A244798 -id:A244798 - cp's OEIS frontend

A244740 Irregular triangular array read by rows: T(n,k) = number of positive integers m such that (prime(n) mod m) = k, for k=1..(-1 + prime(k))/2.

1, 2, 1, 3, 1, 1, 3, 2, 2, 1, 1, 5, 1, 2, 1, 1, 1, 4, 3, 2, 1, 2, 1, 1, 1, 5, 1, 3, 2, 2, 1, 1, 1, 1, 3, 3, 4, 1, 3, 1, 2, 1, 1, 1, 1, 5, 3, 2, 2, 4, 1, 2, 1, 2, 1, 1, 1, 1, 1, 7, 1, 4, 2, 2, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 8, 3, 2, 2, 3, 1, 3, 1, 2, 1, 2, 1

Offset: 1

Clark Kimberling, Jul 06 2014

(sum of numbers in row n) = prime(n+1)-2; (column 1) = A244796; (column 2) = A244797; (column 3) = A244798.

			First 12 rows:
1
2 1
3 1 1
3 2 2 1 1
5 1 2 1 1 1
4 3 2 1 2 1 1 1
5 1 3 2 2 1 1 1 1
3 3 4 1 3 1 2 1 1 1
5 3 2 2 4 1 2 1 2 1 1 1 1 1
7 1 4 2 2 1 3 1 2 1 1 1 1 1 1
8 3 2 2 3 1 3 1 2 1 2 1 1 1 1 1 1 1
7 3 2 1 5 2 2 2 2 1 2 1 2 1 1 1 1 1 1 1
For row 4, count these congruences:
11 = (1 mod m) for m = 2, 5, 10;
11 = (2 mod m) for m = 3, 9;
11 = (3 mod m) for m = 4, 8;
11 = (4 mod m) for m = 7;
11 = (5 mod m) for m = 6, so that (row 4) = (3,2,2,1,1).

Clark Kimberling, Table of n, a(n) for n = 1..500

Cf. A000040, A244796, A244797, A244798, A244799, A244800.

z = 60; f[n_, m_, k_] := f[n, m, k] = If[Mod[Prime[n], m] == k, 1, 0];
t[k_] := t[k] = Table[f[n, m, k], {n, 1, z}, {m, 1, -1 + Prime[n]}];
u = Table[Count[t[k][[i]], 1], {k, 1, 40}, {i, 1, z}];
v = Table[u[[n, k]], {k, 2, 20}, {n, 1, (-1 + Prime[k])/2}]
Flatten[v] (* A244740 *)

A244796 Number of moduli m such that (prime(n) mod m) = 1.

Original entry on oeis.org

0, 1, 2, 3, 3, 5, 4, 5, 3, 5, 7, 8, 7, 7, 3, 5, 3, 11, 7, 7, 11, 7, 3, 7, 11, 8, 7, 3, 11, 9, 11, 7, 7, 7, 5, 11, 11, 9, 3, 5, 3, 17, 7, 13, 8, 11, 15, 7, 3, 11, 7, 7, 19, 7, 8, 3, 5, 15, 11, 15, 7, 5, 11, 7, 15, 5, 15, 19, 3, 11, 11, 3, 7, 11, 15, 3, 5, 17

Offset: 1

Clark Kimberling, Jul 06 2014

Except for the initial 0, this is column 1 of the array at A244740,

			prime(5) = 11 = (1 mod m) for m = 2, 5, 10, so that a(5) = 3.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A244740, A244797, A244798.

z = 300; f[n_, m_] := If[Mod[Prime[n], m] == 1, 1, 0];
t = Table[f[n, m], {n, 1, z}, {m, 1, Prime[n]}];
Table[Count[t[[k]], 1], {k, 1, z}] (* A244796 *)

a(n) = A000005(prime(n) - 1) - 1. - Gionata Neri, Jul 04 2017

A244797 Number of moduli m such that (prime(n) mod m) = 2.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 3, 1, 3, 3, 1, 3, 3, 1, 5, 3, 3, 1, 3, 3, 1, 3, 4, 3, 3, 5, 1, 7, 1, 3, 3, 3, 7, 1, 5, 1, 3, 3, 7, 5, 3, 1, 7, 1, 7, 1, 3, 3, 8, 1, 7, 3, 1, 3, 7, 5, 3, 1, 5, 5, 1, 3, 3, 3, 1, 11, 3, 3, 7, 1, 7, 7, 3, 3, 3, 3, 5, 3, 7, 3, 3, 1, 7, 1, 3, 8

Offset: 1

Clark Kimberling, Jul 06 2014

Except for the initial 0,0, this is column 2 of the array at A244740.

			prime(5) = 11 = (2 mod m) for m = 3, 9 so that a(5) = 2.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A244740, A244796, A244798.

Cf. A032741, A040976.

z = 300; f[n_, m_] := If[Mod[Prime[n], m] == 2, 1, 0];
t = Table[f[n, m], {n, 1, z}, {m, 1, Prime[n]}];
Table[Count[t[[k]], 1], {k, 1, z}] (* A244797 *)

a(n) = A032741(A040976(n)). - Ridouane Oudra, Mar 17 2024

cp's OEIS Frontend

A244740 Irregular triangular array read by rows: T(n,k) = number of positive integers m such that (prime(n) mod m) = k, for k=1..(-1 + prime(k))/2.

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A244796 Number of moduli m such that (prime(n) mod m) = 1.

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A244797 Number of moduli m such that (prime(n) mod m) = 2.

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