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Revision History for A076511 (Underlined text is an addition; strikethrough text is a deletion.)

Showing entries 1-10 | older changes
A076511 Numerator of cototient(n)/totient(n).
(history; published version)
#16 by Michael De Vlieger at Mon Nov 21 09:38:42 EST 2022
STATUS

reviewed

approved

#15 by Joerg Arndt at Mon Nov 21 02:57:40 EST 2022
STATUS

proposed

reviewed

#14 by Amiram Eldar at Mon Nov 21 01:52:59 EST 2022
STATUS

editing

proposed

#13 by Amiram Eldar at Mon Nov 21 01:45:13 EST 2022
FORMULA

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

#12 by Amiram Eldar at Mon Nov 21 01:44:08 EST 2022
FORMULA

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

MATHEMATICA

Table[Numerator[n/EulerPhi[n] - 1], {n, 1, 100}] (* Amiram Eldar, Nov 21 2022 *)

CROSSREFS

Cf. A076512 (denominators), A000010, A009195, A051953, A082695, A109395.

STATUS

approved

editing

#11 by Amiram Eldar at Mon Nov 21 01:41:37 EST 2022
FORMULA

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)

MATHEMATICA

Table[Numerator[n/EulerPhi[n] - 1], {n, 1, 100}] (* Amiram Eldar, Nov 21 2022 *)

CROSSREFS

Cf. A076512 (denominators), A000010, A009195, A051953, A059956, A082695, A109395.

KEYWORD

nonn,frac,changed

STATUS

editing

approved

#10 by Amiram Eldar at Mon Nov 21 01:36:55 EST 2022
FORMULA

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)

MATHEMATICA

Table[Numerator[n/EulerPhi[n] - 1], {n, 1, 100}] (* Amiram Eldar, Nov 21 2022 *)

CROSSREFS

Cf. A076512 (denominators), A000010, A009195, A051953, A059956, A082695, A109395.

STATUS

approved

editing

#9 by Susanna Cuyler at Fri Sep 07 20:30:01 EDT 2018
STATUS

reviewed

approved

#8 by Michel Marcus at Fri Sep 07 10:54:44 EDT 2018
STATUS

proposed

reviewed

Discussion
Fri Sep 07 11:00
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).
13:31
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).
#7 by Antti Karttunen at Fri Sep 07 10:45:49 EDT 2018
STATUS

editing

proposed

Discussion
Fri Sep 07 10:54
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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Last modified August 30 07:09 EDT 2024. Contains 375532 sequences. (Running on oeis4.)