A165430 -id:A165430 - cp's OEIS frontend

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

Eric W. Weisstein, Nov 11 2002

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

			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;

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.

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

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

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

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

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

row(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d);} \\ Michel Marcus, Oct 11 2015

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

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

A188999 Bi-unitary sigma: sum of the bi-unitary divisors of n.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Apr 15 2011

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.

			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.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Krishnaswami Alladi, On arithmetic functions and divisors of higher order, J. Austral. Math. Soc. 23 (series A) (1977), 9-27.
József Sándor and Borislav Crstici, Perfect numbers: Old and new issues; perspectives, in Handbook of number theory, II, p. 45.
László Tóth, On the bi-unitary analogues of Euler's arithmetical function and the gcd-sum function J. Int. Seq. 12 (2009), Article 09.5.2.
Charles 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.

Cf. A222266, A027748, A124010, A034448, A319072.

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

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) ;

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 *)

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

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

from math import prod
from sympy import factorint
def A188999(n): return prod((p**(e+1)-1)//(p-1)-(0 if e&1 else p**(e>>1)) for p,e in factorint(n).items()) # Chai Wah Wu, Dec 28 2024

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
Dirichlet g.f.: zeta(s-1) * zeta(s) * zeta(2*s-1) * Product_{p prime} (1 - 2/p^(2*s-1) + 1/p^(3*s-2) + 1/p^(3*s-1) - 1/p^(4*s-2)). - Amiram Eldar, Aug 28 2023

A222266 Irregular triangle which lists the bi-unitary divisors of n in row n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 2, 3, 6, 1, 7, 1, 2, 4, 8, 1, 9, 1, 2, 5, 10, 1, 11, 1, 3, 4, 12, 1, 13, 1, 2, 7, 14, 1, 3, 5, 15, 1, 2, 8, 16, 1, 17, 1, 2, 9, 18, 1, 19, 1, 4, 5, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 2, 3, 4, 6, 8, 12, 24, 1, 25, 1, 2, 13, 26, 1, 3, 9, 27, 1, 4, 7, 28, 1, 29, 1, 2, 3, 5, 6, 10, 15, 30, 1, 31, 1, 2, 4, 8, 16, 32, 1, 3, 11, 33, 1, 2, 17, 34, 1, 5, 7, 35

Offset: 1

R. J. Mathar, May 05 2013

The bi-unitary divisors of n are the divisors of n such that the largest common unitary divisor of d and n/d is 1, indicated by A165430.
The first difference from the triangle A077609 is in row n=16.
The concept of bi-unitary divisors was introduced by Suryanarayana (1972). - Amiram Eldar, Mar 09 2024

			The table starts
  1;
  1, 2;
  1, 3;
  1, 4;
  1, 5;
  1, 2, 3, 6;
  1, 7;
  1, 2, 4, 8;
  1, 9;
  1, 2, 5, 10;
  1, 11;
  1, 3, 4, 12;
  1, 13;
  1, 2, 7, 14;
  1, 3, 5, 15;
  1, 2, 8, 16;
  1, 17;

Michael De Vlieger, Table of n, a(n) for n = 1..13171 (rows 1 <= n <= 2000).
D. Suryanarayana, The number of bi-unitary divisors of an integer, in: A. A. Gioia and D. L. Goldsmith (eds.), The Theory of Arithmetic Functions, Lecture Notes in Mathematics, Vol 251, Springer, Berlin, Heidelberg, 1972.

Cf. A077609, A165430, A188999 (row sums), A286324 (row lengths).

# Return set of unitary divisors of n.
A077610_row := proc(n)
    local u,d ;
    u := {} ;
    for d in numtheory[divisors](n) do
        if igcd(n/d,d) = 1 then
            u := u union {d} ;
        end if;
    end do:
    u ;
end proc:
# true if d is a bi-unitary divisor of n.
isbiudiv := proc(n,d)
    if n mod d = 0 then
        A077610_row(d) intersect A077610_row(n/d) ;
        if % = {1} then
            true;
        else
            false;
        end if;
    else
        false;
    end if;
end proc:
# Return set of bi-unitary divisors of n
biudivs := proc(n)
    local u,d ;
    u := {} ;
    for d in numtheory[divisors](n) do
        if isbiudiv(n,d) then
            u := u union {d} ;
        end if;
    end do:
    u ;
end proc:
for n from 1 to 35 do
    print(op(biudivs(n))) ;
end do:

