A073605 -id:A073605 - cp's OEIS frontend

A083401 Numbers k such that A073605(k) is prime.

3, 4, 5, 6, 15, 21, 25, 27, 28, 52, 70, 205, 245, 308, 339, 357, 415, 524, 537, 1455, 1500

Offset: 1

Farideh Firoozbakht, Jun 07 2003

No other terms below 2200. [Max Alekseyev, May 11 2009]

			a(1)=3 because A073605(3)=7 is prime and A073605(1) and A073605(2) aren't primes.

Cf. A053664, A073605, A053664.

For[k = 0, k < 875, If[PrimeQ[ChineseRemainder[ -Range[k], Prime[Range[k]]]], Print[k]], k++ ]

a(20)-a(21) from Max Alekseyev, May 11 2009

A083400 Numbers n such that A053664(n) and A073605(n) are primes.

Original entry on oeis.org

3, 4, 5, 6, 27

Offset: 1

Farideh Firoozbakht, Jun 07 2003

			a(4)=6 because A053664(6)=29243 and A073605(6)=787 are primes.

Cf. A053664, A073605, A083376.

For[k = 0, k < 1415, If[PrimeQ[ChineseRemainder[Range[k], Prime[Range[k]]]] && PrimeQ[ChineseRemainder[ -Range[k], Prime[Range[k]]]], Print[k]], k++ ]

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).

Original entry on oeis.org

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

Rémy Sigrist, Jan 13 2018

nonn

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.

			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.

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

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

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

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 *)

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)

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).

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A083401 Numbers k such that A073605(k) is prime.

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A083400 Numbers n such that A053664(n) and A073605(n) are primes.

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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).

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