A077610 Triangle in which n-th row lists unitary divisors of n.

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, 1, 3, 4, 12, 1, 13, 1, 2, 7, 14, 1, 3, 5, 15, 1, 16, 1, 17, 1, 2, 9, 18, 1, 19, 1, 4, 5, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 3, 8, 24, 1, 25, 1, 2, 13, 26, 1, 27, 1, 4, 7, 28, 1, 29, 1, 2, 3, 5, 6, 10, 15, 30

Offset: 1

Author

Eric W. Weisstein, Nov 11 2002

Keywords

• nonn
• tabf

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

Examples

			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;
		

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.
• Index entries for sequences related to enumerating the rationals.

Crossrefs

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

Programs

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

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;
# Alternatively:
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

Mathematica

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

Formula

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