cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A063035 Number of integers m <= 10^n that contain a square factor (i.e., belong to A013929).

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%I A063035 #44 Jul 20 2024 20:05:35
%S A063035 3,39,392,3917,39206,392074,3920709,39207306,392072876,3920729058,
%T A063035 39207289720,392072897726,3920728981706,39207289814053,
%U A063035 392072898145897,3920728981459595
%N A063035 Number of integers m <= 10^n that contain a square factor (i.e., belong to A013929).
%C A063035 Note that "containing a square factor" (A013929) is different from "squareful" (A001694).
%H A063035 Robert G. Wilson v, <a href="/A063035/b063035.txt">Table of n, a(n) for n = 1..36</a>
%F A063035 Limit_{n->oo} a(n)/10^n = A229099. - _Robert G. Wilson v_, Aug 12 2014
%F A063035 a(n) = 10^n - A071172(n). - _Amiram Eldar_, Mar 10 2024
%t A063035 f[n_] := Sum[-MoebiusMu[i]Floor[n/i^2], {i, 2, Sqrt@ n}]; Table[ f[10^n], {n, 0, 14}]
%o A063035 (PARI) { default(realprecision, 50); for (n=1, 100, t=10^n - 1; a=10^n - sum(k=1, sqrt(t), moebius(k)*floor(t/k^2)); write("b063035.txt", n, " ", a) ) } \\ _Harry J. Smith_, Aug 16 2009
%o A063035 (Python)
%o A063035 from math import isqrt
%o A063035 from sympy import mobius
%o A063035 def A063035(n): return (m:=10**n)-sum(mobius(k)*(m//k**2) for k in range(1,isqrt(m)+1)) # _Chai Wah Wu_, Jul 20 2024
%Y A063035 Cf. A001694, A013929, A229099.
%Y A063035 For the complementary counts see A053462 and A071172.
%K A063035 nonn
%O A063035 1,1
%A A063035 _Robert G. Wilson v_, Aug 02 2001
%E A063035 More terms from _Harry J. Smith_, Aug 16 2009
%E A063035 Edited (with a more precise definition and a new value for a(1)) by _N. J. A. Sloane_, Aug 06 2012. As a result of this change, the programs probably now give the wrong value for a(1). The source of the trouble was the ambiguous meaning of squareful - the official definition of squareful is A001694.