A061232 -id:A061232 - cp's OEIS frontend

A294194 a(n) is the sum of primes between n!+1 and (n+1)!.

0, 2, 8, 90, 1493, 40148, 1536501, 78716604, 5275720734, 445867194366, 46449459576595, 5844108759723625, 873766735145244948, 153087877536458860660, 31062245895591356844637, 7225340509057562100376047, 1909774727511978452561271489, 569166269815032401548622424344

Offset: 0

Olivier Gérard, Oct 22 2017

nonn

			a(2) = 3 + 5 = 8.
a(3) = 7 + 11 + 13 + 17 + 19 + 23 = 90.

Amiram Eldar, Table of n, a(n) for n = 0..21 (extended using Kim Walisch's primesum; terms 0..19 from Daniel Suteu)
Kim Walisch, primesum program.

Cf. A063959.
Cf. A007504 (sum of first n primes).
Cf. A061232 (number of primes between n!+1 and (n+1)!).
Cf. A294193 (sum of integers between n!+1 and (n+1)!).
Cf. A294195 (product of primes between n!+1 and (n+1)!).

Table[Plus @@ Table[Prime[i], {i, PrimePi[n!] + 1, PrimePi[(n + 1)!]}], {n, 0, 10}]

a(n) = { my(s = 0); forprime(p=n!+1, (n+1)!, s+=p); s; } \\ Iain Fox, Nov 23 2017

use ntheory ':all'; for(0..100) { print "a($) = ", sum_primes(factorial($)+1, factorial($+1)), "\n"; } # _Daniel Suteu, Nov 15 2018

a(n) = A063959(n+1) - A063959(n). - Amiram Eldar, Jun 14 2024

a(11)-a(13) from Iain Fox, Nov 23 2017
a(14) from Iain Fox, Nov 28 2017
a(15)-a(17) from Daniel Suteu, Nov 15 2018

A294195 Product of all primes between n!+1 and (n+1)!.

Original entry on oeis.org

1, 2, 15, 7436429, 141690116050851861774628683971583952877

Offset: 0

Olivier Gérard, Oct 29 2017

nonn

The next term has 251 digits in base 10.

Cf. A061232 (number of primes between n!+1 and (n+1)!).
Cf. A294193 (sum of integers between n!+1 and (n+1)!).
Cf. A294194 (sum of primes between n!+1 and (n+1)!).
Cf. A294196 (log of product of primes between n!+1 and (n+1)!).

Table[Times @@ Table[Prime[i], {i, PrimePi[n!] + 1, PrimePi[(n + 1)!]}], {n, 0, 4}]

A294196 Floor of log of product of all primes between n!+1 and (n+1)!.

Original entry on oeis.org

2, 15, 87, 579, 4276, 35103, 322168, 3264471, 36285842, 439070392, 5747983086

Offset: 2

Olivier Gérard, Oct 29 2017

			a(3) = floor(log(7*11*13*17*19*23)) = floor(15.82) = 15.

Cf. A061232 (number of primes between n!+1 and (n+1)!).
Cf. A294194 (sum of primes between n!+1 and (n+1)!).
Cf. A294195 (product of primes between n!+1 and (n+1)!).

Table[Floor[Plus @@ Table[Log[Prime[i]*1.], {i, PrimePi[n!] + 1, PrimePi[(n + 1)!]}]], {n, 2, 11}]

a(12) from Daniel Suteu, Nov 15 2018

A346425 a(n) is the greatest number k such that k! <= prime(n).

Original entry on oeis.org

2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Bernard Schott, Jul 16 2021

nonn

Terms 2, 3, 4, 5, ... appear respectively 3, 6, 21, 98, ... times consecutively; indeed, 2 appears A061232(1) + A061232(2) times, then every m >= 3 appears A061232(m) times.

			prime(1) = 2 and the greatest k with k! <= 2 is 2, so a(1) = 2.
prime(4) = 7 and the greatest k with k! <= 7 is 3, so a(4) = 3.
prime(10) = 29 and the greatest k with k! <= 29 is 4 so a(10) = 4.
Rows with n, prime(n), greatest k! <=n, a(n) for n = 1..14
      n        1    2    3    4    5    6    7    8    9   10   11   12   13   14
   prime(n)    2    3    5    7   11   13   17   19   23   29   31   37   41   43
  greatest k!  2    2    2    6    6    6    6    6    6   24   24   24   24   24
    a(n)       2    2    2    3    3    3    3    3    3    4    4    4    4    4

Cf. A000040, A061232, A136437.

a(n) = my(k=1); until (k! > prime(n), k++); k-1; \\ Michel Marcus, Jul 19 2021

a(n)! = A000040(n) - A136437(n).

A294197 Floor of log base n! of product of all primes between n!+1 and (n+1)!.

Original entry on oeis.org

3, 8, 27, 120, 650, 4117, 30380, 255000, 2402333, 25086428, 287582999

Offset: 2

Olivier Gérard, Oct 29 2017

			a(3) = floor(log(7436429)/log(6)) = floor(8.830...) = 8.

Cf. A061232 (number of primes between n!+1 and (n+1)!).
Cf. A294195 (product of primes between n!+1 and (n+1)!).
Cf. A294196 (log of product of primes between n!+1 and (n+1)!).

Table[Floor[1/Log[n!] Plus @@ Table[Log[Prime[i]*1.], {i, PrimePi[n!] + 1, PrimePi[(n + 1)!]}]], {n, 2, 10}]

a(n) = floor(log(vecprod(primes([n!+1, (n+1)!])))/log(n!)); \\ Michel Marcus, Jan 19 2025

a(n) = Sum_{i=primepi(n!)+1..primepi((n+1)!)} log(prime(i)) / log(n!).

a(12) from Jinyuan Wang, Jan 19 2025

cp's OEIS Frontend

A294194 a(n) is the sum of primes between n!+1 and (n+1)!.

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A294195 Product of all primes between n!+1 and (n+1)!.

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A294196 Floor of log of product of all primes between n!+1 and (n+1)!.

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A346425 a(n) is the greatest number k such that k! <= prime(n).

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A294197 Floor of log base n! of product of all primes between n!+1 and (n+1)!.

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Mathematica

PARI

Formula

Extensions