A244800 -id:A244800 - 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 *)

A244799 Number of moduli m such that (prime(n) mod m) is odd, where 1 <= m < prime(n).

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 9, 12, 14, 17, 21, 23, 25, 29, 29, 33, 37, 41, 42, 46, 49, 51, 52, 56, 62, 64, 68, 66, 70, 74, 83, 86, 86, 90, 93, 99, 103, 108, 106, 111, 113, 119, 123, 125, 124, 130, 139, 147, 144, 148, 148, 156, 160, 163, 164, 168, 174, 182, 180, 182

Offset: 1

Clark Kimberling, Jul 06 2014

a(n) + A244800(n) = A000040(n) = prime(n).

			In the following table, mh abbreviates mod(h) and p(n) = prime(n).
n . p(n) . m2 . m3 . m4 . m5 . m6 . m7 . m8 . m9 . m10 . m11 #odd #even
1 . 2 .... 0 .. 0 ........................................... 0 .. 2
2 . 3 .... 0 .. 1 .. 0 ...................................... 1 .. 2
3 . 5 .... 0 .. 1 .. 2 .. 1 .. 0 ............................ 2 .. 3
4 . 7 .... 0 .. 1 .. 1 .. 3 .. 2 .. 1 .. 0 .................. 4 .. 3,
so that A244799 = (0,1,2,4,...) and A244800 = (2,2,3,3,...).

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

Cf. A244800, A000040.

z = 1000; f[n_, m_] := If[OddQ[Mod[Prime[n], m]], 1, 0]
t = Table[f[n, m], {n, 1, z}, {m, 1, Prime[n]}];
Table[Count[t[[k]], 1], {k, 1, z}] (* A244799 *)
Table[With[{p=Prime[n]},Count[Mod[p,Range[p-1]],?OddQ]],{n,60}] (* _Harvey P. Dale, Jul 24 2022 *)

A157654 Triangle T(n, k, m) = 1 if k = 0 or k = n, otherwise m*abs( (n-k)^(m-1) - k^(m-1) ), with m = 2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 4, 0, 4, 1, 1, 6, 2, 2, 6, 1, 1, 8, 4, 0, 4, 8, 1, 1, 10, 6, 2, 2, 6, 10, 1, 1, 12, 8, 4, 0, 4, 8, 12, 1, 1, 14, 10, 6, 2, 2, 6, 10, 14, 1, 1, 16, 12, 8, 4, 0, 4, 8, 12, 16, 1

Offset: 0

Roger L. Bagula, Mar 03 2009

For the cases of m = 0, 1 the triangles reduce to T(n, k, m) = A103451(n, k). - G. C. Greubel, Dec 13 2021

			Triangle begins as:
  1;
  1,  1;
  1,  0,  1;
  1,  2,  2,  1;
  1,  4,  0,  4,  1;
  1,  6,  2,  2,  6,  1;
  1,  8,  4,  0,  4,  8,  1;
  1, 10,  6,  2,  2,  6, 10,  1;
  1, 12,  8,  4,  0,  4,  8, 12,  1;
  1, 14, 10,  6,  2,  2,  6, 10, 14,  1;
  1, 16, 12,  8,  4,  0,  4,  8, 12, 16, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000007, A103451, A244800.

T:= func< n,k,q | k eq 0 or k eq n select 1 else q*Abs( (n-k)^(q-1) - k^(q-1) ) >;
[T(n,k,2): k in [0..n], n in [0..15]]; // G. C. Greubel, Dec 13 2021

T[n_, k_, m_]:= T[n, k, m]= If[k==0 || k==n, 1, m*Abs[(n-k)^(m-1) - k^(m-1)]];
Table[T[n,k,2], {n,0,15}, {k,0,n}]//Flatten

def A157684(n,k,q): return 1 if (k==0 or k==n) else q*abs((n-k)^(q-1) - k^(q-1))
flatten([[A157684(n,k,2) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Dec 13 2021

T(n, k, m) = 1 if k = 0 or k = n, otherwise m*abs( (n-k)^(m-1) - k^(m-1) ), with m = 2.
From G. C. Greubel, Dec 13 2021: (Start)
Sum_{k=0..n} T(n, k, 2) = (-1)*[n==0] + A244800(n-1).
T(2*n, n, 2) = A000007(n). (End)

Edited by G. C. Greubel, Dec 13 2021

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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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A244799 Number of moduli m such that (prime(n) mod m) is odd, where 1 <= m < prime(n).

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A157654 Triangle T(n, k, m) = 1 if k = 0 or k = n, otherwise m*abs( (n-k)^(m-1) - k^(m-1) ), with m = 2, read by rows.

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