|
| |
|
|
A188999
|
|
Bi-unitary sigma: sum of the bi-unitary divisors of n.
|
|
58
|
|
|
|
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-François 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: A324706 A049417 A331110 * A186644 A337177 A125139
Adjacent sequences: A188996 A188997 A188998 * A189000 A189001 A189002
|
|
|
KEYWORD
|
mult,nonn
|
|
|
AUTHOR
|
R. J. Mathar, Apr 15 2011
|
|
|
STATUS
|
approved
|
| |
|
|
