A298155 For any n >= 0 and k > 0, the prime(k)-adic valuation of a(n) equals the prime(k)-adic valuation of n + k (where prime(k) denotes the k-th prime).

1, 6, 5, 28, 3, 2, 11, 4680, 1, 2, 357, 76, 5, 6, 23, 16, 9, 770, 1, 348, 403, 2, 75, 8, 7, 1998, 1, 340, 1353, 86, 19, 672, 235, 26, 9, 4, 1, 36570, 7, 88, 3, 2, 295, 2196, 17, 98, 39, 400, 1943, 114, 11, 8804, 68985, 2, 1, 24, 1, 790, 3, 364, 1909, 3366, 185

Offset: 0

Author

Rémy Sigrist, Jan 13 2018

Keywords

• nonn

Comments

This sequence has similarities with A102370: here, for k > 0, a(n) and n + k have the same prime(k)-adic valuation, there, for k >= 0, A102370(n) and n + k have the same k-th binary digit (the least significant binary digit having index 0).

For any positive number, say k, we can use the Chinese remainder theorem to find a term that is a multiple of k; this term has index < k.

a(n) is even iff n is odd.

See A298161 for the indices of ones in the sequence.

Examples

			For n = 7:
- the 2-adic valuation of 7 + 1 is 3,
- the 3-adic valuation of 7 + 2 is 2,
- the 5-adic valuation of 7 + 3 is 1,
- the 7-adic valuation of 7 + 4 is 0,
- the 11-adic valuation of 7 + 5 is 0,
- the 13-adic valuation of 7 + 6 is 1,
- for k > 6, the prime(k)-adic valuation of 7 + k is 0,
- hence a(7) = 2^3 * 3^2 * 5^1 * 13^1 = 4680.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000040, A002110, A007814, A007949, A060175, A073605, A102370, A112765, A214411, A298161.

Programs

Maple

f:= proc(n) local v, p, k;
  v:= 1: p:= 1:
  for k from 1 do
    p:= nextprime(p);
    if p > n+k then return v fi;
    v:= v * p^padic:-ordp(n+k,p)
  od
end proc:
map(f, [$0..100]); # Robert Israel, Jan 16 2018

Mathematica

f[n_] := Module[{v = 1, p = 1, k}, For[k = 1, True, k++, p = NextPrime[p]; If[p > n + k, Return[v]]; v *= p^IntegerExponent[n + k, p]]];
f /@ Range[0, 100] (* Jean-François Alcover, Jul 30 2020, after Maple *)

PARI

a(n) = my (v=1, k=0); forprime(p=1, oo, k++; if (n+k < p, break); v *= p^valuation(n+k,p)); return (v)

Formula

For any n >= 0:

- a(n) = Product_{ k > 0 } A000040(k)^A060175(n + k, k) (this product is well defined as only finitely many terms are > 1),

- A007814(a(n)) = A007814(n + 1),

- A007949(a(n)) = A007949(n + 2),

- A112765(a(n)) = A112765(n + 3),

- A214411(a(n)) = A214411(n + 4),

- gcd(n, a(n)) = 1.

For any n > 0:

- a(A073605(n)) is a multiple of A002110(n).