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Numbers k at which the ratio (number of squares in the multiplicative group modulo k)/k reaches a new minimum.
0

%I #16 May 28 2018 06:06:20

%S 1,2,3,4,6,8,12,24,60,120,420,840,4620,9240,60060,120120,1021020,

%T 2042040,19399380,38798760,446185740,892371480

%N Numbers k at which the ratio (number of squares in the multiplicative group modulo k)/k reaches a new minimum.

%C I.e., numbers k such that A046073(k)/k < A046073(j)/j for all j < k.

%C Appears to be just the union of 2*A002110, 4*A002110, and {1,3,6}. - _Don Reble_, Nov 26 2017

%e k A046073(k) A046073(k)/k

%e ======= ========== ========================================

%e 1 1 1/1 = 1 = 1.000000000

%e 2 1 1/2 = 1/2 = 0.500000000

%e 3 1 1/3 = 1/3 = 0.333333333...

%e 4 1 1/4 = 1/4 = 0.250000000

%e 6 1 1/6 = 1/6 = 0.166666666...

%e 8 1 1/8 = 1/8 = 0.125000000

%e 12 1 1/12 = 1/12 = 0.083333333...

%e 24 1 1/24 = 1/24 = 0.041666666...

%e 60 2 2/60 = 1/30 = 0.033333333...

%e 120 2 2/120 = 1/60 = 0.016666666...

%e 420 6 6/420 = 1/70 = 0.014285714...

%e 840 6 6/840 = 1/140 = 0.007142857...

%e 4620 30 30/4620 = 1/154 = 0.006493506...

%e 9240 30 30/9240 = 1/308 = 0.003246753...

%e 60060 180 180/60060 = 3/1001 = 0.002997002...

%e 120120 180 180/120120 = 3/2002 = 0.001498501...

%e 1021020 1440 1440/1021020 = 24/17017 = 0.001410354...

%e 2042040 1440 1440/2042040 = 12/17017 = 0.000705177...

%o (PARI) m=oo;for(k=1,oo, m > A046073(k)/k||next;print1(k",");m=A046073(k)/k) \\ _M. F. Hasler_, Nov 27 2017

%Y Cf. A046073.

%K nonn,more

%O 1,2

%A _Jon E. Schoenfield_, Oct 28 2017

%E Terms a(19) .. a(22) from _Joerg Arndt_, Dec 28 2017