cp's OEIS Frontend

A188999 Bi-unitary sigma: sum of the bi-unitary divisors of n.

%I A188999 #49 Feb 16 2025 08:33:14
%S A188999 1,3,4,5,6,12,8,15,10,18,12,20,14,24,24,27,18,30,20,30,32,36,24,60,26,
%T A188999 42,40,40,30,72,32,63,48,54,48,50,38,60,56,90,42,96,44,60,60,72,48,
%U A188999 108,50,78,72,70,54,120,72,120,80,90,60,120,62,96,80,119,84,144,68,90,96,144,72,150,74,114,104,100
%N A188999 Bi-unitary sigma: sum of the bi-unitary divisors of n.
%C A188999 The sequence of bi-unitary perfect numbers obeying a(n) = 2*n consists of only 6, 60, 90 [Wall].
%C A188999 Row sum of row n of the irregular table of the bi-unitary divisors, A222266.
%H A188999 Reinhard Zumkeller, <a href="/A188999/b188999.txt">Table of n, a(n) for n = 1..10000</a>
%H A188999 Krishnaswami Alladi, <a href="http://dx.doi.org/10.1017/S1446788700017304">On arithmetic functions and divisors of higher order</a>, J. Austral. Math. Soc. 23 (series A) (1977), 9-27.
%H A188999 József Sándor and Borislav Crstici, <a href="http://dx.doi.org/10.1007/1-4020-2547-5_1">Perfect numbers: Old and new issues; perspectives</a>, in Handbook of number theory, II, p. 45.
%H A188999 László Tóth, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL12/Toth2/toth5.html">On the bi-unitary analogues of Euler's arithmetical function and the gcd-sum function</a> J. Int. Seq. 12 (2009), Article 09.5.2.
%H A188999 Charles R. Wall, <a href="http://dx.doi.org/10.1090/S0002-9939-1972-0289403-9">Bi-unitary perfect numbers</a>, Proc. Am. Math. Soc. 33 (1) (1972), 39-42.
%H A188999 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BiunitaryDivisor.html">Biunitary Divisor</a>.
%H A188999 Tomohiro Yamada, <a href="https://arxiv.org/abs/1705.00189">2 and 9 are the only biunitary superperfect numbers</a>, arXiv:1705.00189 [math.NT], 2017.
%F A188999 Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if e is odd, a(p^e) = (p^(e+1)-1)/(p-1) -p^(e/2) if e is even.
%F A188999 a(n) = A000203(n) - A319072(n). - _Omar E. Pol_, Sep 29 2018
%F A188999 Dirichlet g.f.: zeta(s-1) * zeta(s) * zeta(2*s-1) * Product_{p prime} (1 - 2/p^(2*s-1) + 1/p^(3*s-2) + 1/p^(3*s-1) - 1/p^(4*s-2)). - _Amiram Eldar_, Aug 28 2023
%e A188999 The divisors of n=16 are d=1, 2, 4, 8 and 16. The greatest common unitary divisor of (1,16) is 1, of (2,8) is 1, of (4,4) is 4, of (8,2) is 1, of (16,1) is 1 (see A165430). So 1, 2, 8 and 16 are bi-unitary divisors of 16, which sum to a(16) = 1 + 2 + 8 + 16 = 27.
%p A188999 A188999 := proc(n) local a,e,p,f; a :=1 ; for f in ifactors(n)[2] do e := op(2,f) ; p := op(1,f) ; if type(e,'odd') then a := a*(p^(e+1)-1)/(p-1) ; else a := a*((p^(e+1)-1)/(p-1)-p^(e/2)) ; end if; end do: a ; end proc:
%p A188999 seq( A188999(n),n=1..80) ;
%t A188999 f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; Table[DivisorSum[n, # &, Last@ Intersection[f@ #, f[n/#]] == 1 &], {n, 76}] (* _Michael De Vlieger_, May 07 2017 *)
%t A188999 a[n_] := If[n==1, 1, Product[{p, e} = pe; If[OddQ[e], (p^(e+1)-1)/(p-1), ((p^(e+1)-1)/(p-1)-p^(e/2))], {pe, FactorInteger[n]}]]; Array[a, 80] (* _Jean-François Alcover_, Sep 22 2018 *)
%o A188999 (Haskell)
%o A188999 a188999 n = product $ zipWith f (a027748_row n) (a124010_row n) where
%o A188999    f p e = (p ^ (e + 1) - 1) `div` (p - 1) - (1 - m) * p ^ e' where
%o A188999            (e', m) = divMod e 2
%o A188999 -- _Reinhard Zumkeller_, Mar 04 2013
%o A188999 (PARI) udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
%o A188999 gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
%o A188999 biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n));
%o A188999 a(n) = vecsum(biudivs(n)); \\ _Michel Marcus_, May 07 2017
%o A188999 (PARI) a(n) = {f = factor(n); for (i=1, #f~, p = f[i,1]; e = f[i,2]; f[i,1] = if (e % 2, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1) -p^(e/2)); f[i,2] = 1;); factorback(f);} \\ _Michel Marcus_, Nov 09 2017
%o A188999 (Python)
%o A188999 from math import prod
%o A188999 from sympy import factorint
%o A188999 def A188999(n): return prod((p**(e+1)-1)//(p-1)-(0 if e&1 else p**(e>>1)) for p,e in factorint(n).items()) # _Chai Wah Wu_, Dec 28 2024
%Y A188999 Cf. A222266, A027748, A124010, A034448, A319072.
%K A188999 mult,nonn,easy
%O A188999 1,2
%A A188999 _R. J. Mathar_, Apr 15 2011