A259850 Numbers k such that k/phi(k) equals sigma(x)/x for some x<=k.

1, 3, 8, 9, 14, 15, 16, 21, 26, 27, 28, 32, 40, 45, 50, 52, 56, 63, 64, 75, 80, 81, 98, 100, 104, 112, 128, 130, 135, 144, 147, 160, 162, 182, 189, 192, 196, 200, 208, 216, 224, 225, 243, 250, 255, 256, 260, 288, 310, 320, 324, 338, 364, 372, 375, 384, 392, 400

Offset

1,2

Comments

This sequence is motivated by the fact that sigma(n)/n and n/phi(n) are both >= 1.

For the first few terms, we get these ratios: 1, 3/2, 2, 3/2, 7/3, 15/8, 2, ....

The ordered list of distinct values up to a given limit is:

up to 10^1: [1, 3/2, 2];

up to 10^2: [1, 3/2, 7/4, 15/8, 2, 13/6, 7/3, 5/2];

up to 10^3: [1, 3/2, 7/4, 15/8, 31/16, 255/128, 2, 13/6, 7/3, 5/2, 91/36, 31/12, 85/32, 65/24, 35/12, 3, 31/10, 13/4];

up to 10^4: [1, 3/2, 7/4, 15/8, 31/16, 255/128, 2, 13/6, 7/3, 5/2, 91/36, 31/12, 85/32, 65/24, 403/144, 1105/384, 35/12, 635/216, 2555/864, 3, 217/72, 127/42, 73/24, 31/10, 51/16, 13/4, 1651/504, 527/160, 403/120, 221/64, 7/2, 127/36, 217/60];

up to 10^5: [1, 3/2, 7/4, 15/8, 31/16, 255/128, 65535/32768, 2, 33/16, 267/128, 13/6, 7/3, 133/54, 5/2, 91/36, 31/12, 85/32, 21845/8192, 65/24, 11/4, 89/32, 403/144, 1105/384, 35/12, 635/216, 2555/864, 3, 217/72, 127/42, 73/24, 665/216, 595/192, 31/10, 19/6, 51/16, 77/24, 1397/432, 13/4, 1651/504, 527/160, 949/288, 403/120, 221/64, 7/2, 127/36, 511/144, 6851/1920, 217/60, 119/32];

tending towards the intersection of the 2 sets {sigma(n)/n} (A017665/A017666) and {n/phi(n)} (A109395/A076512).

If k is a term, then so are all numbers > k with the same set of prime factors as k. - Robert Israel, Mar 09 2023

Links

Robert Israel, Table of n, a(n) for n = 1..1230

Example

1/phi(1) = 1/1 = sigma(1)/1, so 1 is in the sequence.
3/phi(3) = 3/2 = sigma(2)/2, so 3 is in the sequence.
8/phi(8) = 2/1 = sigma(6)/6, so 8 is in the sequence.

Maple

R:= NULL: count:= 0: V:= {}:
for k from 1 while count < 100 do
 V:= V union  {numtheory:-sigma(k)/k};
 if member(k/numtheory:-phi(k), V) then R:= R, k; count:= count+1 fi;
od:
R; # Robert Israel, Mar 08 2023

Prog

(PARI) lista(nn) = {vs = vector(nn, n, sigma(n)/n); ve = vector(nn, n, n/eulerphi(n)); vr = []; for (n=1, #ve, ven = ve[n]; for (m=1, n, if ((vs[m] == ven), print1(n, ", "); break); ); ); }

Crossrefs

Cf. A000010, A109395, A076512.

Cf. A000203, A017665, A017666.

Primitive elements: A361363.

Sequence in context: A066494 A082721 A239388 * A071677 A084747 A286177

Adjacent sequences: A259847 A259848 A259849 * A259851 A259852 A259853

Keyword

nonn

Author

Michel Marcus, Jul 07 2015

Extensions

Name corrected by Michel Marcus, Nov 25 2020

Status

approved