1,3

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

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.

Index entries for sequences related to enumerating the rationals.

d is unitary divisor of n <=> gcd(n, d) = d and gcd(n/d, d) = 1. - Peter Luschny, Jun 13 2025

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;

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;

# Alternative:

isUnitary := (n, d) -> igcd(n, d) = d and igcd(n/d, d) = 1:

aList := n -> select(k -> isUnitary(n, k), [seq(1..n)]): # Peter Luschny, Jun 13 2025

row[n_] := Select[ Divisors[n], GCD[#, n/#] == 1 &]; Table[row[n], {n, 1, 30}] // Flatten (* Jean-François Alcover, Oct 22 2012 *)

(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 2012

(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

(Python)

from math import gcd

def is_unitary(n, d) -> bool: return gcd(n, d) == d and gcd(n//d, d) == 1

def aList(n) -> list[int]: return [k for k in range(1, n+1) if is_unitary(n, k)]

for n in range(1, 31): print(aList(n)) # Peter Luschny, Jun 13 2025

Cf. A037445, A027750, A034444 (row lengths), A034448 (row sums); A206778.

Sequence in context: A222266 A077609 A379027 * A329534 A317746 A364449

Adjacent sequences: A077607 A077608 A077609 * A077611 A077612 A077613

nonn,tabf

Eric W. Weisstein, Nov 11 2002

approved