A092965 -id:A092965 - cp's OEIS frontend

A092969 a(1) = 2; for n>1, a(n) = largest prime of the form n!/k + 1, where k < n, or 0 if no such prime exists.

2, 3, 7, 13, 61, 241, 2521, 20161, 72577, 604801, 39916801, 59875201, 3113510401, 17435658241, 186810624001, 10461394944001, 118562476032001, 0, 24329020081766401, 304112751022080001, 12772735542927360001

Offset: 1

Amarnath Murthy, Mar 26 2004

nonn

Conjecture: There are only finitely many zeros in this sequence. In other words the sequence is identical to A092965 barring a finite set of terms which are zero.

I found zeros for n: 18,51,53,84,95,100,104,106,143,178,180,181,188,202,203,(204). - Robert G. Wilson v, Mar 27 2004

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A092968, A092970.

f[n_] := Block[{k = 1}, While[ !PrimeQ[n!/k + 1], k++ ]; If[k < n, n!/k + 1, 0]]; Table[ f[n], {n, 22}] (* Robert G. Wilson v, Mar 27 2004 *)

a(n)=for (i=1,n,if(isprime(n!/i+1),return((n!/i+1))))

More terms from Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 26 2004

A089136 Primes in the progression (n! + m)/m where n advances by 1 and m resets to 1 upon each prime occurrence.

Original entry on oeis.org

2, 3, 7, 13, 61, 241, 2521, 20161, 72577, 604801, 39916801, 59875201, 3113510401, 17435658241, 186810624001, 10461394944001, 118562476032001, 246245142528001, 24329020081766401, 304112751022080001

Offset: 1

Cino Hilliard, Dec 05 2003

From Martin Fuller, Apr 26 2007: (Start)
Both this sequence and A092965 involve the largest prime of the form (n!/m)+1 but they differ in the allowed values of m. The present sequence allows any integer m dividing n!. But A092965 requires m to be the product of distinct numbers up to n.
I believe that the sequences differ at n=104 and n=106: a(104)=(104!/121)+1, A092965(104)=(104!/266)+1, a(106)=(106!/121)+1, A092965(106)=(106!/133)+1. (End)

			n=7,m=1, (7!+ 1)/1 = 5041 not prime, m advances to 2, (7!+2)/2 = 2521 prime keep it. n advances to 8 and m resets to 1. (8!+ 1)/1 = 61*661 not prime. m advances to 2. (8!+2)/2 = 20161 prime keep it n advances to 9 etc

Martin Fuller, Table of n, a(n) for n = 1..200

Different from A092965 (see Comments).

k = m = 1; Reap[Do[If[PrimeQ[#], k++; m = 1; Sow[#], m++] &[(k! + m)/m], {n, 100}]][[-1, 1]] (* Michael De Vlieger, Apr 16 2024 *)

nfactp2d2(n,m) = { for(x=1,n, for(k=1,m, y=floor((x!+ k)/k); if(isprime(y),print1(y",");break) ) ) }

lista(nn) = my(list = List()); for(x=1, nn, my(k=1, y); while (!isprime(y=floor((x!+ k)/k)), k++); listput(list, y)); Vec(list); \\ Michel Marcus, Apr 16 2024

A092967 Largest prime of the form a squarefree number + 1 where the prime divisors of the squarefree number are < n.

Original entry on oeis.org

2, 3, 7, 7, 31, 31, 211, 211, 211, 211, 2311, 2311, 6007, 6007, 6007, 6007, 102103, 102103, 3233231, 3233231, 3233231, 3233231, 17160991

Offset: 1

Amarnath Murthy, Mar 26 2004

nonn

Conjecture: a(n)-1 has prime(n)-1 divisors. Subsidiary sequence: Number of primes of the form 2*p*q*r*...+1 where p, q, r, etc. are distinct odd primes < n.

			a(13) = 6007 = 2*3*7*11*13 + 1, as 2*5*7*11*13 + 1, etc. are composite.

Cf. A092965, A060957.

Mathematica
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<Ryan Propper, Aug 13 2005 *)
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More terms from Ryan Propper, Aug 13 2005

A371924 a(n) is the least b such that prime(n)-1 divides b!.

