A361363 -id:A361363 - cp's OEIS frontend

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

Michel Marcus, Jul 07 2015

nonn

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

			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.

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

Cf. A000010, A109395, A076512.
Cf. A000203, A017665, A017666.
Primitive elements: A361363.

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

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);););}

Name corrected by Michel Marcus, Nov 25 2020

cp's OEIS Frontend

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

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