A053709 -id:A053709 - cp's OEIS frontend

A053710 Prime-balanced factorials: factorials k! that are the mean of their 2 closest neighboring primes.

6, 120, 3628800, 51090942171709440000

Offset: 1

Labos Elemer, Feb 10 2000

nonn

Values of k are in A053709.

The next two terms are 171! and 190!. - Jud McCranie, Jul 04 2000

			For k = 21, k! = 51090942171709440000, d = 31, and the closest primes to 21! are q = 21! - 31 = 51090942171709439969, p = 21! + 31 = 51090942171709440031.

Amiram Eldar, Table of n, a(n) for n = 1..7

Cf. A033932, A033933, A006990, A037151, A006562, A053709.

Select[Range[25]!,NextPrime[#]-#==#-NextPrime[#,-1]&] (* Harvey P. Dale, May 08 2025 *)

k! = (p+q)/2; p = k! + d, q = k! - d, where p and q are the closest primes to k!.

a(n) = A053709(n)!.

a(3) corrected by Sean A. Irvine, Jan 14 2022

A075409 a(n) is the smallest m such that n!-m and n!+m are both primes.

Original entry on oeis.org

0, 1, 5, 7, 19, 19, 31, 17, 11, 17, 83, 67, 353, 227, 163, 59, 61, 113, 353, 31, 1447, 571, 389, 191, 337, 883, 101, 1823, 659, 709, 163, 1361, 439, 307, 1093, 1733, 2491, 1063, 1091, 1999, 1439, 109, 2753, 607, 2617, 269, 103, 2663, 337, 14447, 2221, 5471, 2887

Offset: 2

Zak Seidov, Sep 18 2002

nonn

For n=3,5,10,21,171,190,348, n! is an interprime, the average of two consecutive primes, see A053709. In general n! may be average of several pairs of primes, in which case the minimal distance is in the sequence. See also n^n and n!! as average of two primes in A075468 and A075410.

According to Goldbach's conjecture, a(n) always exists with a(n) = A047160(n!). - Jens Kruse Andersen, Jul 30 2014

			a(4)=5 because 4!=24 and 19 and 25 are primes with smallest distance 5 from 4!.

Vincenzo Librandi, Table of n, a(n) for n = 2..100

Cf. A033932, A033933, A047160, A053709, A075468, A075410.

smp[n_]:=Module[{m=1,nf=n!},While[!PrimeQ[nf+m]||!PrimeQ[nf-m],m=m+2];m]; Join[{0},Array[smp,60,3]] (* Harvey P. Dale, Apr 18 2014 *)

a(n) = {my (m=0); until (ok, ok = isprime(n!-m) && isprime(n!+m); if (!ok, m++);); return (m);} \\ Michel Marcus, Apr 19 2013

More terms from David Wasserman, Jan 17 2005

A053711 Numbers d such that, for some k, the upper and lower primes closest to k! are k! + d and k! - d.

Original entry on oeis.org

1, 7, 11, 31, 397, 409, 1657, 2131, 7607

Offset: 1

Labos Elemer, Feb 10 2000

This sequence lists d = nextprime(k!) - k! = prevprime(k!) - k! for k in A053709.

			For k = 10, k! = 3628800, d = 11, and the closest primes to 10! are q = 10! - 11 = 3628789 and p = 10! + 11 = 3628811. The differences d are listed here.

Cf. A033932, A033933, A006990, A037151, A006562, A053709, A053710.

Reap[Do[If[SameQ @@ #, Sow@ First[#]] &@ Abs[# - NextPrime[#, {-1, 1}]] &[i!], {i, 200}]][[-1, -1]] (* Michael De Vlieger, Jan 14 2022 *)

a(5)-a(8) from Donovan Johnson, Oct 12 2008

a(9) from Hans Havermann, Aug 15 2014

A053714 Smallest (in magnitude) nonzero number m such that n!+m is prime.

