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A059380 Array of values of Jordan function J_k(n) read by antidiagonals (version 2).

%I A059380 #24 Feb 15 2015 02:14:48
%S A059380 1,1,1,1,3,2,1,7,8,2,1,15,26,12,4,1,31,80,56,24,2,1,63,242,240,124,24,
%T A059380 6,1,127,728,992,624,182,48,4,1,255,2186,4032,3124,1200,342,48,6,1,
%U A059380 511,6560,16256,15624,7502,2400,448,72,4,1,1023,19682
%N A059380 Array of values of Jordan function J_k(n) read by antidiagonals (version 2).
%D A059380 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
%D A059380 R. Sivaramakrishnan, The many facets of Euler's totient. II. Generalizations and analogues, Nieuw Arch. Wisk. (4) 8 (1990), no. 2, 169-187
%H A059380 Enrique Pérez Herrero, <a href="/A059380/b059380.txt">Table of n, a(n) for n = 1..10000</a>
%e A059380 Array begins:
%e A059380 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, ...
%e A059380 1, 3, 8, 12, 24, 24, 48, 48, 72, 72, ...
%e A059380 1, 7, 26, 56, 124, 182, 342, 448, 702, ...
%e A059380 1, 15, 80, 240, 624, 1200, 2400, 3840, ...
%p A059380 J := proc(n,k) local i,p,t1,t2; t1 := n^k; for p from 1 to n do if isprime(p) and n mod p = 0 then t1 := t1*(1-p^(-k)); fi; od; t1; end;
%t A059380 JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/#]&]/;(n>0)&&IntegerQ[n];
%t A059380 A004736[n_]:=Binomial[Floor[3/2+Sqrt[2*n]],2]-n+1;
%t A059380 A002260[n_]:=n-Binomial[Floor[1/2+Sqrt[2*n]],2];
%t A059380 A059380[n_]:=JordanTotient[A002260[n],A004736[n]]; (* _Enrique Pérez Herrero_, Dec 19 2010 *)
%o A059380 (PARI)
%o A059380 jordantot(n,k)=sumdiv(n,d,d^k*moebius(n/d));
%o A059380 A002260(n)=n-binomial(floor(1/2+sqrt(2*n)),2);
%o A059380 A004736(n)=binomial(floor(3/2+sqrt(2*n)),2)-n+1;
%o A059380 A059380(n)=jordantot(A002260(n),A004736(n)); \\ _Enrique Pérez Herrero_, Jan 08 2011
%Y A059380 See A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5). Columns give A000225, A024023, A020522, A024049, A059387, etc.
%Y A059380 Main diagonal gives A067858.
%K A059380 nonn,tabl
%O A059380 1,5
%A A059380 _N. J. A. Sloane_, Jan 28 2001