A206590 -id:A206590 - cp's OEIS frontend

A206588 Number of solutions k of prime(k)=prime(n) (mod n), where 1<=k

0, 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 1, 2, 2, 0, 2, 1, 2, 1, 1, 1, 2, 1, 1, 0, 2, 0, 3, 1, 2, 2, 3, 1, 3, 1, 1, 2, 2, 1, 3, 1, 3, 2, 2, 1, 3, 1, 3, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 0, 1, 1, 0, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 3, 0, 3, 0, 1, 1, 2, 0, 4, 1, 2, 1, 3, 1, 5, 1, 1, 0, 1, 0, 2, 0, 2, 1, 2

Offset: 2

Clark Kimberling, Feb 09 2012

nonn

In the following guide to related sequences, c(n) is the number of solutions (n,k) of s(k)=s(n) (mod n), where 1<=k

s(n).............c(n)

prime(n).........A206588

prime(n+1).......A206589

n^2..............A057918

n^3..............A206590

Fibonacci(n+1)...A206713

2^(n-1)..........A206714

n!...............A072480

n(n+1)/2.........A206824

n^4..............A206825

n(n+1)(n+2)/6....A206826

n(n+1)(2n+1)/6...A206827

C(2n,n)..........A206828

For some choices of s, the limiting frequency of 0's in c appears to be a positive constant.

			For k=1 to 7, the numbers p(8)-p(k) are 17,16,14,12,8,6,4, so that a(8)=2.

Cf. A206589.

f[n_, k_] := If[Mod[Prime[n] - Prime[k], n] == 0, 1, 0];
t[n_] := Flatten[Table[f[n, k], {k, 1, n - 1}]]
a[n_] := Count[Flatten[t[n]], 1]
Table[a[n], {n, 2, 120}]  (* A206588 *)

A206825 Number of solutions (n,k) of k^4=n^4 (mod n), where 1<=k

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 3, 2, 0, 0, 1, 0, 0, 0, 7, 0, 2, 0, 1, 0, 0, 0, 3, 4, 0, 8, 1, 0, 0, 0, 7, 0, 0, 0, 5, 0, 0, 0, 3, 0, 0, 0, 1, 2, 0, 0, 7, 6, 4, 0, 1, 0, 8, 0, 3, 0, 0, 0, 1, 0, 0, 2, 15, 0, 0, 0, 1, 0, 0, 0, 11, 0, 0, 4, 1, 0, 0, 0, 7, 26, 0, 0, 1, 0, 0, 0, 3, 0, 2, 0, 1, 0, 0, 0, 7, 0, 6, 2

Offset: 2

Clark Kimberling, Feb 12 2012

nonn

			8 divides exactly three of the numbers 8^4-k^4 for k = 1, 2 , ..., 7, so that a(8) = 3.

Antti Karttunen, Table of n, a(n) for n = 2..16384

Cf. A206590.

s[k_] := k^4;
f[n_, k_] := If[Mod[s[n] - s[k], n] == 0, 1, 0];
t[n_] := Flatten[Table[f[n, k], {k, 1, n - 1}]]
a[n_] := Count[Flatten[t[n]], 1]
Table[a[n], {n, 2, 120}]  (* A206825 *)

A206825(n) = { my(n4 = n^4); sum(k=1,n-1,!((n4-(k^4))%n)); }; \\ Antti Karttunen, Nov 17 2017

A206826 Number of solutions (n,k) of s(k)=s(n) (mod n), where 1<=k

Original entry on oeis.org

0, 2, 1, 2, 1, 2, 1, 2, 3, 2, 2, 2, 3, 4, 1, 2, 1, 2, 6, 3, 3, 2, 4, 2, 3, 2, 6, 2, 5, 2, 1, 3, 3, 8, 3, 2, 3, 4, 6, 2, 3, 2, 6, 3, 3, 2, 2, 2, 3, 5, 6, 2, 1, 8, 6, 5, 3, 2, 8, 2, 3, 5, 1, 8, 5, 2, 6, 4, 12, 2, 2, 2, 3, 3, 6, 8, 4, 2, 6, 2, 3, 2, 8, 8, 3, 3, 6, 2, 5, 8, 6, 4, 3, 8, 2, 2, 3, 4, 6

Offset: 1

Clark Kimberling, Feb 12 2012

nonn

			5 divides exactly two of the numbers s(n)-s(k) for k=1,2,3,4, so that a(5)=2.

Cf. A206590.

s[k_] := k (k + 1) (k + 2)/6;
f[n_, k_] := If[Mod[s[n] - s[k], n] == 0, 1, 0];
t[n_] := Flatten[Table[f[n, k], {k, 1, n - 1}]]
a[n_] := Count[Flatten[t[n]], 1]
Table[a[n], {n, 2, 120}]  (* A206826 *)

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cp's OEIS Frontend

A206588 Number of solutions k of prime(k)=prime(n) (mod n), where 1<=k

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A206825 Number of solutions (n,k) of k^4=n^4 (mod n), where 1<=k

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A206826 Number of solutions (n,k) of s(k)=s(n) (mod n), where 1<=k

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