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; Table[Function[d, Union@ Flatten@ Select[Transpose@ {d, n/d}, Last@ Intersection[f@ #1, f@ #2] == 1 & @@ # &]]@ Select[Divisors@ n, # <= Floor@ Sqrt@ n &], {n, 35}] (* Michael De Vlieger, May 07 2017 *)

isbdiv(f, d) = {for (i=1, #f~, if(f[i, 2]%2 == 0 && valuation(d, f[i, 1]) == f[i, 2]/2, return(0))); 1;}
row(n) = {my(d = divisors(n), f = factor(n), bdiv = []); for(i=1, #d, if(isbdiv(f, d[i]), bdiv = concat(bdiv, d[i]))); bdiv; } \\ Amiram Eldar, Mar 24 2023

A384047 Triangle read by rows: T(n, k) for 1 <= k <= n is the largest divisor of k that is a unitary divisor of n.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 1, 5, 1, 2, 3, 2, 1, 6, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 3, 4, 1, 3, 1, 4, 3, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13

Offset: 1

Amiram Eldar, May 18 2025

			Triangle begins:
  1
  1, 2
  1, 1, 3
  1, 1, 1, 4
  1, 1, 1, 1, 5
  1, 2, 3, 2, 1, 6
  1, 1, 1, 1, 1, 1, 7
  1, 1, 1, 1, 1, 1, 1, 8
  1, 1, 1, 1, 1, 1, 1, 1, 9
  1, 2, 1, 2, 5, 2, 1, 2, 1, 10

Amiram Eldar, Table of n, a(n) for n = 1..10585 (first 145 rows flattened)

Cf. A005117, A050873, A077610, A145388 (row sums), A165430, A225174, A384046.

Upper right triangle of A322482.

udiv[n_] := Select[Divisors[n], CoprimeQ[#, n/#] &]; T[n_, k_] := Max[Intersection[udiv[n], Divisors[k]]]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten

udiv(n) = select(x -> gcd(x, n/x) == 1, divisors(n));
T(n, k) = vecmax(setintersect(udiv(n), divisors(k)));

T(n, 1) = 1.
T(n, n) = n.
T(n, k) <= A050873(n, k) = gcd(n, k), with equality if n is squarefree (A005117).

A116550 The bi-unitary analog of Euler's totient function of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 3, 6, 7, 8, 6, 10, 8, 12, 9, 9, 15, 16, 12, 18, 14, 14, 15, 22, 17, 24, 18, 26, 21, 28, 15, 30, 31, 23, 24, 25, 29, 36, 27, 28, 31, 40, 21, 42, 35, 34, 33, 46, 36, 48, 36, 37, 42, 52, 39, 42, 46, 42, 42, 58, 34, 60, 45, 51, 63, 50, 35, 66, 56, 51, 38, 70, 62, 72, 54

Offset: 1

Leroy Quet, Mar 16 2006

nonn

a(1)=1; for n>1, a(n) is the number of numbers mA225174(m,n)=1. - N. J. A. Sloane, May 01 2013

			12 = 2^2 * 3^1. Of the positive integers < 12, there are 8 integers where no prime divides these integers the same number of times the prime divides 12: 1, 2 = 2^1, 5 = 5^1, 7 = 7^1, 8 = 2^3, 9 = 3^2, 10 = 2^1 *5^1 and 11 = 11^1. So a(12) = 8. The other positive integers < 12 (3 = 3^1, 4 = 2^2 and 6 = 2^1 * 3^1) each are divisible by at least one prime the same number of times this prime divides 12.

M. Lal, H. Wareham and R. Mifflin, Iterates of the bi-unitary totient function, Utilitas Math., 10 (1976), 347-350.

Donovan Johnson, Table of n, a(n) for n = 1..10000
László Tóth, On the Bi-Unitary Analogues of Euler's Arithmetical Function and the Gcd-Sum Function, JIS, Vol. 12 (2009), Article 09.5.2, function phi**(n).

Cf. A225174, A005424, A225175, A225176, A000010, A047994.