Original entry on oeis.org

1, 1, 1, -1, 7, -1, -1, 23, -13, 11, 1, -1, -23, -1, 43, 23, 31, 37, 89, 29, 31, 31, -89, -73, 41, -37, 1, 67, -31, -1, -61, -1, -1, 97, 61, -127, 1, -1, -73, 53, 1, -79, 71, 47, -53, -89, -79, 53, -59, 61, -179, 53, -59, -127, -61, 149, 107, -109, -137, -139, -71, -71, -101, 67, -127, 283, 73, 83, -103, -97, -751, 101

Offset: 1

Labos Elemer, Feb 10 2000

sign

a(n) is the defined, nonzero (thus excluding a(1) and a(2) of A033933) minimum of A033932(n) and A033933(n) multiplied by -1 if that minimum is not A033932(n). If n!+m and n!-m are equidistant primes (A053709), we have (arbitrarily) chosen positive m.

			For n=4, the possible m are -1 (24-1) and +5 (24+5). The former is closer to 4! so a(4) is -1.
For n=5, the possible m are -7 (120-7) and +7 (120+7). Being equidistant to 5!, a(5) is +7.

Hans Havermann, Table of n, a(n) for n = 1..2000

Cf. A006990, A037151, A033932, A033933, A053709, A056752 (unsigned version with a different second term).

Edited by Hans Havermann, Jul 23 2014

A075275 Numbers k such that k!! is an interprime, i.e., the average of two successive primes.

Original entry on oeis.org

5, 7, 10, 11, 22, 41, 67, 76, 91, 96, 163, 245, 299, 341, 434, 510, 535, 800, 935, 1401, 1403, 1747

Offset: 1

Zak Seidov, Sep 12 2002

The parity of k is opposite to the parity of the differences.

a(23) > 3000. - Michael S. Branicky, Jan 20 2025

			5 is a term because 5!! = 15 is the average of two successive primes, 13 and 17;
163 is a term because 163!! is the average of two successive primes, 163!! -+ 128.

Cf. A053709. The differences between k!! and its neighboring primes are in A075453.

PrevPrim[n_] := Block[{k = n - 1}, While[ !PrimeQ[k], k-- ]; k]; NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k] Do[ a = n!!; If[ 2a == PrevPrim[a] + NextPrim[a], Print[n]], {n, 3, 762}]

Edited, corrected and extended by Robert G. Wilson v, Sep 16 2002

a(18)-a(22) from Michael S. Branicky, Jan 19 2025

A053712 Lower balancing primes to prime-balanced factorials.

Original entry on oeis.org

5, 113, 3628789, 51090942171709439969

Offset: 1

Labos Elemer, Feb 10 2000

nonn

The next two terms are 171!-397 and 190!-409. - Jud McCranie, Jul 04 2000

			113 is balancing 5! = 120 from below, where 5! = 120 is a balanced factorial.

Amiram Eldar, Table of n, a(n) for n = 1..7

Cf. A033932, A033933, A037151, A006990, A006562, A053709, A053710, A053711, A053713.

a(n) = A053709(n)! - A053711(n) = A053710(n) - A053711(n). - Amiram Eldar, Mar 10 2025

A053713 Upper balancing primes to prime-balanced factorials.

Original entry on oeis.org

7, 127, 3628811, 51090942171709440031

Offset: 1

Labos Elemer, Feb 10 2000

nonn

The next two terms are 171!+397 and 190!+409, which are too large to include. - Jud McCranie, Jul 04 2000

			127 is balancing 5! = 120 from above, where 5! = 120 is a balanced factorial.

Amiram Eldar, Table of n, a(n) for n = 1..7

Cf. A033932, A033933, A037151, A006990, A006562, A053709, A053710, A053711, A053712.

a(n) = A053709(n)! + A053711(n) = A053710(n) + A053711(n). - Amiram Eldar, Mar 10 2025

A074165 Numbers n such that n!!! is an interprime.

