A297934 Triangular array T(n, k), read by rows: least common prime factor of n and k, or 0 if n and k are coprime.

0, 2, 0, 0, 0, 0, 2, 3, 2, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 2, 0, 0, 3, 0, 0, 3, 0, 0, 2, 0, 2, 5, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 2, 0, 2, 0, 2, 3, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 2, 7, 2, 0, 2, 0, 2, 0, 0, 3, 0, 5, 3, 0, 0, 3, 5

Offset: 3

Author

Felix Fröhlich, Jan 09 2018

Keywords

• nonn
• tabl
• easy

Comments

n is prime (A000040) if and only if Sum_{i=2..n-1} T(n, i) = 0.

n is a prime power (A025475) if and only if for any two x, y such that both T(n, x), T(n, y) > 0 also T(n, x) = T(n, y).

Examples

			============================================================
. n \ k |  2   3   4   5   6   7   8   9  10  11  12  13  14
--------|---------------------------------------------------
.   3   |  0
.   4   |  2   0
.   5   |  0   0   0
.   6   |  2   3   2   0
.   7   |  0   0   0   0   0
.   8   |  2   0   2   0   2   0
.   9   |  0   3   0   0   3   0   0
.  10   |  2   0   2   5   2   0   2   0
.  11   |  0   0   0   0   0   0   0   0   0
.  12   |  2   3   2   0   2   0   2   3   2   0
.  13   |  0   0   0   0   0   0   0   0   0   0   0
.  14   |  2   0   2   0   2   7   2   0   2   0   2   0
.  15   |  0   3   0   5   3   0   0   3   5   0   3   0   0
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 3..11028 (rows 3 <= n <= 150).

Crossrefs

Cf. A000040, A025475, A081520.

Programs

Mathematica

Table[If[CoprimeQ[n, k], 0, First@ Intersection[FactorInteger[n][[All, 1]], FactorInteger[k][[All, 1]] ]], {n, 3, 15}, {k, 2, n - 1}] // Flatten (* Michael De Vlieger, Jan 23 2018 *)

PARI

t(n, k) = if(gcd(n, k) > 1, my(f=factor(n)[, 1]~, g=factor(k)[, 1]~); return(vecmin(setintersect(f, g)))); 0
trianglerows(n) = for(x=3, n+2, for(y=2, x-1, print1(t(x, y), ", ")); print(""))
trianglerows(13) \\ print upper 13 rows of triangle

Extensions

Value of T(12, 6) and PARI program corrected by Felix Fröhlich, Jan 23 2018