088256.

P:= 2: count:= 0:
for n from 2 to 500 do
  P:= P*ithprime(n);
  # first try d=1
  if isprime(P+1) then
    good:= isprime(P-1);
  elif isprime(P-1) then good:= false
  else
    for d from ithprime(n+1) by 2 do
      if igcd(d,P) > 1 then next fi;
      if isprime(P+d) then
        good:= isprime(P-d); break
      elif isprime(P-d) then
        good:= false; break
      fi
    od;
  fi;
  if good then
     count:= count+1;
     A[count]:= n;
  fi
od:
seq(A[i],i=1..count);  # Robert Israel, Aug 29 2016

prim[n_] := Times @@ Prime[Range[n]]; Select[Range[2, 100], Total[NextPrime[(p = prim[#]), {-1, 1}]] == 2*p &] (* Amiram Eldar, May 19 2024 *)

a002110(n) = prod(k=1, n, prime(k));
for(n=2, 1e3, if((nextprime(a002110(n)) - a002110(n)) == (a002110(n) - precprime(a002110(n))), print1(n, ", ")))

A340041 The prime gap, divided by two, which surrounds p#.

Original entry on oeis.org

1, 1, 6, 1, 9, 24, 23, 40, 51, 37, 60, 36, 68, 87, 66, 84, 99, 95, 115, 88, 117, 143, 51, 177, 182, 168, 139, 243, 221, 193, 204, 516, 260, 154, 182, 306, 239, 216, 191, 211, 303, 263, 672, 303, 615, 417, 312, 378, 275, 375, 322, 445, 312, 294, 354, 492, 399, 348, 461

Offset: 2

Robert G. Wilson v, Jan 22 2021

nonn

If p and q are consecutive primes, we say here that there is a gap of q-p. (Other sequences use different definitions of "gap".) - N. J. A. Sloane, Mar 07 2021

Records: 1, 6, 9, 24, 40, 51, 60, 68, 87, 99, 115, 117, 143, 177, 182, 243, 516, 672, 855, 915, 925, 1100, 1139, 1620, 1863, 2272, 2842, 4177, 4190, 5025, 5692, 6254, 6413, 6879, 7914, 8026, 9928, 10604, ..., .

			For a(1), there are two contiguous primes {2, 3} with 2 being 2#. The prime gap is 1. However, the two primes do not surround 2#, so a(1) like A340013(2) is undefined.
For a(2), the prime gap contains {5, 6, 7}, with 3# = 6 in the middle. The prime gap is 2, therefore a(2) = 1;
For a(3), the prime gap contains {29, 30, 31}, with 5# = 30  in the middle. The prime gap is 2, therefore a(3) = 1.
For a(4), the prime gap contains {199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211}, with 7# =  205 in the middle. The prime gap is 12, therefore a(4) = 6. etc.

Cf. A006862, A007014, A038711, A060270, A340013 (analog for n!).

a[n_] := Block[{p = Times @@ Prime@ Range@ n}, (NextPrime[p, 1] - NextPrime[p, -1])/2]; a[1] = 0; Array[a, 60]

a(n) = (A006862(n) - A007014(n))/2 = (A038711(n) + A060270(n))/2.

a(n) = A058044(n)/2. - Hugo Pfoertner, Jan 22 2021

cp's OEIS Frontend

A340007 Number of times the n-th prime (=A000040(n)) occurs in A038711.

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

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A060270 Distance of n-th primorial from previous prime.

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A340006 Number of times the n-th prime (=A000040(n)) occurs in A060270.

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A060269 Distance of n-th primorial from closest prime.

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A350460 Positive integers k such that if p is the next prime > k then p - k is prime.

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