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.
%I A085021 #22 Jun 09 2025 00:50:41
%S A085021 0,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,2,1,2,2,1,2,1,2,1,1,2,3,1,1,1,1,1,
%T A085021 2,2,2,1,2,1,2,1,3,2,2,1,3,2,1,2,3,3,3,2,3,1,2,2,2,2,1,1,2,2,1,2,2,3,
%U A085021 1,2,3,2,3,2,2,3,1,1,3,1,3,2,2,2,1,1,2,2,1,1,3,4,1,2,3,2,2,1,3,4
%N A085021 Number of prime factors of cyclotomic(n,2), which is A019320(n), the value of the n-th cyclotomic polynomial evaluated at x=2.
%C A085021 The Mobius transform of this sequence yields A046051, the number of prime factors of Mersenne number 2^n-1.
%C A085021 The number of prime factors in the primitive part of 2^n-1. - _T. D. Noe_, Jul 19 2008
%H A085021 Max Alekseyev, <a href="/A085021/b085021.txt">Table of n, a(n) for n = 1..1206</a> (first 500 term from T. D. Noe)
%H A085021 Vjekoslav Kovač and Florian Luca, <a href="https://arxiv.org/abs/2506.04883">On the number of divisors of Mersenne numbers</a>, arXiv:2506.04883 [math.NT], 2025. See Table 2 p. 9.
%H A085021 H. Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/cn/">Factorization of Cyclotomic Numbers</a>
%H A085021 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CyclotomicPolynomial.html">Cyclotomic Polynomial</a>
%H A085021 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MoebiusTransform.html">Möbius Transform</a>
%e A085021 a(11) = 2 because cyclotomic(11,2) = 2047, which has two factors: 23 and 89.
%t A085021 Join[{0}, Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 2]]][[2]], {n, 2, 100}]]
%o A085021 (PARI) a(n) = #factor(polcyclo(n, 2))~; \\ _Michel Marcus_, Mar 06 2015
%Y A085021 omega(Phi(n,x)): this sequence (x=2), A085028 (x=3), A085029 (x=4), A085030 (x=5), A085031 (x=6), A085032 (x=7), A085033 (x=8), A085034 (x=9), A085035 (x=10).
%Y A085021 Cf. A005420, A019320, A046051, A046801, A059499, A064078, A112927, A212953.
%K A085021 nonn
%O A085021 1,11
%A A085021 _T. D. Noe_, Jun 19 2003