A265109 -id:A265109 - cp's OEIS frontend

A035346 Let F(n) = Q(n) - P(n) be the Fortunate numbers (A005235); sequence gives n such that F(n) = prime(n+1).

1, 2, 3, 6, 7, 8, 14, 16, 17, 21, 73, 801, 1971, 3332, 3469, 3509, 4318, 7986, 41292

Offset: 1

Positive n such that A002110(n) + A000040(n+1) is prime. - Robert Israel, Dec 02 2015

Subsequence of A265109. - Altug Alkan, Dec 02 2015

			a(10) = 21 because A002110(21) + prime(22) = 40729680599249024150621323549 = 2*3*5*...*67*71*73 + 79 is prime.

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.
S. W. Golomb, The evidence for Fortune's conjecture, Math. Mag. 54 (1981), 209-210.

Cf. A000040, A002110, A005235, A006862, A035345, A265109.

p:= 3:
A[1]:= 1:
count:= 1:
Primorial:= 2:
for n from 2 to 1000 do
  Primorial:= Primorial*p;
  p:= nextprime(p);
  if isprime(Primorial + p) then
    count:= count+1;
    A[count]:= n;
  fi
od:
seq(A[i],i=1..count); # Robert Israel, Dec 02 2015

Select[Range@ 801, PrimeQ[Product[Prime@ k, {k, #}] + Prime[# + 1]] &] (* Michael De Vlieger, Dec 02 2015 *)

lista(nn) = {s = 1; for(k=1, nn, s *= prime(k); if(ispseudoprime(s + prime(k+1)), print1(k, ", ")); ); } \\ Altug Alkan, Dec 02 2015

a(10)-a(11) were found by Labos Elemer, May 02 2000
a(12) from Ralf Stephan, Oct 20 2002
Offset changed by Altug Alkan, Dec 02 2015
a(13) from Michael De Vlieger, Dec 02 2015
a(14)-a(18) from Altug Alkan, Dec 02 2015
a(19) from Henri Lifchitz, Nov 08 2024

A100380 a(n) = least k such that prime(n) + A002110(k) is prime.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 4, 2, 1, 2, 2, 1, 3, 2, 2, 1, 2, 2, 1, 2, 3, 2, 5, 2, 1, 2, 1, 3, 5, 3, 2, 1, 4, 1, 2, 2, 3, 2, 2, 1, 3, 1, 2, 1, 3, 3, 2, 1, 4, 2, 1, 3, 2, 2, 2, 1, 2, 2, 1, 3, 4, 2, 1, 4, 3, 2, 3, 1, 3, 2, 3, 2, 2, 3, 2, 3, 4, 3, 3, 1, 4, 1, 2, 5, 2, 3, 2, 1, 4, 4, 3, 5, 3, 4, 2, 4, 1, 4, 2

Offset: 1

Pierre CAMI, Dec 30 2004

Conjecture: every prime number can be written as +- p(n) -+ p(k)# where p(i)=i-th prime, p(i)#=i-th primorial.
The sequence grows remarkably slowly. The largest number occurring within the first 50000 elements is 90. - Stefan Steinerberger, Apr 10 2006
a(1) = 0 is the minimum value of a(n). It is also unrepeated in this sequence. - Altug Alkan, Dec 02 2015

			prime(8)=19;
19 + 2 = 21 = 3*7,
19 + 6 = 25 = 5*5, and
19 + 30 = 49 = 7*7, but
19 + 210 = 229, which is prime; 210=prime(4)#, so a(8)=4.

Robert Israel, Table of n, a(n) for n = 1..10000 (corrected by Ray Chandler, Jan 19 2019)

Cf. A002110, A265109.

primorial:= proc(n) option remember: ithprime(n)*procname(n-1) end proc:
primorial(0):= 1:
f:= proc(n) local k, p;
  p:= ithprime(n);
  for k from 0 do if isprime(p+primorial(k)) then return k fi od:
end proc:
map(f, [$1..100]);# Robert Israel, Aug 27 2015

Table[k := 0;While[Not[PrimeQ[Prime[n]+Product[Prime[i],{i,1,k}]]],k++ ];k,{n,1, 100}] (* Stefan Steinerberger, Apr 10 2006 *)

primo(n) = prod(i=1, n, prime(i));
a(n) = {k=0; while(!isprime(prime(n)+primo(k)), k++); k;} \\ Michel Marcus, Aug 27 2015

from itertools import count, islice
from sympy import isprime, prime, primorial
def A002110(n): return primorial(n) if n > 0 else 1
def A100380(n):
    pn = prime(n)
    return next(k for k in count(0) if isprime(pn+A002110(k)))
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Jan 10 2025

More terms from Stefan Steinerberger, Apr 10 2006

a(1) = 0 added and name edited by Altug Alkan, Dec 02 2015

A380027 a(n) is the largest prime p such that p - a(n-1) is a primorial, starting with a(1) = 2.

Original entry on oeis.org

2, 3, 5, 11, 41, 9699731

Offset: 1

Hayden Chesnut, Jan 09 2025

nonn

From Michael S. Branicky, Jan 11 2025: (Start)
The corresponding k are such that 0 <= k < PrimePi(P), so a(n-1)+1 <= a(n) <= a(n-1)+primorial(PrimePi(a(n-1))-1).
a(7) >= 9699731 + primorial(452), which is prime and has 1351 digits, so it is too large to include, even in a b-file. (End)

			a(3) = 5
For primes less than 5+5#:
31 - 5 = 26 is not in A002110
...
13 - 5 = 8 is not in A002110
11 - 5 = 6 is in A002110 so a(4) = 11

Cf. A002110, A265109, A380026.

from itertools import count, islice
from sympy import isprime, primepi, primorial
def A002110(n): return primorial(n) if n > 0 else 1
def agen(an=2): # generator of terms
    while True:
        yield an
        an = next(s for k in range(primepi(an)-1, -1, -1) if isprime(s:=an+A002110(k)))
print(list(islice(agen(), 6))) # Michael S. Branicky, Jan 11 2025

a(n) = a(n-1) + A002110(A265109(A000720(a(n-1)))), for n > 1. - Michael S. Branicky, Jan 10 2025

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A035346 Let F(n) = Q(n) - P(n) be the Fortunate numbers (A005235); sequence gives n such that F(n) = prime(n+1).

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A100380 a(n) = least k such that prime(n) + A002110(k) is prime.

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Maple

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A380027 a(n) is the largest prime p such that p - a(n-1) is a primorial, starting with a(1) = 2.

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Python

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