# returns the greatest common unitary divisor of m and n, A225174(m,n)
f:=proc(m,n)
   local i,ans;
   ans:=1;
   for i from 1 to min(m,n) do
     if ((m mod i) = 0) and (igcd(i,m/i) = 1)  then
       if ((n mod i) = 0) and (igcd(i,n/i) = 1)  then ans:=i; fi;
     fi;
   od;
ans; end;
A116550:=proc(n)
  global f; local ct,m;
  ct:=0;
  if n = 1 then RETURN(1) else
  for m from 1 to n-1 do
    if f(m,n)=1 then ct:=ct+1; fi;
  od:
  fi;
  ct;
end; # N. J. A. Sloane, May 01 2013
A116550 := proc(n)
    local a,k;
    a := 0 ;
    for k from 1 to n do
        if A165430(k,n) = 1 then
            a := a+1 ;
        end if ;
    end do:
    a ;
end proc: # R. J. Mathar, Jul 21 2016

a[1] = 1; a[n_] := With[{pp = Power @@@ FactorInteger[n]}, Count[Range[n], m_ /; Intersection[pp, Power @@@ FactorInteger[m]] == {}]]; Table[a[n], {n, 1, 90}] (* Jean-François Alcover, Sep 05 2013 *)
phi[x_, n_] := DivisorSum[n, MoebiusMu[#]*Floor[x/#] &]; a[n_] := DivisorSum[n, (-1)^PrimeNu[#]*phi[n/#, #] &, CoprimeQ[#, n/#] &]; Array[a, 100] (* Amiram Eldar, Jul 16 2022 *)

udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
a(n) = if (n==1, 1, sum(k=1, n-1, gcud(n, k) == 1)); \\ Michel Marcus, Nov 09 2017

phi(x,n) = sumdiv(n, d, moebius(d)*floor(x/d));
a(n) = sumdiv(n, d, (gcd(d, n/d) == 1) * (-1)^omega(d) * phi(n/d,d)); \\ Amiram Eldar, Jul 16 2022

For n>1, if n = product{p=primes,p|n} p^b(n,p), where each b(n,p) is a positive integer, then a(n) is number of positive integers m, m < n, such that each b(m,p) does not equal b(n,p).

a(n) = Sum_{d|n, gcd(d,n/d)=1} (-1)^omega(d) * phi(x, n), where phi(x, n) = #{1 <= k <= x, gcd(k, n) = 1} = Sum_{d|n} mu(d) * floor(x/d) (Tóth, 2009). - Amiram Eldar, Jul 16 2022

More terms from R. J. Mathar, Jan 23 2008

Entry revised by N. J. A. Sloane, May 01 2013

A225174 Square array read by antidiagonals: T(m,n) = greatest common unitary divisor of m and n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 5, 1, 3, 1, 1

Offset: 1

N. J. A. Sloane, May 01 2013

			Array begins
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, ...
1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3, ...
1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, ...
1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, ...
1, 2, 3, 1, 1, 6, 1, 1, 1, 2, 1, 3, ...
1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, ...
...
The unitary divisors of 3 are 1 and 3, those of 6 are 1,2,3,6; so T(6,3) = T(3,6) = 3.

M. Lal, H. Wareham and R. Mifflin, Iterates of the bi-unitary totient function, Utilitas Math., 10 (1976), 347-350.

Antti Karttunen, Table of n, a(n) for n = 1..20100; the first 200 antidiagonals of the array

See A034444, A077610 for unitary divisors of n.

Different from A059895.

# returns the greatest common unitary divisor of m and n
f:=proc(m,n)
local i,ans;
ans:=1;
for i from 1 to min(m,n) do
if ((m mod i) = 0) and (igcd(i,m/i) = 1)  then
   if ((n mod i) = 0) and (igcd(i,n/i) = 1)  then ans:=i; fi;
fi;
od;
ans; end;

f[m_, n_] := Module[{i, ans=1}, For[i=1, i<=Min[m, n], i++, If[Mod[m, i]==0 && GCD[i, m/i]==1, If[Mod[n, i]==0 && GCD[i, n/i]==1, ans=i]]]; ans];
Table[f[m-n+1, n], {m, 1, 14}, {n, 1, m}] // Flatten (* Jean-François Alcover, Jun 19 2018, translated from Maple *)

