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.
A214089(14) = 1430083494841, n#_14 = 13082761331670030, and (1430083494841^2 - 1) / (4 * 13082761331670030) = 39080794, so a(14) = 39080794.
A215085 := proc(n) (A214089(n)^2-1)/4/A002110(n) ; end proc: # R. J. Mathar, Aug 21 2012
from itertools import product from sympy import sieve, prime, isprime, primorial from sympy.ntheory.modular import crt def A215085(n): return ( 1 if n == 1 else ( int( min( filter( isprime, ( crt(tuple(sieve.primerange(prime(n) + 1)), t)[0] for t in product((1, -1), repeat=n) ), ) ) ) ** 2 - 1 ) // 4 // primorial(n) ) # Chai Wah Wu, May 31 2022 for n in range(1, 16): print(A215085(n), end=", ")
n smallest k k*(k+1) prime factorization 1 1 2 2 2 2*3 3 5 2*3*5 4 14 2*3*5*7 5 384 2^7*3*5*7*11 6 1715 2^2*3*7^3*11*13 7 714 2*3*5*7*11*13*17 8 633555 2^2*3^3*5*7*11^3*13*17*19^2
a193314 n = head [k | k <- [1..], let kk' = a002378 k,
mod kk' (a002110 n) == 0, a006530 kk' == a000040 n]
-- Reinhard Zumkeller, Jun 14 2015
f[n_] := Block[{k = 1, p = Fold[ Times, 1, Prime@ Range@ n], tst = Prime@ Range@ n},While[ First@ Transpose@ FactorInteger[ k*p]!=tst || IntegerQ@ Sqrt[ 4k*p+1], k++]; Floor@ Sqrt[k*p]]; Array[f, 8]
(* the search for a(9), I also used *) lst = {}; p = Prime@ Range@ 9; Do[ q = {a, b, c, d, e, f, g, h, i}; If[ IntegerQ[ Sqrt[4Times @@ (p^q) + 1]], r = Floor@ Sqrt@ Times @@ (p^q); Print@ r; AppendTo[lst, r]], {i, 9}, {h, 9}, {g, 9}, {f, 10}, {e, 11}, {d, 14}, {c, 16}, {b, 24}, {a, 8}]
a(n)={
my(v=[Mod(0,1)],u,P=1,t,g,k);
forprime(p=2,prime(n),
P*=p;
u=List();
for(i=1,#v,
listput(u,chinese(v[i],Mod(-1,p)));
listput(u,chinese(v[i],Mod(0,p)))
);
v=0;v=Vec(u)
);
v=vecsort(lift(v));
while(1,
for(i=1,#v,
t=(v[i]+k)*(v[i]+k+1)/P;
if(!t,next);
while((g=gcd(P,t))>1, t/=g);
if (t==1, return(v[i]+k))
);
k += P
)
}; \\ Charles R Greathouse IV, Aug 18 2011
2, 3, and 5 are the first three primes. The first oblong number that is a multiple of all three first primes is 30. So, a(3) = 30. The first oblong number that is a multiple of primorial(5) = 2310 is 19*2310 = 43890, so a(5) = 43890.
a002110(n) = prod(i=1,n, prime(i)) \\ after Washington Bomfim in A002110 a344005(n) = for(m=1, oo, if((m*(m+1))%n==0, return(m))) a(n) = my(m=a344005(a002110(n))); m*(m+1) \\ Felix Fröhlich, May 31 2022
from sympy import integer_nthroot, primorial
def oblong(n): r = integer_nthroot(n, 2)[0]; return r*(r+1) == n
def a(n):
m = psharp = primorial(n)
while not oblong(m): m += psharp
return m
print([a(n) for n in range(1, 11)]) # Michael S. Branicky, May 25 2022
# faster alternative using Python 3.8+ A344005(n) by Chai Wah Wu from sympy import primorial def a(n): return (m := A344005(primorial(n)))*(m+1) print([a(n) for n in range(1, 18)]) # Michael S. Branicky, May 26 2022
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