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%I #23 Mar 09 2023 13:52:56
%S 1,3,8,9,14,15,16,21,26,27,28,32,40,45,50,52,56,63,64,75,80,81,98,100,
%T 104,112,128,130,135,144,147,160,162,182,189,192,196,200,208,216,224,
%U 225,243,250,255,256,260,288,310,320,324,338,364,372,375,384,392,400
%N Numbers k such that k/phi(k) equals sigma(x)/x for some x<=k.
%C This sequence is motivated by the fact that sigma(n)/n and n/phi(n) are both >= 1.
%C For the first few terms, we get these ratios: 1, 3/2, 2, 3/2, 7/3, 15/8, 2, ....
%C The ordered list of distinct values up to a given limit is:
%C up to 10^1: [1, 3/2, 2];
%C up to 10^2: [1, 3/2, 7/4, 15/8, 2, 13/6, 7/3, 5/2];
%C 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];
%C 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];
%C 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];
%C tending towards the intersection of the 2 sets {sigma(n)/n} (A017665/A017666) and {n/phi(n)} (A109395/A076512).
%C 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
%H Robert Israel, <a href="/A259850/b259850.txt">Table of n, a(n) for n = 1..1230</a>
%e 1/phi(1) = 1/1 = sigma(1)/1, so 1 is in the sequence.
%e 3/phi(3) = 3/2 = sigma(2)/2, so 3 is in the sequence.
%e 8/phi(8) = 2/1 = sigma(6)/6, so 8 is in the sequence.
%p R:= NULL: count:= 0: V:= {}:
%p for k from 1 while count < 100 do
%p V:= V union {numtheory:-sigma(k)/k};
%p if member(k/numtheory:-phi(k), V) then R:= R,k; count:= count+1 fi;
%p od:
%p R; # _Robert Israel_, Mar 08 2023
%o (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);););}
%Y Cf. A000010, A109395, A076512.
%Y Cf. A000203, A017665, A017666.
%Y Primitive elements: A361363.
%K nonn
%O 1,2
%A _Michel Marcus_, Jul 07 2015
%E Name corrected by _Michel Marcus_, Nov 25 2020
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