A108552 -id:A108552 - cp's OEIS frontend

A060462 Integers k such that k! is divisible by k*(k+1)/2.

1, 3, 5, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 23, 24, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94

Offset: 1

Michel ten Voorde, Apr 09 2001

nonn

k! / (k-th triangular number) is an integer.
a(n) = A072668(n) for n>0.
From Bernard Schott, Dec 11 2020: (Start)
Numbers k such that Sum_{j=1..k} j divides Product_{j=1..k} j.
k is a term iff k != p-1 with p is an odd prime (see De Koninck & Mercier reference).
The ratios obtained a(n)!/T(a(n)) = A108552(n). (End)

			5 is a term because 5*4*3*2*1 = 120 is divisible by 5 + 4 + 3 + 2 + 1 = 15.

Jean-Marie De Koninck & Armel Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 181 pp. 31 and 163, Ellipses, Paris, 2004.
Joseph D. E. Konhauser et al., Which Way Did The Bicycle Go?, Problem 98, pp. 29; 145-146, MAA Washington DC, 1996.

Harry J. Smith, Table of n, a(n) for n = 1..2001 [offset adapted by _Georg Fischer_, Jan 04 2021]

Cf. A000142, A000217, A072668, A108552.

for n from 1 to 300 do if n! mod (n*(n+1)/2) = 0 then printf(`%d,`,n) fi:od:

Select[Range[94], Mod[#!, #*(# + 1)/2] == 0 &] (* Jayanta Basu, Apr 24 2013 *)

{ f=1; t=0; n=-1; for (m=1, 4000, f*=m; t+=m; if (f%t==0, write("b060462.txt", n++, " ", m)); if (n==2000, break); ) } \\ Harry J. Smith, Jul 05 2009

from sympy import composite
def A060462(n): return composite(n-1)-1 if n>1 else 1 # Chai Wah Wu, Aug 02 2024

Corrected and extended by Henry Bottomley and James Sellers, Apr 11 2001

Offset corrected by Alois P. Heinz, Dec 11 2020

A116536 Numbers that can be expressed as the ratio of the product and the sum of consecutive prime numbers starting from 2.

Original entry on oeis.org

1, 3, 125970, 1278362451795, 305565807424800745258151050335, 2099072522743338791053378243660769678400212601239922213271230, 330455532167461882998265688366895823334392289157931734871641555

Offset: 1

Paolo P. Lava & Giorgio Balzarotti, Mar 27 2006

Let prime(i) denote the i-th prime (A000040). Let F(m) = (Product_{i=1..m} prime(i)) / (Sum_{i=1..m} prime(i)). Sequence gives integer values of F(m) and A051838 gives corresponding values of m. - N. J. A. Sloane, Oct 01 2011

			a(1) = 1 because 2/2 = 1.
a(2) = 3 because (2*3*5)/(2+3+5) = 30/10 = 3.
a(3) = 125970 because (2*3*5*7*11*13*17*19)/(2+3+5+7+11+13+17+19) = 9699690/77 = 125790.

G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 158.

Amiram Eldar, Table of n, a(n) for n = 1..81 (terms 1..42 from Vincenzo Librandi)

Cf. A001414, A108552, A067111, A051838, A140763, A141092, A159578.

Subsequence of A325307.

import Data.Maybe (catMaybes)
a116536 n = a116536_list !! (n-1)
a116536_list = catMaybes $ zipWith div' a002110_list a007504_list where
   div' x y | m == 0    = Just x'
            | otherwise = Nothing where (x',m) = divMod x y
-- Reinhard Zumkeller, Oct 03 2011

[p/s: n in [1..40] | IsDivisibleBy(p,s) where p is &*[NthPrime(i): i in [1..n]] where s is &+[NthPrime(i): i in [1..n]]];  // Bruno Berselli, Sep 30 2011

P:=proc(n) local i,j, pp,sp; pp:=1; sp:=0; for i from 1 by 1 to n do pp:=pp*ithprime(i); sp:=sp+ithprime(i); j:=pp/sp; if j=trunc(j) then print(j); fi; od; end: P(100);

seq = {}; sum = 0; prod = 1; p = 1; Do[p = NextPrime[p]; prod *= p; sum += p; If[Divisible[prod, sum], AppendTo[seq, prod/sum]], {50}]; seq (* Amiram Eldar, Nov 02 2020 *)

a(n) = A002110(A051838(n)) / A007504(A051838(n)). - Reinhard Zumkeller, Oct 03 2011

a(n) = A159578(n)/A001414(A159578(n)). - Amiram Eldar, Nov 02 2020

A130318 Integer values of k!!/S(k), where S(k) is the sum of all odd numbers less than or equal to k, if k is odd, or the sum of all even numbers less than or equal to k, if k is even.

