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Triangle in which n-th row lists unitary divisors of n.
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%I #43 Mar 09 2024 05:59:29

%S 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,

%T 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,

%U 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

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

%C n-th row = n-th row of A165430 without repetitions. - _Reinhard Zumkeller_, Mar 04 2013

%C 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

%C 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

%H Reinhard Zumkeller, <a href="/A077610/b077610.txt">Rows n=1..1000 of triangle, flattened</a>

%H Eckford Cohen, <a href="https://doi.org/10.1007/BF01180473">Arithmetical functions associated with the unitary divisors of an integer</a>, Mathematische Zeitschrift, Vol. 74 (1960), pp. 66-80.

%H R. Vaidyanathaswamy, <a href="https://doi.org/10.1090/S0002-9947-1931-1501607-1">The theory of multiplicative arithmetic functions</a>, Transactions of the American Mathematical Society, Vol. 33, No. 2 (1931), pp. 579-662.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnitaryDivisor.html">Unitary Divisor</a>.

%H <a href="/index/Ra#rational">Index entries for sequences related to enumerating the rationals</a>.

%e 1;

%e 1, 2;

%e 1, 3;

%e 1, 4;

%e 1, 5;

%e 1, 2, 3, 6;

%e 1, 7;

%e 1, 8;

%e 1, 9;

%e 1, 2, 5, 10;

%e 1, 11;

%p with(numtheory);

%p # returns the number of unitary divisors of n and a list of them, from _N. J. A. Sloane_, May 01 2013

%p f:=proc(n)

%p local ct,i,t1,ans;

%p ct:=0; ans:=[];

%p t1:=divisors(n);

%p for i from 1 to nops(t1) do

%p d:=t1[i];

%p if igcd(d,n/d)=1 then ct:=ct+1; ans:=[op(ans),d]; fi;

%p od:

%p RETURN([ct,ans]);

%p end;

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

%o (Haskell)

%o a077610 n k = a077610_row n !! k

%o a077610_row n = [d | d <- [1..n], let (n',m) = divMod n d,

%o m == 0, gcd d n' == 1]

%o a077610_tabf = map a077610_row [1..]

%o -- _Reinhard Zumkeller_, Feb 12 02

%o (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]))))

%o v=[];for(n=1,20,v=concat(v,row(n)));v \\ _Charles R Greathouse IV_, Sep 02 2015

%o (PARI) row(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d);} \\ _Michel Marcus_, Oct 11 2015

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

%K nonn,tabf

%O 1,3

%A _Eric W. Weisstein_, Nov 11 2002