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 A061020 #71 Jun 30 2025 20:04:27
%S A061020 1,-1,-2,3,-4,2,-6,-5,7,4,-10,-6,-12,6,8,11,-16,-7,-18,-12,12,10,-22,
%T A061020 10,21,12,-20,-18,-28,-8,-30,-21,20,16,24,21,-36,18,24,20,-40,-12,-42,
%U A061020 -30,-28,22,-46,-22,43,-21,32,-36,-52,20,40,30,36,28,-58,24,-60,30,-42,43,48,-20,-66,-48,44,-24,-70,-35
%N A061020 Negate primes in factorizations of divisors of n, then sum.
%C A061020 Analog of sigma function A000203(n) with primes negated.
%C A061020 Unsigned sequence |a(n)| (A206369) gives the number of numbers 1 <= k <= n for which GCD(k,n) is a square. |a(n)| = Sum_{d|n} d*(-1)^bigomega(n/d). - _Vladeta Jovovic_, Dec 29 2002
%H A061020 T. D. Noe, <a href="/A061020/b061020.txt">Table of n, a(n) for n = 1..10000</a>
%H A061020 Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, <a href="https://doi.org/10.2140/involve.2022.15.251">Finding structure in sequences of real numbers via graph theory: a problem list</a>, Involve, a Journal of Mathematics, Vol. 15, No. 2 (2022), pp. 251-270; <a href="https://arxiv.org/abs/2012.04625">arXiv preprint</a>, arXiv:2012.04625 [math.CO], 2020-2021.
%H A061020 László Tóth, <a href="http://arxiv.org/abs/1111.4842">A survey of the alternating sum-of-divisors function</a>, arXiv:1111.4842 [math.NT], 2011-2014.
%F A061020 Replace each divisor d of n by A061019[d] and sum. Replace p^q with (1-(-p)^(q+1))/(1+p) in prime factorization of n.
%F A061020 Inverse mobius transform of A061019. In other words a(n) = Sum_{d|n} d*(-1)^bigomega(d), where bigomega(n) = A001222(n).
%F A061020 a(n) = Sum_{d|n} d*mu(core(d)) where core(x) = A007913(x) is the smallest number such that x*core(x) is a square. - _Benoit Cloitre_, Apr 07 2002
%F A061020 G.f.: A(x) = Sum_{k>=1} lambda(k)*k*x^k/(1 - x^k) where lambda(k) is the Liouville function, A008836. - _Stuart Clary_, Apr 15 2006
%F A061020 G.f.: A(x) is x times the logarithmic derivative of A118206(x). - _Stuart Clary_, Apr 15 2006
%F A061020 Dirichlet g.f.: zeta(s)*zeta(2 s - 2)/zeta(s - 1). - _Stuart Clary_, Apr 15 2006
%F A061020 a(n) = Sum_{d|n} d*lambda(d), where lambda(n) is A008836(n). - _Enrique Pérez Herrero_, Aug 29 2013
%e A061020 a(12) = 1-2-3+4+6-12 = (1-2+4)*(1-3) = -6.
%p A061020 with(numtheory):
%p A061020 A061020 := proc(n) local d; add(d*(-1)^bigomega(d), d=divisors(n)) end:
%p A061020 seq(A061020(n), n=1..72); # _Peter Luschny_, Aug 29 2013
%t A061020 nmax = 72; Drop[ CoefficientList[ Series[ Sum[ LiouvilleLambda[k] k x^k/(1 - x^k), {k, 1, nmax} ], {x, 0, nmax} ], x ], 1 ] (* _Stuart Clary_, Apr 15 2006, updated by _Jean-François Alcover_, Dec 04 2017 *)
%t A061020 f[p_, e_] := ((-p)^(e+1)-1)/(-p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 24 2023 *)
%o A061020 (PARI) for(n=1,100,print1(sumdiv(n,d,(d)*moebius(core(d))),","))
%o A061020 (PARI) a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)/(1+p*X))[n]) \\ _Ralf Stephan_
%o A061020 (PARI) A061020(n) = {my(f=factorint(n)); prod(k=1, #f[,2], ((-f[k,1])^(f[k,2]+1)-1)/(-f[k,1]-1))} \\ _Andrew Lelechenko_, Apr 22 2014
%o A061020 (Haskell)
%o A061020 a061020 = sum . map a061019 . a027750_row
%o A061020 -- _Reinhard Zumkeller_, Feb 08 2012
%Y A061020 Cf. A000203, A061019, A076792, A206369.
%Y A061020 Cf. A027750, A007913.
%K A061020 easy,sign,mult
%O A061020 1,3
%A A061020 _Marc LeBrun_, Apr 13 2001