Original entry on oeis.org

1, 1, 4, 128, 11520, 143360, 425425, 2064384, 619315200, 280284364800, 6801567252480, 27512370460575, 178541140377600, 152355106455552000, 167834385271436083200, 6074006324109115392000, 29734853645550994565625, 231916605102348042240000, 392866729043377583554560000

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, May 23 2007

For n >= 8, a(n) ends with 0 or 5.

			6 --> 6!! = 48; 6 + 4 + 2 = 12; 48/12 = 4.
17 --> 17!! = 34459425; 17 + 15 + 13 + 11 + 9 + 7 + 5 + 3 + 1 = 81; 34459425/81 = 425425.

Jayanta Basu, Table of n, a(n) for n = 0..100

Cf. A108552, A000290, A005408, A130319.

Cf. A002620, A006882.

f:= n-> `if`(irem(doublefactorial(n), floor((n+1)^2/4), 'r')=0, r, [][]):
map(f, [$1..50])[];  # Alois P. Heinz, Mar 16 2024

Select[Table[Times @@ (t = If[OddQ[n], Range[1, n, 2], Range[2, n, 2]])/Plus @@ t, {n, 41}], IntegerQ] (* Jayanta Basu, Aug 12 2013 *)

Integers of the form k!!/((k+1)/2)^2, for k odd and k!!/(k*(k+2)/4) for k even. [corrected by Jon E. Schoenfield, Mar 16 2024]

A130319 Numbers k for which k!!/S(k) is an integer, where S(k) is the sum of all odd numbers less than or equal to k, if k is odd, or the sum of all even numbers less than or equal to k, if k is even.

Original entry on oeis.org

1, 2, 6, 10, 14, 16, 17, 18, 22, 26, 28, 29, 30, 34, 38, 40, 41, 42, 46, 48, 49, 50, 52, 53, 54, 58, 62, 64, 65, 66, 68, 69, 70, 74, 76, 77, 78, 82, 86, 88, 89, 90, 94, 96, 97, 98, 100, 101, 102, 106, 108, 109, 110, 112, 113, 114, 118, 122, 124, 125, 126, 128

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, May 23 2007

			6 --> 6!! = 48; 6 + 4 + 2 = 12; 48/12 = 4.
17 --> 17!! = 34459425; 17 + 15 + 13 + 11 + 9 + 7 + 5 + 3 + 1 = 81; 34459425/81 = 425425.

Harvey P. Dale, Table of n, a(n) for n = 0..10000

Cf. A108552, A000290, A005408, A130318.

Cf. A002620, A006882.

q:= n-> irem(doublefactorial(n), floor((n+1)^2/4))=0:
select(q, [$1..200])[];  # Alois P. Heinz, Mar 16 2024

r[n_] := If[OddQ[n], Range[1, n, 2], Range[2, n, 2]]; Select[Range[100], Divisible[Times @@ (x = r[#]), Plus @@ x] &] (* Jayanta Basu, Aug 12 2013 *)
Select[Range[100],If[OddQ[#],Divisible[#!!,((#+1)/2)^2],Divisible[#!!,(#(#+2))/4]]&] (* Harvey P. Dale, Nov 30 2016 *)

A130332 Integer values of n!!/sum(i=0..n,n), with n>=1.

Original entry on oeis.org

1, 1, 21, 1485, 6144, 225225, 17694720, 59520825, 6539968512, 24325703325, 145332633600, 14230536445125, 2596962041856000, 11288163762500625, 78354054748569600, 11665426077721040625, 86068915523813376000

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, May 24 2007

After the ninth term all the other numbers end in 0 or 5.

			5!! = 5*3*1 = 15; 5+4+3+2+1 = 15; 15/15 = 1.
13!! = 13*11*9*7*5*3*1 = 135135; 13+12+11+10+9+8+7+6+5+4+3+2+1 = 91; 135135/91 = 1485.

Harvey P. Dale, Table of n, a(n) for n = 0..466

Cf. A130318, A126695, A108552.

P:=proc(n) local a,i,j,k,w; for i from 1 by 1 to n do k:=i; w:=i-2; while w>0 do k:=k*w; w:=w-2; od; j:=sum('w','w'=1..i); a:=k/j; if trunc(a)=a then print(a) fi; od; end: P(100);

Select[Table[n!!/((n(n+1))/2),{n,50}],IntegerQ] (* Harvey P. Dale, Jul 24 2019 *)

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A060462 Integers k such that k! is divisible by k*(k+1)/2.

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A116536 Numbers that can be expressed as the ratio of the product and the sum of consecutive prime numbers starting from 2.

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A130318 Integer values of k!!/S(k), where S(k) is the sum of all odd numbers less than or equal to k, if k is odd, or the sum of all even numbers less than or equal to k, if k is even.

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A130319 Numbers k for which k!!/S(k) is an integer, where S(k) is the sum of all odd numbers less than or equal to k, if k is odd, or the sum of all even numbers less than or equal to k, if k is even.

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A130332 Integer values of n!!/sum(i=0..n,n), with n>=1.

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