A082470 -id:A082470 - cp's OEIS frontend

A092789 a(n) = smallest prime of the form prime(n)+m! for some m >= 0.

3, 5, 7, 13, 13, 19, 19, 43, 29, 31, 37, 43, 43, 67, 53, 59, 61, 67, 73, 73, 79, 103, 89, 113, 103, 103, 109, 109, 229, 137, 151, 137, 139, 163, 151, 157, 163, 283, 173, 179, 181, 363061, 193, 199, 199, 223, 331, 229, 229, 349, 239, 241, 5281, 257, 263, 269, 271

Offset: 1

Cino Hilliard, Apr 14 2004

nonn

n! + p is composite for n >= p since p divides n! for n >= p.

Is it known that such a prime always exists? If not the definition should say "or -1 if no such prime exists". - N. J. A. Sloane, Aug 11 2011

Cf. A175193, A175194, A082470.

SmallestP:=function(p) for m in [0..p-1] do q:=p+Factorial(m); if IsPrime(q) then return q; end if; end for; return -1; end function; [ SmallestP(NthPrime(n)): n in [1..80] ]; // Klaus Brockhaus, Mar 02 2010

A092789 := proc(n) local q,m ; for m from 0 do q := ithprime(n)+m! ; if isprime(q) then return q; end if; end do ; end proc:
seq(A092789(n),n=1..80) ; # R. J. Mathar, Mar 02 2010

nfactpm3(n) = { forprime(p=1,n, c=0; for(x=0,n,y=x!+p;if(isprime(y),c++;print1(y",");break)); ) }

Definition and offset corrected following a suggestion from Leroy Quet. - Klaus Brockhaus, Mar 02 2010

A092790 a(n) = (n+1)*phi(n-1)/2.

Original entry on oeis.org

2, 5, 6, 14, 8, 27, 20, 33, 24, 65, 28, 90, 48, 68, 72, 152, 60, 189, 88, 138, 120, 275, 104, 270, 168, 261, 180, 434, 128, 495, 272, 350, 288, 444, 228, 702, 360, 492, 336, 860, 264, 945, 460, 564, 528, 1127, 400, 1071, 520, 848, 648, 1430, 504, 1140, 696, 1062, 840, 1769

Offset: 3

N. J. A. Sloane, Nov 04 2008

[The old entry with this sequence number was a duplicate of A082470.]
Prepending [0, 3] and setting offset = 0 the sequence becomes the row sums of A378068. - Peter Luschny, Dec 27 2024
a(n) is the sum of row n-1 of A078401. - Amiram Eldar, May 12 2025

Amiram Eldar, Table of n, a(n) for n = 3..10000
Jeffrey Jaffe, Permutation numbers, Math. Mag., 49 (1976), 80-84.

Cf. A000010 (phi), A078401, A378068.

Table[(n+1) EulerPhi[n-1]/2,{n,3,60}] (* Harvey P. Dale, Apr 22 2012 *)

a(n) = (n+1)*eulerphi(n-1)/2; \\ Michel Marcus, Sep 18 2017

A175193 a(n) is the smallest positive integer such that (the n-th prime)+a(n)! is prime, or -1 if no such prime exists.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 2, 3, 3, 2, 4, 3, 3, 2, 3, 3, 2, 3, 4, 3, 4, 3, 2, 3, 2, 5, 4, 4, 3, 2, 4, 2, 3, 3, 5, 3, 3, 2, 9, 2, 3, 2, 4, 5, 3, 2, 5, 3, 2, 7, 3, 3, 3, 2, 3, 3, 2, 4, 4, 3, 2, 4, 8, 3, 5, 2, 4, 3, 4, 3, 3, 5, 3, 5, 4, 5, 4, 2, 5, 2, 3, 4, 3, 5, 3, 2, 4, 4, 4, 5, 7, 4, 3, 6, 2, 4, 3, 4, 3, 3, 2, 3

Offset: 1

Leroy Quet, Mar 01 2010

nonn

A175194(n) = a(n)!.

			From _Michael De Vlieger_, Nov 24 2017: (Start)
Records and their indices in a(n):
    i          n   a(n)
   --------------------
    1        1      1
    2        2      2
    3        4      3
    4        8      4
    5       29      5
    6       42      9
    7      233     10
    8      254     42
    9     4508     49
   10     7003    124
   11     7385    276
   12    60650    311
   13    97146    542
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, First positions of values and records in a(n).

Cf. A082470, A092789, A175194.

Fold[Append[#1, SelectFirst[Range[12], Function[k, PrimeQ[Prime[#2] + k!]]]] &, {1}, Range[2, 105]] (* Michael De Vlieger, Nov 24 2017 *)

a(n) = {my(k = 1, p = prime(n)); while (!isprime(p + k!), k++); k;} \\ Michel Marcus, Nov 25 2017

A092789(n) = A000040(n) + a(n)!.

Extended by Ray Chandler, Mar 04 2010

A352912 Irregular triangle read by rows: row n (n>=1) lists the primes of the form prime(n) + k! for k >= 0.

