A353527 -id:A353527 - cp's OEIS frontend

A353526 The smallest prime not dividing n, reduced modulo 4.

2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2

Offset: 1

Antti Karttunen, Apr 24 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A010873, A053669, A353486, A353516, A353528, A353529.

Cf. A007395, A353527 (bisections).

a[n_] := Module[{p = 2}, While[Divisible[n, p], p = NextPrime[p]]; Mod[p, 4]]; Array[a, 100] (* Amiram Eldar, Jul 25 2022 *)

A053669(n) = forprime(p=2, , if(n%p, return(p))); \\ From A053669
A353526(n) = (A053669(n)%4);

a(n) = A010873(A053669(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} ((p mod 4)*(p-1)/(Product_{q prime, q <= p} q)) = 2.2324714414... . - Amiram Eldar, Jul 25 2022

A353487 a(n) = A276086(2*n) mod 4, where A276086 is the primorial base exp-function.

Original entry on oeis.org

1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 3

Offset: 0

Antti Karttunen, Apr 24 2022

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to primorial base

Even bisection of A353486. Odd bisection of A353516. Sequence A353517 shifted once left.

Cf. A010873, A276086, A353465, A353466, A353488, A353489, A353527.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A353487(n) = (A276086(2*n)%4);

a(n) = A353486(2*n) = A010873(A276086(2*n)).
a(n) = A353516(2*n + 1).
a(n) = A353517(1+n). [See comments in A353516 for a proof]
For n >= 1, a(n) = (A353517(n) * A353527(n)) mod 4.

A353517 The largest proper divisor of A276086(2*n) reduced modulo 4, where A276086(n) the primorial base exp-function.

Original entry on oeis.org

1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1

Offset: 0

Antti Karttunen, Apr 24 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to primorial base

Even bisection of A353516. Sequence A353487 shifted one term right.

Cf. A010873, A032742, A276086, A324895, A353527.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324895(n) = A032742(A276086(n));
A353517(n) = (A324895(2*n)%4);

a(n) = A353516(2*n) = A010873(A324895(2*n)).
For n >= 1, a(n) = (A353487(n) * A353527(n)) mod 4.
For n >= 1, a(n) = A353487(n-1). [See A353516 for a proof]

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A353526 The smallest prime not dividing n, reduced modulo 4.

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A353487 a(n) = A276086(2*n) mod 4, where A276086 is the primorial base exp-function.

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A353517 The largest proper divisor of A276086(2*n) reduced modulo 4, where A276086(n) the primorial base exp-function.

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