A396581 - OEIS

%I #12 Jun 02 2026 19:46:11

%S 1,2,1,3,4,6,1,3,6,7,10,12,1,4,6,10,11,14,17,20,1,4,7,10,15,16,20,24,

%T 27,30,1,5,9,12,15,21,22,26,30,34,38,42,1,5,9,13,17,22,28,29,34,38,44,

%U 47,52,56,1,6,10,16,19,24,29,36,37,42,48,52,57,63,67,72

%N The fractional part of A000037(n) is the a(n)-th smallest fractional part of all sqrt(m) where m is nonsquare and 1 <= m <= A000037(n).

%H David A. Corneth, <a href="/A396581/b396581.txt">Table of n, a(n) for n = 1..10100</a>

%e a(5) = 4 as A000037(5) = 7 and the nonsquare numbers m such that 1 <= m <= 7 are 2, 3, 5, 6, 7. Listing and sorting the fractional parts of sqrt(m) for these numbers gives [0.23..., 5], [0.41..., 2], [0.44..., 6], [0.64..., 7], [0.73..., 3] and 7 is 4th from the left.

%o (PARI) a(n) = {n += (1 + sqrtint(4*n)) \ 2; my(l = List()); for(i = 1, n, if(!issquare(i), listput(l, [frac(sqrt(i)), i]))); listsort(l); select(x->x[2] == n, l, 1)[1]}

%Y Cf. A000037.

%K nonn

%O 1,2

%A _David A. Corneth_, May 29 2026