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 A059379 Array of values of Jordan function J_k(n) read by antidiagonals (version 1). 22
 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; text; internal format)
 OFFSET 1,4 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. LINKS Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000 FORMULA 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 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, ... 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; #alternative A059379 := proc(n, k)     add(d^k*numtheory[mobius](n/d), d=numtheory[divisors](n)) ; end proc: seq(seq(A059379(d-k, k), k=1..d-1), d=2..12) ; # R. J. Mathar, Nov 23 2018 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 Pérez Herrero, Dec 19 2010 *) 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 Pérez Herrero, Jan 08 2011 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. Main diagonal gives A067858. Sequence in context: A183759 A101477 A077887 * A065487 A025258 A118846 Adjacent sequences:  A059376 A059377 A059378 * A059380 A059381 A059382 KEYWORD nonn,tabl AUTHOR N. J. A. Sloane, Jan 28 2001 STATUS approved

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Last modified October 21 14:18 EDT 2019. Contains 328301 sequences. (Running on oeis4.)