up_to = 20100; \\ = binomial(200+1,2)
A225174sq(m,n) = { my(a=min(m,n),b=max(m,n),md=0); fordiv(a,d,if(0==(b%d)&&1==gcd(d,a/d)&&1==gcd(d,b/d),md=d)); (md); };
A225174list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, if(i++ > up_to, return(v)); v[i] = A225174sq((a-(col-1)),col))); (v); };
v225174 = A225174list(up_to);
A225174(n) = v225174[n]; \\ Antti Karttunen, Nov 28 2018

T(m,n) = T(n,m) = A165430(n,m).

A337490 a(0)=1; for n > 0, a(n) = the greatest common divisor (GCD) of n and the sum of all previous terms if the GCD is not already in the sequence; otherwise a(n) = a(n-1) + n.

Original entry on oeis.org

1, 2, 4, 7, 11, 5, 6, 13, 21, 30, 10, 21, 33, 46, 14, 29, 45, 62, 18, 37, 57, 78, 22, 45, 69, 94, 26, 53, 81, 110, 140, 171, 203, 236, 270, 305, 341, 378, 416, 39, 79, 120, 162, 205, 249, 294, 340, 387, 3, 52, 102, 17, 69, 122, 176, 231, 287, 344, 402, 461, 521, 582, 644, 707, 771, 836, 902, 969

Offset: 0

Scott R. Shannon, Aug 29 2020

nonn

The sequence displays the unusual behavior of decreasing 53 times in the first 1975 terms, due to the existence of a GCD which has not previously appeared in the sequence, but then not decreasing again for n up to at least 100 million. In this period there are 37 repeated terms, the first being 21 at n=11 and the last 161202 at n=2054. In the same range many values do not appear, for example 16,23,28,32,36. It is unknown when the sequence decreases again, or if all values eventually appear. The 100 millionth term is 4999999948050717.

See the companion sequence A333980 for the sum of the terms from a(0) to a(n).

			a(2) = 4 as the sum of all previous terms is a(0)+a(1) = 3, and the GCD of 3 and 2 is 1, which has already appeared in the sequence. Therefore a(2) = a(1) + n = 2 + 2 = 4.
a(4) = 11 as the sum of all previous terms is a(0)+...+a(3) = 14, and the GCD of 14 and 4 is 2. However 2 has already appeared so a(4) = a(3) + n = 7 + 4 = 11.
a(5) = 5 as the sum of all previous terms is a(0)+...+a(4) = 25, and the GCD of 25 and 5 is 5, and as 5 has not previous appeared a(5) = 5.

Scott R. Shannon, Table of n, a(n) for n = 0..10000
Scott R. Shannon, Graph of the terms for n=0..2500. This includes the last known decrease in the sequence, n(1974) = 42.
Scott R. Shannon, Graph of the terms for n=0..10000000.

Cf. A333980, A333826 (same rules but starting a(1)=1), A165430, A064814, A082299, A005132, A336957.

lista(nn) = {my(va = vector(nn), s=0); va[1] = 1; s += va[1]; for (n=2, nn, my(g = gcd(n-1, s)); if (#select(x->(x==g), va), va[n] = va[n-1]+n-1, va[n] = g); s += va[n];); va;} \\ Michel Marcus, Sep 05 2020

A200723 The sum of integers k from 1 to n such that the greatest common unitary divisor of k and n is 1.

Original entry on oeis.org

1, 1, 3, 6, 10, 10, 21, 28, 36, 32, 55, 53, 78, 66, 69, 120, 136, 112, 171, 144, 153, 170, 253, 211, 300, 240, 351, 300, 406, 237, 465, 496, 384, 416, 445, 539, 666, 522, 558, 633, 820, 444, 903, 780, 772, 770, 1081, 887, 1176, 912, 951, 1104

Offset: 1

R. J. Mathar, Nov 21 2011

Previous name was "Bi-unitary Euler function of n".

a(n) is the sum over all entries equal to 1 in row A165430(n,), weighted by the column index.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)
László Tóth, On the bi-unitary analogues of Euler's arithmetical function and the gcd-sum function J. Int. Seq. 12 (2009), Article 09.5.2.

