A109395 -id:A109395

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A331176

a(n) = n - n/gcd(n, phi(n)), where phi is Euler totient function.

+10
1

0, 0, 0, 2, 0, 3, 0, 6, 6, 5, 0, 9, 0, 7, 0, 14, 0, 15, 0, 15, 14, 11, 0, 21, 20, 13, 24, 21, 0, 15, 0, 30, 0, 17, 0, 33, 0, 19, 26, 35, 0, 35, 0, 33, 30, 23, 0, 45, 42, 45, 0, 39, 0, 51, 44, 49, 38, 29, 0, 45, 0, 31, 56, 62, 0, 33, 0, 51, 0, 35, 0, 69, 0, 37, 60, 57, 0, 65, 0, 75, 78, 41, 0, 77, 0, 43, 0, 77, 0, 75

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

FORMULA

a(n) = n - A109395(n).

MATHEMATICA

Table[n-n/GCD[n, EulerPhi[n]], {n, 90}] (* Harvey P. Dale, Aug 20 2020 *)

PROG

(PARI) A331176(n) = (n - (n/gcd(n, eulerphi(n))));

CROSSREFS

Cf. A000010, A009195, A109395.

Cf. A003277 (indices of zeros).

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jan 11 2020

STATUS

approved

A342867

a(n) is the least number k such that the continued fraction for phi(k)/k contains exactly n elements.

+10
1

1, 2, 3, 15, 35, 33, 65, 215, 221, 551, 455, 2001, 3417, 3621, 11523, 16705, 16617, 69845, 107545, 157285, 324569, 358883, 1404949, 1569295, 3783970, 3106285, 7536065, 12216295, 10589487, 24038979, 57759065, 51961945, 177005465, 131462695, 741703701, 1467144445

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

a(n) is the least number k such that A342866(k) = n.

All the terms above 3 are composite numbers.

LINKS

Table of n, a(n) for n=1..36.

FORMULA

a(2) = 2 since 2 is the least number k such that A342866(k) = 2.

MATHEMATICA

f[n_] := Length @ ContinuedFraction[EulerPhi[n]/n]; seq[max_] := Module[{s = Table[0, {max}], c = 0, n = 1, i}, While[c < max, i = f[n]; If[i <= max && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[20]

PROG

(PARI) a(n) = my(k=1); while (#contfrac(eulerphi(k)/k) != n, k++); k; \\ Michel Marcus, Mar 30 2021

CROSSREFS

Cf. A000010, A007694, A076512, A109395, A342866.

Cf. A071865 (similar, with sigma(k)/k).