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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).

%I A298155 #30 Jul 30 2020 07:18:00
%S A298155 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,
%T A298155 1998,1,340,1353,86,19,672,235,26,9,4,1,36570,7,88,3,2,295,2196,17,98,
%U A298155 39,400,1943,114,11,8804,68985,2,1,24,1,790,3,364,1909,3366,185
%N 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).
%C A298155 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).
%C A298155 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.
%C A298155 a(n) is even iff n is odd.
%C A298155 See A298161 for the indices of ones in the sequence.
%H A298155 Rémy Sigrist, <a href="/A298155/b298155.txt">Table of n, a(n) for n = 0..10000</a>
%F A298155 For any n >= 0:
%F A298155 - a(n) = Product_{ k > 0 } A000040(k)^A060175(n + k, k) (this product is well defined as only finitely many terms are > 1),
%F A298155 - A007814(a(n)) = A007814(n + 1),
%F A298155 - A007949(a(n)) = A007949(n + 2),
%F A298155 - A112765(a(n)) = A112765(n + 3),
%F A298155 - A214411(a(n)) = A214411(n + 4),
%F A298155 - gcd(n, a(n)) = 1.
%F A298155 For any n > 0:
%F A298155 - a(A073605(n)) is a multiple of A002110(n).
%e A298155 For n = 7:
%e A298155 - the 2-adic valuation of 7 + 1 is 3,
%e A298155 - the 3-adic valuation of 7 + 2 is 2,
%e A298155 - the 5-adic valuation of 7 + 3 is 1,
%e A298155 - the 7-adic valuation of 7 + 4 is 0,
%e A298155 - the 11-adic valuation of 7 + 5 is 0,
%e A298155 - the 13-adic valuation of 7 + 6 is 1,
%e A298155 - for k > 6, the prime(k)-adic valuation of 7 + k is 0,
%e A298155 - hence a(7) = 2^3 * 3^2 * 5^1 * 13^1 = 4680.
%p A298155 f:= proc(n) local v, p, k;
%p A298155   v:= 1: p:= 1:
%p A298155   for k from 1 do
%p A298155     p:= nextprime(p);
%p A298155     if p > n+k then return v fi;
%p A298155     v:= v * p^padic:-ordp(n+k,p)
%p A298155   od
%p A298155 end proc:
%p A298155 map(f, [$0..100]); # _Robert Israel_, Jan 16 2018
%t A298155 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]]];
%t A298155 f /@ Range[0, 100] (* _Jean-François Alcover_, Jul 30 2020, after Maple *)
%o A298155 (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)
%Y A298155 Cf. A000040, A002110, A007814, A007949, A060175, A073605, A102370, A112765, A214411, A298161.
%K A298155 nonn
%O A298155 0,2
%A A298155 _Rémy Sigrist_, Jan 13 2018