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 A300382 Dirichlet series for a cubic module of rank 6. 1
 1, 0, 0, 8, 6, 0, 0, 0, 10, 0, 24, 0, 0, 0, 0, 32, 0, 0, 40, 48, 0, 0, 0, 0, 30, 0, 0, 0, 60, 0, 64, 0, 0, 0, 0, 80, 0, 0, 0, 0, 84, 0, 0, 192, 60, 0, 0, 0, 51, 0, 0, 0, 0, 0, 144, 0, 0, 0, 120, 0, 124, 0, 0, 130, 0, 0, 0, 0, 0, 0, 144, 0, 0, 0, 0, 320, 0, 0, 160, 192, 91, 0, 0, 0, 0, 0, 0, 0, 180, 0, 0, 0, 0, 0, 240, 0, 0, 0, 240, 239, 204, 0, 0, 0, 0, 0, 0, 0, 220, 0, 0, 0, 0, 0, 0, 480, 0, 0, 0, 0, 405 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Submitted as a substitute for A031365 which appears to display a faulty A031365(16)=24 in the version published 1997. LINKS Table of n, a(n) for n=1..121. M. Baake, Solution of the coincidence problem in dimensions d<=4, arxiv:math/0605222 (2006), (5.12) MAPLE read("transforms") : # expansion of 1/(1-5^(-s)) in (5.10) L1 := [1, seq(0, i=2..200)] : for k from 1 do if 5^k <= nops(L1) then L1 := subsop(5^k=1, L1) ; else break ; end if; end do: # multiplication with 1/(1-p^(-2s)) in (5.10) for i from 1 do p := ithprime(i) ; if modp(p, 5) = 2 or modp(p, 5)=3 then Laux := [1, seq(0, i=2..200)] : for k from 1 do if p^(2*k) <= nops(Laux) then Laux := subsop(p^(2*k)=1, Laux) ; else break ; end if; end do: L1 := DIRICHLET(L1, Laux) ; end if; if p > nops(L1) then break; end if; end do: # multiplication with 1/(1-p^(-s))^2 in (5.10) for i from 1 do p := ithprime(i) ; if modp(p, 5) = 1 or modp(p, 5)=4 then Laux := [1, seq(0, i=2..200)] : for k from 1 do if p^k <= nops(Laux) then Laux := subsop(p^k=k+1, Laux) ; else break ; end if; end do: L1 := DIRICHLET(L1, Laux) ; end if; if p > nops(L1) then break; end if; end do: # this is now Zeta_L(s), seems to be A035187 # print(L1) ; # generate Zeta_L(s-1) L1shft := [seq(op(i, L1)*i, i=1..nops(L1))] ; # generate 1/Zeta_L(s) L1x := add(op(i, L1)*x^(i-1), i=1..nops(L1)) : taylor(1/L1x, x=0, nops(L1)) : L1i := gfun[seriestolist](%) ; # generate 1/Zeta_L(2s) L1i2 := [1, seq(0, i=2..nops(L1))] ; for k from 2 to nops(L1i) do if k^2 < nops(L1i2) then L1i2 := subsop(k^2=op(k, L1i), L1i2) ; else break ; end if; end do: # generate Zeta_L(s)*Zeta_L(s-1) DIRICHLET(L1, L1shft) ; # generate Zeta_L(s)*Zeta_L(s-1)/Zeta_L(2s) L1 := DIRICHLET(%, L1i2) ; # generate 1/(1+4^(-s)) Laux := [1, seq(0, i=2..nops(L1))] : for k from 1 do if 4^k <= nops(Laux) then Laux := subsop(4^k=(-1)^k, Laux) ; else break; end if ; end do: # generate Zeta_L(s)*Zeta_L(s-1)/Zeta_L(2s)/(1+4^(-s)) L1 := DIRICHLET(L1, Laux) ; # generate 1+4^(1-s) Laux := [1, seq(0, i=2..3), 4, seq(0, i=5..nops(L1))] ; DIRICHLET(L1, Laux) ; # R. J. Mathar, Mar 04 2018 CROSSREFS Cf. A031365. Sequence in context: A010119 A010116 A031365 * A004013 A010118 A226317 Adjacent sequences: A300379 A300380 A300381 * A300383 A300384 A300385 KEYWORD nonn,less AUTHOR R. J. Mathar, Mar 04 2018 STATUS approved

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Last modified November 30 21:14 EST 2023. Contains 367462 sequences. (Running on oeis4.)