cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A061214 Product of composite numbers between the n-th and (n+1)st primes.

Original entry on oeis.org

1, 4, 6, 720, 12, 3360, 18, 9240, 11793600, 30, 45239040, 59280, 42, 91080, 311875200, 549853920, 60, 1072431360, 328440, 72, 2533330800, 531360, 4701090240, 60072730099200, 970200, 102, 1157520, 108, 1367520, 1063186156509747740870400000, 2146560, 43191973440
Offset: 1

Views

Author

Amarnath Murthy, Apr 21 2001

Keywords

Examples

			a(4) = 8 * 9 * 10 = 720. 7 is the fourth prime and 11 is the fifth prime. a(5) = 12 as 11 and 13 both are primes.

Links

Crossrefs

Cf. A006530, A052248, A052297, A072472, A361760, A361761.
Cf. A046933 and A054265 (number and sum of these composites).

Programs

  • Haskell
    a061214 n = a061214_list !! (n-1)
a061214_list = f a000040_list where
   f (p:ps'@(p':ps)) = (product [p+1..p'-1]) : f ps'
-- Reinhard Zumkeller, Jun 22 2011
  • Maple
    A061214 := proc(n)
    local k ;
    product(k,k=ithprime(n)+1..ithprime(n+1)-1) ;
end proc: # R. J. Mathar, Apr 23 2013
  • Mathematica
    Table[Times@@Range[Prime[n]+1,Prime[n+1]-1],{n,30}] (* Harvey P. Dale, Jun 14 2011 *)
Times@@Range[#[[1]]+1,#[[2]]-1]&/@Partition[Prime[Range[40]],2,1] (* Harvey P. Dale, Apr 23 2022 *)
  • PARI
    { n=0; q=2; forprime (p=3, prime(2001), a=1; for (i=q + 1, p - 1, a*=i); q=p; write("b061214.txt", n++, " ", a) ) } \\ Harry J. Smith, Jul 19 2009
  • PARI
    v=primes(100);for(i=1,#v-1,v[i]=prod(j=v[i]+1,v[i+1]-1,j));vecextract(v,"1..-2") \\ Charles R Greathouse IV, Feb 27 2012
  • Python
    from math import prod
from sympy import prime
def A061214(n): return prod(i for i in range(prime(n)+1,prime(n+1))) # Chai Wah Wu, Jul 10 2022

Formula

A006530(a(n)) = A052248(n) for n > 1. - Reinhard Zumkeller, Jun 22 2011

Extensions

More terms from James Sellers, Apr 24 2001
Better definition from T. D. Noe, Jan 21 2008

A361760 a(n) = Product_{i=prime(n)..prime(n+1)-1} i.

Original entry on oeis.org

2, 12, 30, 5040, 132, 43680, 306, 175560, 271252800, 870, 1402410240, 2193360, 1722, 3916440, 14658134400, 29142257760, 3540, 65418312960, 22005480, 5112, 184933148400, 41977440, 390190489920, 5346472978828800, 94109400, 10302, 119224560, 11556, 149059680, 120140035685601494718355200000, 272613120
Offset: 1

Views

Author

Karl-Heinz Hofmann, Mar 23 2023

Keywords

Links

Crossrefs

Cf. A061214, A072472, A361761.

Programs

  • PARI
    a(n) = my(x=1); for(i=prime(n), prime(n+1)-1, x*=i); x; \\ Michel Marcus, Mar 28 2023
  • Python
    from sympy import prod, sieve
def A361760(n): return prod(range(sieve[n], sieve[n+1]))

Formula

a(n) = A000040(n)*A061214(n).

A361761 a(n) = Product_{i=prime(n)..prime(n+1)} i.

Original entry on oeis.org

6, 60, 210, 55440, 1716, 742560, 5814, 4037880, 7866331200, 26970, 51889178880, 89927760, 74046, 184072680, 776881123200, 1719393207840, 215940, 4383026968320, 1562389080, 373176, 14609718723600, 3484127520, 34726953602880, 518607878946393600, 9505049400, 1061106
Offset: 1

Views

Author

Karl-Heinz Hofmann, Mar 23 2023

Keywords

Links

Crossrefs

Cf. A006094, A061214, A072472, A112231, A361760.

Programs

  • PARI
    a(n) = my(x=1); for(i=prime(n), prime(n+1), x*=i); x; \\ Michel Marcus, Mar 28 2023
  • Python
    from sympy import prod, sieve
def A361761(n): return prod(range(sieve[n],sieve[n+1]+1))

Formula

a(n) = A006094(n)*A061214(n).

A109919 a(1) = 1, then product of consecutive composite numbers sandwiched between primes.

Original entry on oeis.org

1, 2, 1, 3, 4, 5, 6, 7, 720, 11, 12, 13, 3360, 17, 18, 19, 9240, 23, 11793600, 29, 30, 31, 45239040, 37, 59280, 41, 42, 43, 91080, 47, 311875200, 53, 549853920, 59, 60, 61, 1072431360, 67, 328440, 71, 72, 73, 2533330800, 79, 531360, 83, 4701090240, 89
Offset: 1

Views

Author

Amarnath Murthy, Jul 16 2005

Keywords

Comments

a(1) = a(3) = 1 as empty product is defined to be 1.
The odd numbered terms are in A061214. - T. D. Noe, Oct 02 2012

Crossrefs

Cf. A109920.
Cf. A072472.
Cf. A061214 (product of composite numbers between primes).

Programs

  • Maple
    A109919 := proc(n) local p; if n mod 2 = 0 then ithprime(n/2) ; elif n = 1 then 1 ; else p := ithprime((n-1)/2) ; mul(i,i=p+1..nextprime(p)-1) ; fi ; end: for n from 1 to 80 do printf("%d, ",A109919(n)) ; od ; # R. J. Mathar, May 02 2007

Formula

a(2n) = prime(n) and a(2n+1)= product of composite numbers between prime(n) and prime(n+1).
a(2n) = A000040(n). a(2n+1) = A072472(n)/A000040(n+1). - R. J. Mathar, May 02 2007

Extensions

More terms from R. J. Mathar, May 02 2007

A260754 a(n) = prime(n+1)! / prime(n).

Original entry on oeis.org

3, 40, 1008, 5702400, 566092800, 27360571392000, 7155594141696000, 1360632459941314560000, 384424434510421824110592000000, 283546160488893890266398720000000
Offset: 1

Views

Author

Altug Alkan, Aug 20 2015

Keywords

Examples

			a(2) = 5! / 3 = 40.

Crossrefs

Cf. A000040, A039716, A072472.

Programs

Formula

a(n) = prime(n+1)! / prime(n) = A039716(n+1) / A000040(n).
10^(n-1)|a(n+3) for n>=0. - G. C. Greubel, Aug 20 2015
Showing 1-5 of 5 results.

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