Original entry on oeis.org

3, 3, 5, 7, 11, 29, 13, 31, 127, 727, 13, 17, 131, 5051, 3628811, 19, 37, 733, 19, 23, 41, 137, 362897, 39916817, 43, 139, 739, 5059, 3628819, 39916819, 87178291219, 29, 47, 743, 40343, 362903, 20922789888023, 31, 53, 149, 39916829, 479001629, 2432902008176640029, 37, 151, 751, 40351, 362911, 39916831, 355687428096031, 51090942171709440031, 1124000727777607680031

Offset: 1

Editors of OEIS, based on a suggestion from Hemjyoti Nath, Apr 16 2022

			The initial rows, prefixed by prime(n), are:
[2]: 3, 3,
[3]: 5,
[5]: 7, 11, 29,
[7]: 13, 31, 127, 727,
[11]: 13, 17, 131, 5051, 3628811,
[13]: 19, 37, 733,
[17]: 19, 23, 41, 137, 362897, 39916817,
[19]: 43, 139, 739, 5059, 3628819, 39916819, 87178291219,
[23]: 29, 47, 743, 40343, 362903, 20922789888023,
[29]: 31, 53, 149, 39916829, 479001629, 2432902008176640029,
[31]: 37, 151, 751, 40351, 362911, 39916831, 355687428096031, 51090942171709440031, 1124000727777607680031,
[37]: 43, 61, 157, 757, 5077, 40357, 39916837, 6402373705728037, 2432902008176640037, 51090942171709440037, 8683317618811886495518194401280000037,
...

Michael S. Branicky, Table of n, a(n) for n = 1..1019

Cf. A352913 (last term in each row), A082470 (lengths of rows).

forprime(p=2,59,print1([p],": ");for(k=0,p,if(ispseudoprime(p+k!),print1(p+k!,", ")));print())

from sympy import isprime, prime
from itertools import count, islice
def agen(): # generator of terms
    for n in count(1):
        pn, fk = prime(n), 1
        for k in range(1, pn+1):
            if isprime(pn + fk): yield pn + fk
            fk *= k
print(list(islice(agen(), 51))) # Michael S. Branicky, Apr 16 2022

A175194 a(n) = the smallest factorial such that (the n-th prime)+a(n) is prime, or -1 if no such prime exists.

Original entry on oeis.org

1, 2, 2, 6, 2, 6, 2, 24, 6, 2, 6, 6, 2, 24, 6, 6, 2, 6, 6, 2, 6, 24, 6, 24, 6, 2, 6, 2, 120, 24, 24, 6, 2, 24, 2, 6, 6, 120, 6, 6, 2, 362880, 2, 6, 2, 24, 120, 6, 2, 120, 6, 2, 5040, 6, 6, 6, 2, 6, 6, 2, 24, 24, 6, 2, 24, 40320, 6, 120, 2, 24, 6, 24, 6, 6, 120, 6, 120, 24, 120, 24, 2, 120, 2

Offset: 1

Leroy Quet, Mar 01 2010

nonn

a(n) = A175193(n)!.

Cf. A082470, A092789, A175193.

A092789(n) = A000040(n) + a(n).

Extended by Ray Chandler, Mar 04 2010

A352913 a(n) = largest prime of the form prime(n) + k! (k >= 0).

Original entry on oeis.org

3, 5, 29, 727, 3628811, 733, 39916817, 87178291219, 20922789888023, 2432902008176640029, 1124000727777607680031, 8683317618811886495518194401280000037, 15511210043330985984000041, 523022617466601111760007224100074291200000043, 2658271574788448768043625811014615890319638528000000047

Offset: 1

Editors of OEIS, based on a suggestion from Hemjyoti Nath, Apr 16 2022

nonn

Michael S. Branicky, Table of n, a(n) for n = 1..93

These are the final entries in the rows of the triangle in A352912. See also A082470.

from sympy import isprime, prime
from itertools import count, islice
def agen(): # generator of terms
    for n in count(1):
        pn, fk = prime(n), 1
        for k in range(1, pn+1):
            if isprime(pn + fk): yield pn + fk
            fk *= k
print(list(islice(agen(), 51))) # Michael S. Branicky, Apr 16 2022

A384181 Primes p such that k! + p or |k! - p| is composite for all k >= 0.

Original entry on oeis.org

2, 3, 71, 97, 179, 181, 211, 223, 251, 283, 431, 503, 577, 827, 857, 971, 1019, 1021, 1109, 1213, 1249, 1259, 1279, 1289, 1373, 1427, 1429, 1483, 1571, 1609, 1619, 1637, 1699, 1709, 1759, 1801, 2053, 2129, 2141, 2213, 2269, 2281, 2293, 2297, 2339, 2381, 2477, 2503

Offset: 1

Gonzalo Martínez, May 21 2025

nonn

It is unknown whether there exists a prime p such that k! + p is composite for all k > = 0 (see A082470).

Every prime p in this list satisfies that at least one of the numbers k! + p, |k! - p| is composite; i.e., they cannot both be prime, for k >= 0.

			71 is in this sequence, since k! + 71 is prime only when k = 2, 5, 9, 14, 22, 43, 53 and 55, but |k! - 71| is composite for such values of k.

Cf. A082470, A352912.

from sympy import isprime, primerange, factorial
def ok(p):
    return not any(isprime((fk := factorial(k)) + p) and isprime(abs(fk - p)) for k in range(1, p))
print([p for p in primerange(2, 500) if ok(p)])

a(17)-a(23) from Sean A. Irvine, May 28 2025

a(24)-a(48) from Michael S. Branicky, May 29 2025

cp's OEIS Frontend

A092789 a(n) = smallest prime of the form prime(n)+m! for some m >= 0.

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Maple

PARI

Extensions

A092790 a(n) = (n+1)*phi(n-1)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A175193 a(n) is the smallest positive integer such that (the n-th prime)+a(n)! is prime, or -1 if no such prime exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A352912 Irregular triangle read by rows: row n (n>=1) lists the primes of the form prime(n) + k! for k >= 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Python

A175194 a(n) = the smallest factorial such that (the n-th prime)+a(n) is prime, or -1 if no such prime exists.

Views

Author

Keywords

Comments

Crossrefs

Formula

Extensions

A352913 a(n) = largest prime of the form prime(n) + k! (k >= 0).

Views

Author

Keywords

Links

Crossrefs

Programs

Python

A384181 Primes p such that k! + p or |k! - p| is composite for all k >= 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Python

Extensions