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A188999 Bi-unitary sigma: sum of the bi-unitary divisors of n. 48
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, 42, 40, 40, 30, 72, 32, 63, 48, 54, 48, 50, 38, 60, 56, 90, 42, 96, 44, 60, 60, 72, 48, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The sequence of bi-unitary perfect numbers obeying a(n) = 2*n consists of only 6, 60, 90 [Wall].

Row sum of row n of the irregular table of the bi-unitary divisors, A222266.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

K. Alladi, On arithmetic functions and divisors of higher order, J. Austral. Math. Soc. 23 (series A) (1977) 9-27.

J. Sandor, B. Crstici, Perfect numbers: Old and new issues; perspectives, in Handbook of number theory, II, p 45.

L. Toth, On the bi-unitary analogues of Euler's arithmetical function and the gcd-sum function J. Int. Seq. 12 (2009) # 09.5.2

C. R. Wall, Bi-unitary perfect numbers, Proc. Am. Math. Soc. 33 (1) (1972) 39-42.

Eric Weisstein's World of Mathematics, Biunitary Divisor

Tomohiro Yamada, 2 and 9 are the only biunitary superperfect numbers, arXiv:1705.00189 [math.NT], 2017.

FORMULA

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.

a(n) = A000203(n) - A319072(n). - Omar E. Pol, Sep 29 2018

EXAMPLE

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.

MAPLE

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:

seq( A188999(n), n=1..80) ;

MATHEMATICA

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 *)

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-Fran├žois Alcover, Sep 22 2018 *)

PROG

(Haskell)

a188999 n = product $ zipWith f (a027748_row n) (a124010_row n) where

   f p e = (p ^ (e + 1) - 1) `div` (p - 1) - (1 - m) * p ^ e' where

           (e', m) = divMod e 2

-- Reinhard Zumkeller, Mar 04 2013

(PARI) udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }

gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));

biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n));

a(n) = vecsum(biudivs(n)); \\ Michel Marcus, May 07 2017

(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

CROSSREFS

Cf. A222266, A027748, A124010, A034448, A319072.

Sequence in context: A327668 A324706 A049417 * A186644 A125139 A107224

Adjacent sequences:  A188996 A188997 A188998 * A189000 A189001 A189002

KEYWORD

mult,nonn

AUTHOR

R. J. Mathar, Apr 15 2011

STATUS

approved

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Last modified October 15 19:25 EDT 2019. Contains 328037 sequences. (Running on oeis4.)