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A059379
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Array of values of Jordan function J_k(n) read by antidiagonals (version 1).
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17
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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, 1, 4, 48, 182, 624, 992, 728, 127, 1, 6, 48, 342, 1200, 3124, 4032, 2186, 255, 1, 4, 72, 448, 2400, 7502, 15624, 16256, 6560, 511, 1, 10, 72, 702, 3840
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
R. Sivaramakrishnan, "The many facets of Euler's totient. II. Generalizations and analogues", Nieuw Arch. Wisk. (4) 8 (1990), no. 2, 169-187.
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LINKS
| Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000
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FORMULA
| J_k(n) = sum( d divides n, d^k*mu(n/d)) - Benoit Cloitre (benoit7848c(AT)orange.fr) and Michael Orrison (orrison(AT)math.hmc.edu), Jun 07 2002
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EXAMPLE
| Array begins:
1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, ...
1, 3, 8, 12, 24, 24, 48, 48, 72, 72, ...
1, 7, 26, 56, 124, 182, 342, 448, 702, ...
1, 15, 80, 240, 624, 1200, 2400, 3840, ...
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MAPLE
| 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;
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MATHEMATICA
| JordanTotient[n_, k_:1]:=DivisorSum[n, #^k*MoebiusMu[n/#]&]/; (n>0)&&IntegerQ[n];
A004736[n_]:=Binomial[Floor[3/2+Sqrt[2*n]], 2]-n+1;
A002260[n_]:=n-Binomial[Floor[1/2+Sqrt[2*n]], 2];
A059379[n_]:=JordanTotient[A004736[n], A002260[n]]; (* Enrique Perez Herrero, Dec 19 2010 *)
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PROG
| (PARI)
jordantot(n, k)=sumdiv(n, d, d^k*moebius(n/d));
A002260(n)=n-binomial(floor(1/2+sqrt(2*n)), 2);
A004736(n)=binomial(floor(3/2+sqrt(2*n)), 2)-n+1;
A059379(n)=jordantot(A004736(n), A002260(n)); \\ Enrique Perez Herrero, Jan 08 2011
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CROSSREFS
| 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.
Sequence in context: A183759 A101477 A077887 * A065487 A025258 A118846
Adjacent sequences: A059376 A059377 A059378 * A059380 A059381 A059382
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KEYWORD
| nonn,tabl
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jan 28 2001
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