A147693 Irregular triangle read by rows: T(n, k) = n mod prime(k), n >= 2, 1 <= k <= PrimePi(n), where PrimePi(n) = A000720(n).

0, 1, 0, 0, 1, 1, 2, 0, 0, 0, 1, 1, 1, 2, 0, 0, 2, 3, 1, 1, 0, 4, 2, 0, 1, 0, 3, 1, 2, 1, 4, 0, 0, 0, 2, 5, 1, 1, 1, 3, 6, 2, 0, 0, 2, 4, 0, 3, 1, 1, 0, 0, 1, 4, 2, 0, 1, 1, 2, 5, 3, 1, 2, 2, 3, 6, 4, 0, 0, 0, 3, 4, 7, 5, 1, 1, 1, 4, 5, 8, 6, 2, 0, 0, 2, 0, 6, 9, 7, 3, 1, 1, 0, 1, 0, 10, 8, 4, 2

Offset: 2

Author

Reikku Kulon, Nov 10 2008

Keywords

• easy
• nonn
• tabf

Comments

Equivalently, we define table, P, with columns numbered by the primes (2, 3, 5, ...) instead of 1, 2, 3, ... so that P(n, p) = n mod p.

P begins with P(2, 2).

A complex pattern emerges if values in the triangle are taken modulo 2.

Rows are unique. Row n has length A000720(n). - Jason Kimberley, Nov 2012

Examples

			Triangle P begins:
       2 3 5 7
     +---------
   2 | 0
   3 | 1 0
   4 | 0 1
   5 | 1 2 0
   6 | 0 0 1
   7 | 1 1 2 0
   8 | 0 2 3 1
   9 | 1 0 4 2
  10 | 0 1 0 3
  ...
Each row can be produced from the previous row by adding one to each number and resetting to zero any which would equal their column number. A row number n is prime, initiating a new column numbered n, iff P(n, p) is nonzero for all prime p < n; P(n, n) is then 0.
		

References

• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Exercise 3.7.22 on page 125.

Links

• Jason Kimberley, Rows n = 2..294 of irregular triangle, flattened
• Eric Weisstein's World of Mathematics, Redheffer Matrix

Crossrefs

Cf. A000720, A002321, A046992, A083058.

Programs

Magma

A147693 :=
func< n | [n mod p:p in PrimesUpTo(n)] >;
[A147693(n):n in[2..19]]; // Jason Kimberley, Nov 28 2012

Mathematica

row[n_]:=Table[Mod[n,Prime[i]], {i, PrimePi[n]}]; Array[row, 20, 2]//Flatten (* Stefano Spezia, Jul 17 2025 *)

Formula

a(A046992(n-1)+i) = T(n,i) = n mod A000040(i), where 1 <= i <= A000720(n). - Jason Kimberley, Nov 21 2012

Extensions

Edited by Peter Munn, May 25 2025