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 A193805 Square array read by antidiagonals: S(n,k) is the number of m which are prime to n and are not strong divisors of k. 4
 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 4, 2, 2, 1, 1, 2, 3, 1, 1, 1, 1, 6, 2, 3, 2, 2, 1, 1, 4, 5, 2, 2, 2, 1, 1, 1, 6, 4, 5, 2, 4, 1, 2, 1, 1, 4, 5, 3, 4, 1, 2, 2, 1, 1, 1, 10, 4, 6, 4, 5, 2, 4, 2, 2, 1, 1, 4, 9, 3, 4, 3, 3, 2, 2, 1, 1, 1, 1, 12, 4, 9, 4, 5, 3, 6, 2 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS We say d is a strong divisor of n iff d is a divisor of n and d > 1. Let phi(n) be Euler's totient function. Then phi(n) = S(n,1) = S(n,n). Thus S(n,k) can be regarded as a generalization of the totient function. LINKS Peter Luschny, Euler's totient function EXAMPLE [x][1][2][3][4][5][6][7][8] [1] 1, 1, 1, 1, 1, 1, 1, 1 [2] 1, 1, 1, 1, 1, 1, 1, 1 [3] 2, 1, 2, 1, 2, 1, 2, 1 [4] 2, 2, 1, 2, 2, 1, 2, 2 [5] 4, 3, 3, 2, 4, 2, 4, 2 [6] 2, 2, 2, 2, 1, 2, 2, 2 [7] 6, 5, 5, 4, 5, 3, 6, 4 [8] 4, 4, 3, 4, 3, 3, 3, 4 Triangle k=1..n, n>=1: [1]           1 [2]          1, 1 [3]        2, 1, 2 [4]       2, 2, 1, 2 [5]     4, 3, 3, 2, 4 [6]    2, 2, 2, 2, 1, 2 [7]  6, 5, 5, 4, 5, 3, 6 [8] 4, 4, 3, 4, 3, 3, 3, 4 Triangle n=1..k, k>=1: [1]           1 [2]          1, 1 [3]        1, 1, 2 [4]       1, 1, 1, 2 [5]     1, 1, 2, 2, 4 [6]    1, 1, 1, 1, 2, 2 [7]  1, 1, 2, 2, 4, 2, 6 [8] 1, 1, 1, 2, 2, 2, 4, 4 S(15, 22) = card({1,4,7,8,13,14}) = 6 because the coprimes of 15 are {1,2,4,7,8,11,13,14} and the strong divisors of 22 are {2, 11, 22}. MAPLE strongdivisors := n -> numtheory[divisors](n) minus {1}: coprimes := n -> select(k->igcd(k, n)=1, {\$1..n}): S := (n, k) -> nops(coprimes(n) minus strongdivisors(k)): seq(seq(S(n-k+1, k), k=1..n), n=1..13);  # Square array by antidiagonals. seq(print(seq(S(n, k), k=1..n)), n=1..8); # Lower triangle. seq(print(seq(S(n, k), n=1..k)), k=1..8); # Upper triangle. MATHEMATICA s[n_, k_] := Complement[ Select[ Range[n], GCD [#, n] == 1 &], Rest[ Divisors[k]]] // Length; Table[ s[n-k+1, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 25 2013 *) PROG (PARI) S(n, k)=eulerphi(n)-sumdiv(k, d, gcd(d, n)==1 && d1) for(s=2, 15, for(k=1, s-1, print1(S(s-k, k)", "))) \\ Charles R Greathouse IV, Aug 01 2016 CROSSREFS Cf. A000010, A051953, A193804. Sequence in context: A143258 A027199 A140218 * A159704 A101428 A307223 Adjacent sequences:  A193802 A193803 A193804 * A193806 A193807 A193808 KEYWORD nonn,nice,tabl AUTHOR Peter Luschny, Aug 05 2011 STATUS approved

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Last modified May 26 18:08 EDT 2020. Contains 334630 sequences. (Running on oeis4.)