Original entry on oeis.org

1, 2, 4, 3, 5, 4, 6, 6, 11, 7, 5, 6, 5, 7, 23, 13, 29, 5, 11, 7, 6, 13, 41, 11, 8, 10, 17, 53, 9, 7, 7, 13, 17, 23, 37, 10, 13, 9, 83, 43, 89, 6, 19, 8, 14, 11, 7, 37, 113, 19, 29, 17, 6, 15, 10, 131, 67, 9, 23, 7, 47, 73, 17, 31, 13, 79, 11, 7, 173, 29, 11

Offset: 1

Samuel Harkness, Apr 12 2024

nonn

This list is connected to Pollard's p-1 algorithm, using the version of the algorithm iterating over all positive integers. Say a large number m has two distinct prime factors q and r, and using Pollard's p-1 algorithm someone wishes to obtain the prime factors. Say q = 223 and r = 307. As prime(48) = 223 and a(48) = 37, given a random "a" coprime to m the factor 223 will be discovered in 37 steps. Also, as prime(63) = 307 and a(63) = 17, given a random "a" coprime to m the factor 307 will be discovered in 17 steps. Note that after 37 steps both factors will be discovered, so the algorithm will return m, failing to discover either prime factor. Therefore, when 17 <= b < 37 the prime factor 307 will be discovered. Note that on rare occasions, for a given "a" value, by chance p divides (a^b! - 1), so it is possible that for some "a" values the actual b value will be less. But, for any "a" value and prime p = prime(n), it is guaranteed that b <= a(n).

			For n = 25, prime(25) = 97, so we will use p = 97. Then the prime factorization of p - 1  is p - 1 = 2^5 * 3. Note that for p - 1 to divide b!, the exponents for all prime factors in b! must be greater than or equal to the exponents for all prime factors in the prime factorization of p - 1. We find that 8! = 2^7 * 3^2 * 5 * 7 is the least b such that this is true, so a(25) = 8.

Samuel Harkness, Table of n, a(n) for n = 1..10000
Wikipedia, Pollard's p-1 algorithm

Cf. A000040, A002034, A023503, A092965.

a371924[p_] :=
 Module[{a, d, f, u, v}, f = FactorInteger[p - 1]; d = {};
  For[a = 1, a <= Length[f], a++,
   u = f[[a]];
   v = u[[1]]^u[[2]];
   i = 1;
   While[! Divisible[(u[[1]]*i)!, v], i++]; AppendTo[d, u[[1]]*i]];
  Return[Max[d]]]
list = {};
For[p = 1, p <= 71, p++,
 AppendTo[list, {p, a371924[Prime[p]]}]]
Print[list]

a(n) = my(b=1, q=prime(n)-1); while (b! % q, b++); b; \\ Michel Marcus, Apr 15 2024

from sympy import prime
def A371924(n):
    m = prime(n)-1
    b, k = 1, 1%m
    while k:
        b += 1
        k = k*b%m
    return b # Chai Wah Wu, Apr 25 2024

A206762 a(n) is the least number from 1,2,...,n-1, such that n!/a(n)+1 is prime, and a(n)=0, if such number does not exist.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 2, 5, 6, 1, 8, 2, 5, 7, 2, 3, 0, 5, 8, 4, 16, 3, 8, 4, 10, 1, 13, 8, 2, 19, 4, 11, 11, 7, 3, 1, 12, 13, 4, 1, 24, 2, 8, 5, 9, 25, 16, 2, 12, 0, 26, 0, 17, 22, 44, 22, 37, 7, 48, 4, 37, 18, 7, 39, 16, 19, 7, 15, 19, 36, 30, 1, 14, 15, 16, 1

Offset: 2

Vladimir Shevelev, Feb 12 2012

nonn

The sequence of primes n!/a(n)+1, when a(n)>0, is increasing.

Cf. A089136, A092265, A092965, A092969, A092970.

Table[s = Select[Range[n - 1], PrimeQ[n!/# + 1] &, 1]; If[s == {}, 0, s[[1]]], {n, 2, 100}] (* T. D. Noe, Feb 13 2012 *)

cp's OEIS Frontend

A092969 a(1) = 2; for n>1, a(n) = largest prime of the form n!/k + 1, where k < n, or 0 if no such prime exists.

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A089136 Primes in the progression (n! + m)/m where n advances by 1 and m resets to 1 upon each prime occurrence.

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A092967 Largest prime of the form a squarefree number + 1 where the prime divisors of the squarefree number are < n.

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Extensions

A371924 a(n) is the least b such that prime(n)-1 divides b!.

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Python

A206762 a(n) is the least number from 1,2,...,n-1, such that n!/a(n)+1 is prime, and a(n)=0, if such number does not exist.

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