Original entry on oeis.org

4, 6, 13, 15, 16, 23, 28, 46, 126, 148, 285

Offset: 1

Robert G. Wilson v, Sep 16 2002

nonn

No additional terms up to n = 700. - Harvey P. Dale, Aug 02 2018

Cf. A053709 & A074176.

PrevPrim[n_] := Block[{k = n - 1}, While[ ! PrimeQ[k], k-- ]; k]; NextPrim[n_] := Block[{k = n + 1}, While[ ! PrimeQ[k], k++ ]; k]; NFactorialM[n_, m_] := Block[{k = n, p = n}, While[k > m, k -= m; p = p*k]; p]; Do[ a = NFactorialM[n, 3]; If[2a == PrevPrim[a] + NextPrim[a], Print[n]], {n, 3, 500}]
npipQ[n_]:=Module[{p=Times@@Range[n,1,-3]},p==Mean[{NextPrime[ p], NextPrime[ p,-1]}]]; Select[Range[300],npipQ] (* Harvey P. Dale, Aug 02 2018 *)

A350782 a(n) is the number of pairs of primes (p,q), p < q, such that (p + q)/2 = n!.

Original entry on oeis.org

1, 5, 18, 60, 315, 1615, 9928, 70437, 637504, 5829386, 64647125, 722750400

Offset: 3

Jon E. Schoenfield, Jan 15 2022

a(n) is the number of pairs of primes that are equidistant from n!.

Equivalently, a(n) is the number of positive integers d such that n! - d and n! + d are primes. For the smallest such d, iff there are no primes in the open interval (n! - d, n! + d), then n is a term in A053709, n! is a term in A053710, and d is a term in A053711.

			For n = 4, n! = 24, from which 5 pairs of primes are equidistant; in order of increasing distance, these are (19, 29), (17, 31), (11, 37), (7, 41), and (5, 43), so a(4) = 5. (A prime (23) lies between 19 and 29, so n=4 is not a term of A053709.)
For n = 5, n! = 120, from which 18 pairs of primes are equidistant: (113, 127), (109, 131), (103, 137), (101, 139), (89, 151), (83, 157), (73, 167), (67, 173), (61, 179), (59, 181), (47, 193), (43, 197), (41, 199), (29, 211), (17, 223), (13, 227), (11, 229), and (7, 233), so a(5) = 18. (The pair least distant from 120 is (113, 127), and there are no primes between 113 and 127, so n = 5 is a term of A053709, n! = 120 is a term of A053710, and d = 120 - 113 = 127 - 120 = 7 is a term of A053711.)

Cf. A002375, A053709, A053710, A053711.

Table[Length@IntegerPartitions[2n!,2,Prime@Range@PrimePi[2n!]],{n,3,9}] (* Giorgos Kalogeropoulos, Jan 16 2022 *)

from sympy import isprime, nextprime, factorial
def A350782(n):
    m, p, c = factorial(n), 3, 0
    while p <= m:
        if isprime(2*m-p):
            c += 1
        p = nextprime(p)
    return c # Chai Wah Wu, Jan 16 2022

a(n) = A002375(n!).

a(14) from Martin Ehrenstein, Jan 25 2022

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A053710 Prime-balanced factorials: factorials k! that are the mean of their 2 closest neighboring primes.

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A075409 a(n) is the smallest m such that n!-m and n!+m are both primes.

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A053711 Numbers d such that, for some k, the upper and lower primes closest to k! are k! + d and k! - d.

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A053714 Smallest (in magnitude) nonzero number m such that n!+m is prime.

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A075275 Numbers k such that k!! is an interprime, i.e., the average of two successive primes.

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A053712 Lower balancing primes to prime-balanced factorials.

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A053713 Upper balancing primes to prime-balanced factorials.

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A074165 Numbers n such that n!!! is an interprime.

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