A009194 -id:A009194 - cp's OEIS frontend

A300230 Restricted growth sequence transform of A286570, combining A009194(n) and A046523(n), i.e., gcd(n,sigma(n)) and the prime signature of n.

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

Offset: 1

Antti Karttunen, Mar 01 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A009194, A046523, A286570.

Cf. also A300223, A300226, A300229, A300231, A300232, A300233, A300235.

rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A009194(n) = gcd(n, sigma(n));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from A046523
A286570(n) = (1/2)*(2 + ((A046523(n)+A009194(n))^2) - A046523(n) - 3*A009194(n));
write_to_bfile(1,rgs_transform(vector(65537,n,A286570(n))),"b300230.txt");

A355925 Square array A(n, k) = A009194(A246278(n, k)), read by falling antidiagonals.

Original entry on oeis.org

1, 1, 1, 6, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 2, 15, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 28, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Jul 22 2022

			The top left corner of the array:
   k=  1  2  3  4  5  6  7  8  9 10  11  12 13  14 15 16 17 18  19  20 21
  2k=  2  4  6  8 10 12 14 16 18 20  22  24 26  28 30 32 34 36  38  40 42
-----+-----------------------------------------------------------------------
   1 | 1, 1, 6, 1, 2, 4, 2, 1, 3, 2,  2, 12, 2, 28, 6, 1, 2, 1,  2, 10, 6,
   2 | 1, 1, 3, 1, 1, 3, 3, 1, 1, 1,  1, 15, 3,  3, 3, 1, 1, 1,  3,  1, 3,
   3 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 5,  1, 1, 1, 1, 1,  5,  1, 7,
   4 | 1, 1, 1, 1, 7, 1, 1, 1, 7, 7,  1,  1, 1,  1, 7, 1, 1, 7,  1,  7, 1,
   5 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1, 19, 1, 1, 1, 1,  1,  1, 1,
   6 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1, 17, 1,  1, 1, 1, 1, 1,  1,  1, 1,
   7 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1,  1, 1, 1, 1, 1,  1,  1, 1,
   8 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 19,  1, 1,  1, 1, 1, 1, 1,  1,  1, 1,
   9 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1,  1, 1, 1, 1, 1,  1,  1, 1,
  10 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1,  1, 1, 1, 1, 1,  1,  1, 1,
  11 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1, 37, 1,  1, 1, 1, 1, 1, 31,  1, 1,
  12 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1,  1, 1, 1, 1, 1,  1,  1, 1,
  13 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1,  1, 1, 1, 1, 1,  1,  1, 1,
  14 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1,  1, 1, 1, 1, 1,  1,  1, 1,
  15 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1, 61, 1, 1, 1, 1,  1,  1, 1,
  16 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1,  1, 1, 1, 1, 1,  1,  1, 1,
  17 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1,  1, 1, 1, 1, 1,  1,  1, 1,
  18 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1,  1, 1, 1, 1, 1,  1,  1, 1,
  19 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1,  1, 1, 1, 1, 1,  1,  1, 1,
  20 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1,  1, 1, 1, 1, 1,  1,  1, 1,
  21 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1,  1, 1, 1, 1, 1,  1,  1, 1,

Cf. A000203, A009194, A246278.

Cf. also A341605, A341606, A341607, A341608, A341626, A341627, A355924, A355927 for related arrays of similar construction.

up_to = 105;
A009194(n) = gcd(n, sigma(n));
A246278sq(row,col) = if(1==row,2*col, my(f = factor(2*col)); for(i=1, #f~, f[i,1] = prime(primepi(f[i,1])+(row-1))); factorback(f));
A355925sq(row,col) = A009194(A246278sq(row,col));
A355925list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A355925sq(col,(a-(col-1))))); (v); };
v355925 = A355925list(up_to);
A355925(n) = v355925[n];

A(n, k) = A009194(A246278(n, k)).
A(n, k) = gcd(A246278(n,k), A355927(n, k)).
A(n, k) = A355927(n, k) / A341605(n, k).
A(n, k) = A246278(n, k) / A341606(n, k).

