A081520 -id:A081520 - cp's OEIS frontend

A081518 Final term in row n of A081520.

1, 2, 6, 6, 20, 8, 42, 14, 24, 15, 110, 16, 156, 22, 30, 30, 272, 26, 342, 32, 48, 38, 506, 34, 120, 46, 78, 48, 812, 39, 930, 62, 81, 62, 110, 52, 1332, 70, 99, 65, 1640, 57, 1806, 78, 95, 86, 2162, 70, 336, 82, 135, 94, 2756, 80, 198, 96, 152, 110, 3422, 81, 3660, 118

Offset: 1

Amarnath Murthy, Mar 27 2003

nonn

a(p) = p(p-1) if p is a prime.

a(p^k) = p (p^k - 1) if p is a prime and k >= 1. - Robert Israel, May 21 2015

Ivan Neretin, Table of n, a(n) for n = 1..1000

Cf. A081519, A081520.

A:= n -> select(t->igcd(t,n)>1,[$1..n^2])[n-1]:
1, seq(A(n),n=2..100); # Robert Israel, May 21 2015

Table[Join[{1}, Select[Range[n^2], GCD[n, #] > 1 &, n - 1]][[-1]], {n, 1, 60}](* Ivan Neretin, May 21 2015 *)

More terms from Ryan Propper, Nov 05 2005

A081519 Sum of terms in row n of A081520.

Original entry on oeis.org

1, 3, 10, 13, 51, 24, 148, 57, 109, 77, 606, 100, 1015, 161, 226, 241, 2313, 231, 3250, 318, 493, 425, 5820, 415, 1501, 605, 1054, 664, 11775, 593, 14416, 993, 1344, 1061, 1896, 946, 24643, 1337, 1931, 1302, 33621, 1204, 38830, 1737, 2124, 1985, 50808

Offset: 1

Amarnath Murthy, Mar 27 2003

nonn

a(p) = p^2*(p-1)/2 + 1, p is prime.

a(p^k) = p^(k+1)*(p^k-1)/2 + 1 if p is prime and k >= 1. - Robert Israel, May 21 2015

Ivan Neretin, Table of n, a(n) for n = 1..1000

Cf. A081518, A081520.

A:= proc(n) local L;
  L:= select(t->igcd(t,n)>1,[$1..n^2])[1 .. n-1];
  1+convert(L,`+`)
end proc:
seq(A(n),n=1..100); # Robert Israel, May 21 2015

Table[Plus @@ Join[{1}, Select[Range[n^2], GCD[n, #] > 1 &, n - 1]], {n, 1, 47}](* Ivan Neretin, May 21 2015 *)

More terms from Ryan Propper, Nov 05 2005

A121998 Table, n-th row gives numbers between 1 and n that have a common factor with n.

Original entry on oeis.org

2, 3, 2, 4, 5, 2, 3, 4, 6, 7, 2, 4, 6, 8, 3, 6, 9, 2, 4, 5, 6, 8, 10, 11, 2, 3, 4, 6, 8, 9, 10, 12, 13, 2, 4, 6, 7, 8, 10, 12, 14, 3, 5, 6, 9, 10, 12, 15, 2, 4, 6, 8, 10, 12, 14, 16, 17, 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 19, 2, 4, 5, 6, 8, 10, 12, 14, 15, 16, 18, 20, 3, 6, 7, 9, 12, 14, 15

Offset: 2

Franklin T. Adams-Watters, Sep 11 2006

Row n contains numbers m <= n such that gcd(m,n) > 1, i.e., numbers m in the cototient of n. - Michael De Vlieger, Mar 13 2018

			2;
3;
2,4;
5;
2,3,4,6;
7;
...

Michael De Vlieger, Table of n, a(n) for n = 2..12949 (rows 2 <= n <= 256).

Cf. A051953 (row lengths), A038566, A081520, A133995 (nondivisors in the cototient of n).

Table[Select[Range@ n, ! CoprimeQ[#, n] &], {n, 20}] // Flatten (* Michael De Vlieger, Mar 13 2018 *)

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.

Original entry on oeis.org

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

Felix Fröhlich, Jan 09 2018

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).

			============================================================
. 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

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

Cf. A000040, A025475, A081520.

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 *)

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

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

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A081518 Final term in row n of A081520.

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A081519 Sum of terms in row n of A081520.

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A121998 Table, n-th row gives numbers between 1 and n that have a common factor with n.

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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.

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