Cf. A116550, A165430, A188999

a200723 = sum . zipWith (*) [1..] . map a063524 . a165430_row
-- Reinhard Zumkeller, Mar 04 2013

A200723 := proc(n)
        local a,k ;
        a := 0 ;
        for k from 1 to n do
                if A165430(k,n) = 1 then
                        a := a+ k ;
                end if;
        end do;
        a ;
end proc:
seq(A200723(n),n=1..80) ;

T[n_, k_] := Module[{d = Divisors[GCD[n, k]]}, Max[Select[d, CoprimeQ[#, k/#] && CoprimeQ[#, n/#] &]]]; a[n_] := Sum[k * Boole[T[n, k] == 1], {k, 1, n}]; Array[a, 100] (* Amiram Eldar, May 23 2025 *)

udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
a(n) = sum(k=1, n, if (vecmax(setintersect(udivs(n), udivs(k))) == 1, k)); \\ Michel Marcus, Jun 28 2023

a(6) = 1*1 + 4*1 +5*1 = 10 corresponding to the three 1's in row 6 of A165430.

New name from Amiram Eldar, May 23 2025

A333980 Partial sums of A337490.

Original entry on oeis.org

1, 3, 7, 14, 25, 30, 36, 49, 70, 100, 110, 131, 164, 210, 224, 253, 298, 360, 378, 415, 472, 550, 572, 617, 686, 780, 806, 859, 940, 1050, 1190, 1361, 1564, 1800, 2070, 2375, 2716, 3094, 3510, 3549, 3628, 3748, 3910, 4115, 4364, 4658, 4998, 5385, 5388, 5440, 5542, 5559, 5628, 5750

Offset: 0

Scott R. Shannon, Sep 04 2020

nonn

See A337490 for an explanation of the sequence.

Scott R. Shannon, Table of n, a(n) for n = 0..10000

Cf. A337490, A165430, A064814, A082299, A005132, A336957.

A275254 The bi-unitary gcd-sum function.

Original entry on oeis.org

1, 3, 5, 7, 9, 14, 13, 15, 17, 25, 21, 30, 25, 36, 43, 31, 33, 47, 37, 57, 61, 58, 45, 64, 49, 69, 53, 82, 57, 108, 61, 63, 99, 91, 113, 99, 73, 102, 117, 117, 81, 163, 85, 132, 141, 124, 93, 130, 97, 135, 155, 157, 105, 146, 181

Offset: 1

R. J. Mathar, Jul 21 2016

nonn

Row sums of A165430.

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, On the Bi-Unitary Analogues of Euler's Arithmetical Function and the Gcd-Sum Function, JIS 12 (2009), Article 09.5.2, function P**(n).

Cf. A013661, A165430, A256392.

Pstarstar := proc(n)
    add(A165430(k,n),k=1..n) ;
end proc:

phi[x_, n_] := Sum[Boole[GCD[k, n] == 1], {k, 1, x}]; uphi[1]=1; uphi[n_] := Times @@ (-1 + Power @@@ FactorInteger[n]); a[n_] := DivisorSum[n, uphi[#] * phi[n/#, #] &, GCD[#, n/#] == 1 &]; Array[a, 100] (* Amiram Eldar, Sep 09 2019 *)

a(n) = Sum_{k=1..n} A165430(n,k).

Sum_{k=1..n} a(k) = c * n^2 * log(n) / 2 + O(n^2), where c = Product_{p prime} (1 - (3*p-1)/(p^2*(p+1))) = zeta(2) * Product_{p prime} (1 - (2*p-1)^2/p^4) = A013661 * A256392 = 0.35823163000196141456... . - Amiram Eldar, Dec 22 2023

cp's OEIS Frontend

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A188999 Bi-unitary sigma: sum of the bi-unitary divisors of n.

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A222266 Irregular triangle which lists the bi-unitary divisors of n in row n.

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A384047 Triangle read by rows: T(n, k) for 1 <= k <= n is the largest divisor of k that is a unitary divisor of n.

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A116550 The bi-unitary analog of Euler's totient function of n.

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A225174 Square array read by antidiagonals: T(m,n) = greatest common unitary divisor of m and n.

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