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A059379 Array of values of Jordan function J_k(n) read by antidiagonals (version 1). 22

%I

%S 1,1,1,2,3,1,2,8,7,1,4,12,26,15,1,2,24,56,80,31,1,6,24,124,240,242,63,

%T 1,4,48,182,624,992,728,127,1,6,48,342,1200,3124,4032,2186,255,1,4,72,

%U 448,2400,7502,15624,16256,6560,511,1,10,72,702,3840

%N Array of values of Jordan function J_k(n) read by antidiagonals (version 1).

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

%D R. Sivaramakrishnan, "The many facets of Euler's totient. II. Generalizations and analogues", Nieuw Arch. Wisk. (4) 8 (1990), no. 2, 169-187.

%H Enrique Pérez Herrero, <a href="/A059379/b059379.txt">Table of n, a(n) for n = 1..10000</a>

%F J_k(n) = sum( d divides n, d^k*mu(n/d)) - _Benoit Cloitre_ and Michael Orrison (orrison(AT)math.hmc.edu), Jun 07 2002

%e Array begins:

%e 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, ...

%e 1, 3, 8, 12, 24, 24, 48, 48, 72, 72, ...

%e 1, 7, 26, 56, 124, 182, 342, 448, 702, ...

%e 1, 15, 80, 240, 624, 1200, 2400, 3840, ...

%p 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;

%p #alternative

%p A059379 := proc(n,k)

%p add(d^k*numtheory[mobius](n/d),d=numtheory[divisors](n)) ;

%p end proc:

%p seq(seq(A059379(d-k,k),k=1..d-1),d=2..12) ; # _R. J. Mathar_, Nov 23 2018

%t JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/#]&]/;(n>0)&&IntegerQ[n];

%t A004736[n_]:=Binomial[Floor[3/2+Sqrt[2*n]],2]-n+1;

%t A002260[n_]:=n-Binomial[Floor[1/2+Sqrt[2*n]],2];

%t A059379[n_]:=JordanTotient[A004736[n],A002260[n]]; (* _Enrique Pérez Herrero_, Dec 19 2010 *)

%o (PARI)

%o jordantot(n,k)=sumdiv(n,d,d^k*moebius(n/d));

%o A002260(n)=n-binomial(floor(1/2+sqrt(2*n)),2);

%o A004736(n)=binomial(floor(3/2+sqrt(2*n)),2)-n+1;

%o A059379(n)=jordantot(A004736(n),A002260(n)); \\ _Enrique Pérez Herrero_, Jan 08 2011

%Y 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 Main diagonal gives A067858.

%K nonn,tabl

%O 1,4

%A _N. J. A. Sloane_, Jan 28 2001

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Last modified November 17 16:08 EST 2019. Contains 329241 sequences. (Running on oeis4.)