A100858 -id:A100858 - cp's OEIS frontend

A100604 Numbers k such that (prime(k)-1)! + prime(k)^3 is prime.

2, 3, 4, 28, 894

Offset: 1

Jonathan Vos Post, Nov 30 2004

k = {2, 3, 4, 28} yields primes p(k) = {3, 5, 7, 107}. There are no more such k up to k=100. Verified by Ray Chandler.
a(5) > 600. - Jinyuan Wang, Apr 10 2020
a(6) > 2500. - Michael S. Branicky, Jul 02 2024

			a(3) = 4 because (prime(4)-1)! + prime(4)^3 = (7-1)! + 7^3 = 720 + 343 = 1063 is the 3rd prime of this form.

Cf. A100605, A100858.

Select[Range[30],PrimeQ[(Prime[#]-1)!+Prime[#]^3]&] (* Harvey P. Dale, Jul 13 2022 *)

is(k) = ispseudoprime((prime(k)-1)! + prime(k)^3); \\ Jinyuan Wang, Apr 10 2020

Numbers k such that (prime(k)-1)! + prime(k)^3 is prime, where prime(k) is the k-th prime.

a(5) from Michael S. Branicky, Jul 01 2024

A235267 Places n such that 1 + q!/p! is prime, where p = prime(n) and q = prime(n + 1).

Original entry on oeis.org

3, 5, 33, 35, 43, 69, 86, 106, 108, 116, 124, 127, 171, 174, 182, 200, 201, 238, 255, 256, 270, 271, 277, 294, 310, 318, 323, 332, 356, 384, 388, 390, 392, 397, 398, 402, 409, 469, 494, 495, 520, 542, 551, 562, 572, 582, 606, 632, 633, 645, 649, 652, 671, 672

Offset: 1

K. D. Bajpai, Jan 05 2014

nonn

			3 is in the sequence because (prime(3)! + prime(4)!)/prime(3)! =  (5! + 7!)/5! = (120 + 5040)/120 =  43 which is prime.

K. D. Bajpai, Table of n, a(n) for n = 1..1510

Cf. A000040, A100858.

isA235267 := proc(n)
    local p,q ;
    p := ithprime(n) ;
    q := nextprime(p) ;
    if isprime(1+q!/p!) then
        true;
    else
        false;
    end if;
end proc:
for n from 1 do
    if isA235267(n) then
        print(n) ;
    end if;
end do:

A235392 Primes of the form (p! + q!)/ p! where p= prime(k) and q= prime(k+1), in order of increasing k.

Original entry on oeis.org

43, 157, 19183, 22651, 37057, 121453, 7923366007441921, 4496830293424385744456428801, 45045561823582321, 412807, 49907098805169447878401, 34672666242568358583785606401, 1041421

Offset: 1

K. D. Bajpai, Jan 09 2014

nonn

The 6th term has 6 digits; the 44th term has 44 digits.

The 685th term has 349 digits.

			43 is in the sequence because (5! + 7!)/ 5! = (120 + 5040)/120 = 43 which is prime and 5 and 7 are consecutive primes.
157 is in the sequence because (11! + 13!)/ 11! = (39916800 + 6227020800)/ 39916800 = 157 which is prime and 11 and 13 are consecutive primes.

K. D. Bajpai, Table of n, a(n) for n = 1..1000

Cf. A000040 (prime numbers).

Cf. A100858 (primes:(p-1)! + p).

KD := proc() local a,b,d; a:=ithprime(n); b:=ithprime(n+1); d:=(a! + b!)/ a!; if isprime(d) then RETURN (d); fi; end: seq(KD(), n=1..300);

Select[((#[[1]]!+#[[2]]!)/#[[1]]!&/@Partition[Prime[Range[ 300]], 2,1]), PrimeQ] (* Harvey P. Dale, Mar 07 2019 *)

A100603 Numbers k such that (prime(k)-1)! + prime(k)^4 is prime.

Original entry on oeis.org

1, 2, 4, 67, 212, 1615, 2570

Offset: 1

Jonathan Vos Post, Nov 30 2004

k = {1, 2, 4, 67} yields primes p(k) = {2, 3, 7, 331}. There are no more such k up to k=100. Computed in collaboration with Ray Chandler.

Terms a(5) and greater are only probable primes. - Iain Fox, Mar 05 2018

			a(3) = 4 because (prime(4)-1)! + prime(4)^4 = (7-1)! + 7^4 = 720 + 2401 = 3121 is the 3rd prime of this form.

Cf. A100858.

lst={};Do[p=Prime[n];If[PrimeQ[(p-1)!+p^4], AppendTo[lst, n]], {n, 10^2}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 08 2008 *)

is(k) = my(p=prime(k)); ispseudoprime((p-1)! + p^4) \\ Iain Fox, Mar 05 2018

Numbers k such that (prime(k)-1)! + prime(k)^4 is prime, where prime(k) is the k-th prime.

a(5) from Iain Fox, Mar 05 2018
a(6) from Iain Fox, Mar 11 2018
a(7) from Michael S. Branicky, May 29 2025

cp's OEIS Frontend

A100604 Numbers k such that (prime(k)-1)! + prime(k)^3 is prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A235267 Places n such that 1 + q!/p! is prime, where p = prime(n) and q = prime(n + 1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

A235392 Primes of the form (p! + q!)/ p! where p= prime(k) and q= prime(k+1), in order of increasing k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A100603 Numbers k such that (prime(k)-1)! + prime(k)^4 is prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions