A109395 -id:A109395 - cp's OEIS frontend

A260142 Denominators of the distinct common values of sigma(n)/n and m/phi(m) in the order which they occur when n and m increase.

1, 2, 1, 3, 8, 4, 2, 6, 1, 24, 36, 10, 128, 12, 4, 32, 16, 12, 42, 16, 72, 384, 120, 144, 2, 24, 64, 864, 36, 216, 60, 160, 504, 192, 16, 288, 54, 6, 128, 24, 144, 1920, 4, 32768, 32, 32, 216, 432, 8192, 20, 48, 1296, 1080, 1760, 4320, 384, 704, 1728, 10, 360, 4, 2816, 80

Offset: 1

Michel Marcus, Jul 17 2015

To be considered as common, a value must have appeared for some N in both sequences sigma(n)/n (A017665/A017666) and n/eulerphi(n) (A109395/A076512), with 1<=n<=N.

			sigma(n)/n starts: 1/1, 3/2, 4/3, 7/4, 6/5, 2/1, 8/7, 15/8, 13/9, 9/5, ...
m/phi(m) starts:   1/1, 2/1, 3/2, 2/1, 5/4, 3/1, 7/6,  2/1,  3/2, 5/2, ...
The 1st common value is 1/1 = sigma(1)/1 = 1/eulerphi(1).
The 2nd common value is 3/2 = 3/eulerphi(3) = sigma(2)/2.
The 3rd common value is 2/1 = sigma(6)/6 = 2/eulerphi(2).
The sequence of ratios begin: 1, 3/2, 2, 7/3, 15/8, 7/4, 5/2, 13/6, 3, 65/24, 91/36, 31/10, 255/128, 31/12, ...
So this sequence begins 1, 2, 1, ...

Cf. A259850, A260141 (numerators).

already(vsv, val, vsi, n) = {pos=vecsearch(vsv, val); if (pos, until(vsv[pos] < val, pos--); pos++; pos = vsi[pos] <= n); pos;}
lista(nn) = {vrat = [1]; vsrat = [1]; ve = vector(nn, k, k/eulerphi(k)); vs = vector(nn, k, sigma(k)/k); vesv = vecsort(ve); vesi = vecsort(ve,,1); vssv = vecsort(vs); vssi = vecsort(vs,,1); print1(1, ", "); for (n=2, nn, rn = vs[n]; if (!vecsearch(vsrat, rn) && (already(vesv, rn, vesi, n)), print1(denominator(rn), ", "); vrat = concat(vrat, rn); vsrat = vecsort(vrat,,8), rn = ve[n]; if (!vecsearch(vsrat, rn) && (already(vssv, rn, vssi, n)), print1(denominator(rn), ", "); vrat = concat(vrat, rn); vsrat = vecsort(vrat,,8););););}

A280990 Least prime p such that n divides phi(p*n).

Original entry on oeis.org

2, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 31, 2, 17, 3, 19, 5, 7, 11, 23, 3, 5, 13, 3, 7, 29, 31, 31, 2, 67, 17, 71, 3, 37, 19, 13, 5, 41, 7, 43, 11, 31, 23, 47, 3, 7, 5, 103, 13, 53, 3, 11, 7, 19, 29, 59, 31, 61, 31, 7, 2, 131, 67, 67, 17, 139, 71, 71, 3, 73, 37, 31, 19, 463

Offset: 1

Altug Alkan, Jan 12 2017

nonn

n*a(n) are 2, 4, 9, 8, 25, 18, 49, 16, 27, 50, 121, 36, 169, 98, 465, 32, 289, ...

a(n) <= A034694(A007947(n)). If n is in A050384 then a(n) = A034694(n). - Robert Israel, Jan 12 2017

			a(15) = 31 because 15 does not divide phi(p*15) for p < 31 where p is prime and phi(31*15) = 2*4*30 is divisible by 15.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000010, A007694, A007947, A034694, A050384, A109395.

Cf. A000079, A065119, A086761: for those n such that a(n)=2,3,5. - Michel Marcus, Jan 20 2017

f:= proc(n) local p;
    p:= 2;
    while numtheory:-phi(p*n) mod n <> 0 do p:= nextprime(p) od:
    p
end proc:
map(f, [$1..100]); # Robert Israel, Jan 12 2017

lpp[n_]:=Module[{p=2},While[Mod[EulerPhi[p*n],n]!=0,p=NextPrime[p]];p]; Array[lpp,80] (* Harvey P. Dale, Sep 26 2020 *)

a(n)=my(k = 1); while (eulerphi(prime(k)*n) % n != 0, k++); prime(k);

a(n)=my(t=n/gcd(eulerphi(n),n)); if(t==1, return(2)); forstep(p=if(t%2,2*t,t)+1, if(isprime(t), t, oo),lcm(t,2), if(isprime(p), return(p))); t \\ Charles R Greathouse IV, Jan 20 2017

a(p^k) = p for all primes p and k >= 1. - Robert Israel, Jan 12 2017

a(n) << n^5 by Xylouris' improvement to Linnik's theorem. - Charles R Greathouse IV, Jan 20 2017

A331176 a(n) = n - n/gcd(n, phi(n)), where phi is Euler totient function.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 11 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000010, A009195, A109395.

Cf. A003277 (indices of zeros).

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

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

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

A342867 a(n) is the least number k such that the continued fraction for phi(k)/k contains exactly n elements.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 27 2021

nonn

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

All the terms above 3 are composite numbers.

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

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

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]

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

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

cp's OEIS Frontend

A260142 Denominators of the distinct common values of sigma(n)/n and m/phi(m) in the order which they occur when n and m increase.

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A280990 Least prime p such that n divides phi(p*n).

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A331176 a(n) = n - n/gcd(n, phi(n)), where phi is Euler totient function.

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Mathematica

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Formula

A342867 a(n) is the least number k such that the continued fraction for phi(k)/k contains exactly n elements.

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Programs

Mathematica

PARI

Formula