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#16 by Michael De Vlieger at Mon Nov 21 09:38:42 EST 2022
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#15 by Joerg Arndt at Mon Nov 21 02:57:40 EST 2022
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#14 by Amiram Eldar at Mon Nov 21 01:52:59 EST 2022
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#13 by Amiram Eldar at Mon Nov 21 01:45:13 EST 2022
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| FORMULA
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Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} A076512a(k)/aA076512(k) = zeta(2)*zeta(3)/zeta(6) - 1 = A082695 - 1 = 0.943596... . Amiram Eldar, Nov 21 2022
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#12 by Amiram Eldar at Mon Nov 21 01:44:08 EST 2022
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| FORMULA
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Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} A076512(k)/a(k) = zeta(2)*zeta(3)/zeta(6) - 1 = A082695 - 1 = 0.943596... . Amiram Eldar, Nov 21 2022
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| MATHEMATICA
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Table[Numerator[n/EulerPhi[n] - 1], {n, 1, 100}] (* Amiram Eldar, Nov 21 2022 *)
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| CROSSREFS
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Cf. A076512 (denominators), A000010, A009195, A051953, A082695, A109395.
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| STATUS
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approved
editing
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#11 by Amiram Eldar at Mon Nov 21 01:41:37 EST 2022
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| FORMULA
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From Amiram Eldar, Nov 21 2022: (Start)
Asymptotic means:
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A076512(k) = 6/Pi^2 = 0.607927... (A059956).
Limit_{m->oo} (1/m) * Sum_{k=1..m} A076512(k)/a(k) = zeta(2)*zeta(3)/zeta(6) = 1.943596... (A082695). (End)
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| MATHEMATICA
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Table[Numerator[n/EulerPhi[n] - 1], {n, 1, 100}] (* Amiram Eldar, Nov 21 2022 *)
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| CROSSREFS
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Cf. A076512 (denominators), A000010, A009195, A051953, A059956, A082695, A109395.
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| KEYWORD
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nonn,frac,changed
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| STATUS
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editing
approved
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#10 by Amiram Eldar at Mon Nov 21 01:36:55 EST 2022
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| FORMULA
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From Amiram Eldar, Nov 21 2022: (Start)
Asymptotic means:
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A076512(k) = 6/Pi^2 = 0.607927... (A059956).
Limit_{m->oo} (1/m) * Sum_{k=1..m} A076512(k)/a(k) = zeta(2)*zeta(3)/zeta(6) = 1.943596... (A082695). (End)
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| MATHEMATICA
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Table[Numerator[n/EulerPhi[n] - 1], {n, 1, 100}] (* Amiram Eldar, Nov 21 2022 *)
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| CROSSREFS
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Cf. A076512 (denominators), A000010, A009195, A051953, A059956, A082695, A109395.
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| STATUS
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approved
editing
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#9 by Susanna Cuyler at Fri Sep 07 20:30:01 EDT 2018
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#8 by Michel Marcus at Fri Sep 07 10:54:44 EDT 2018
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Discussion
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Fri Sep 07
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| Antti Karttunen: That's a problem for representation. For the better looks, the plot-script should have an option of selecting value k up to which only n=1..k would be plotted. For the slowness, I think the scatterplot png-images generated by the plot utility should be cached, not recreated every time, as it happens to be now (I'm not sure about this, however).
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| Antti Karttunen: In any case, most of these "number theoretic sequences" look very similar. If sone nice specimen is found, then a separate scatterplot-image could be attached (with the right parameters, and maybe even colors, as many fine images of Remy Sigrist have).
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#7 by Antti Karttunen at Fri Sep 07 10:45:49 EDT 2018
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Discussion
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Fri Sep 07
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| Michel Marcus: I find 65537 a bit big for the graphs : takes a long time to display, and scatterplot is very black
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