OFFSET
1,3
COMMENTS
n-th row = n-th row of A165430 without repetitions. - Reinhard Zumkeller, Mar 04 2013
Denominators of sequence of all positive rational numbers ordered as follows: let m = p(i(1))^e(i(1))*...*p(i(k))^e(i(k)) be the prime factorization of m. Let S(m) be the vector of rationals p(i(k+1-j))^e(i(k+1-j))/p(i(j))^e(i(j)) for j = 1..k. The sequence (a(n)) is the concatenation of vectors S(m) for m = 1, 2, ...; for numerators see A229994. - Clark Kimberling, Oct 31 2013
The concept of unitary divisors was introduced by the Indian mathematician Ramaswamy S. Vaidyanathaswamy (1894-1960) in 1931. He called them "block factors". The term "unitary divisor" was coined by Cohen (1960). - Amiram Eldar, Mar 09 2024
LINKS
Reinhard Zumkeller, Rows n=1..1000 of triangle, flattened
Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Mathematische Zeitschrift, Vol. 74 (1960), pp. 66-80.
R. Vaidyanathaswamy, The theory of multiplicative arithmetic functions, Transactions of the American Mathematical Society, Vol. 33, No. 2 (1931), pp. 579-662.
Eric Weisstein's World of Mathematics, Unitary Divisor.
EXAMPLE
1;
1, 2;
1, 3;
1, 4;
1, 5;
1, 2, 3, 6;
1, 7;
1, 8;
1, 9;
1, 2, 5, 10;
1, 11;
MAPLE
with(numtheory);
# returns the number of unitary divisors of n and a list of them, from N. J. A. Sloane, May 01 2013
f:=proc(n)
local ct, i, t1, ans;
ct:=0; ans:=[];
t1:=divisors(n);
for i from 1 to nops(t1) do
d:=t1[i];
if igcd(d, n/d)=1 then ct:=ct+1; ans:=[op(ans), d]; fi;
od:
RETURN([ct, ans]);
end;
MATHEMATICA
row[n_] := Select[ Divisors[n], GCD[#, n/#] == 1 &]; Table[row[n], {n, 1, 30}] // Flatten (* Jean-François Alcover, Oct 22 2012 *)
PROG
(Haskell)
a077610 n k = a077610_row n !! k
a077610_row n = [d | d <- [1..n], let (n', m) = divMod n d,
m == 0, gcd d n' == 1]
a077610_tabf = map a077610_row [1..]
-- Reinhard Zumkeller, Feb 12 02
(PARI) row(n)=my(f=factor(n), k=#f~); Set(vector(2^k, i, prod(j=1, k, if(bittest(i, j-1), 1, f[j, 1]^f[j, 2]))))
v=[]; for(n=1, 20, v=concat(v, row(n))); v \\ Charles R Greathouse IV, Sep 02 2015
(PARI) row(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); } \\ Michel Marcus, Oct 11 2015
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Eric W. Weisstein, Nov 11 2002
STATUS
approved