A300231 Filter sequence combining A001065(n) and A009194(n), the sum of proper divisors of n and gcd(n,sigma(n)).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 21, 2, 22, 23, 24, 19, 25, 2, 13, 26, 27, 2, 28, 2, 29, 30, 31, 2, 32, 33, 34, 12, 35, 2, 36, 26, 37, 38, 39, 2, 40, 2, 41, 42, 43, 44, 45, 2, 46, 47, 48, 2, 49, 2, 50, 51, 52, 44, 53, 2, 54, 55, 56, 2, 57, 38, 35, 30, 58, 2, 59, 60, 32, 61, 62, 63, 64

Offset: 1

Antti Karttunen, Mar 01 2018

nonn

Restricted growth sequence transform of P(A001065(n), A009194(n)), where P(a,b) is a two-argument form of A000027 used as a Cantor pairing function N x N -> N.

			a(27) = a(35) (= 19) because A001065(27) = A001065(35) = 13 and A009194(27) = A009194(35) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001065, A009194.

Cf. also A295885, A300223, A300226, A300229, A300230, A300232, A300233, A300235.

rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A001065(n) = (sigma(n)-n);
A009194(n) = gcd(n, sigma(n));
Aux300231(n) = (1/2)*(2 + ((A001065(n)+A009194(n))^2) - A001065(n) - 3*A009194(n));
write_to_bfile(1,rgs_transform(vector(65537,n,Aux300231(n))),"b300231.txt");

A300233 Filter sequence combining A051953(n) and A009194(n), cototient of n and gcd(n,sigma(n)).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 13, 2, 15, 16, 17, 14, 18, 2, 19, 2, 20, 21, 22, 23, 24, 2, 25, 26, 27, 2, 28, 2, 29, 30, 31, 2, 32, 33, 34, 35, 36, 2, 37, 26, 38, 39, 40, 2, 41, 2, 42, 43, 44, 45, 46, 2, 47, 48, 49, 2, 50, 2, 51, 52, 53, 45, 54, 2, 55, 43, 56, 2, 57, 39, 58, 59, 60, 2, 61, 62, 60, 63, 55, 64, 65, 2, 66

Offset: 1

Antti Karttunen, Mar 01 2018

nonn

Restricted growth sequence transform of P(A051953(n), A009194(n)), where P(a,b) is a two-argument form of A000027 used as a Cantor pairing function N x N -> N.

			a(20) = a(22) (= 13) because A051953(20) = A051953(22) = 12 and A009194(20) = A009194(22) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A009194, A051953.

Cf. also A295885, A300223, A300226, A300229, A300230, A300231, A300232, A300235.

rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A009194(n) = gcd(n, sigma(n));
A051953(n) = (n - eulerphi(n));
Aux300233(n) = (1/2)*(2 + ((A051953(n)+A009194(n))^2) - A051953(n) - 3*A009194(n));
write_to_bfile(1,rgs_transform(vector(65537,n,Aux300233(n))),"b300233.txt");

A324396 a(1) = 0; for n > 1, a(n) = A009194(A156552(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 3, 3, 1, 1, 4, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 12, 1, 2, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 3, 2, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 10, 1, 1, 1, 6, 1, 1, 1, 12, 5, 6, 15, 1, 3, 3, 1, 2, 3, 4, 5, 1, 1, 2, 3, 1, 3, 1, 1, 6

Offset: 1

Antti Karttunen, Mar 05 2019

nonn

Cf. A009194, A156552, A323243, A323244, A324051, A324394, A324397, A324398.

A009194(n) = gcd(n, sigma(n));
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A324396(n) = if(1==n,0,A009194(A156552(n)));

a(1) = 0; for n > 1, a(n) = A009194(A156552(n)).

a(n) = gcd(A156552(n), A323243(n)).

A326194 Number of iterations of x -> A009194(x) needed to reach a fixed point when starting from x = n, where A009194(x) = gcd(x, sigma(x)).

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 0, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 2, 1, 3, 2, 2, 1, 2, 2, 3, 1, 1, 2, 1, 1, 1, 1, 2, 2

Offset: 1

Antti Karttunen, Aug 24 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000203, A007691 (positions of zeros), A009194, A326195, A326196.

A326194(n) = { my(u=gcd(n,sigma(n))); if(u==n,0,1+A326194(u)); };

If gcd(n,sigma(n)) = n, then a(n) = 0, otherwise a(n) = 1 + a(gcd(n,sigma(n))).

a(n) < A326196(n).

A331744 Lexicographically earliest infinite sequence such that a(i) = a(j) => A009194(i) = A009194(j) and A323901(i) = A323901(j) for all i, j.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 5, 1, 6, 7, 8, 9, 3, 10, 11, 1, 12, 13, 14, 7, 15, 16, 17, 18, 19, 7, 17, 20, 6, 21, 22, 1, 23, 24, 25, 6, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 9, 30, 19, 37, 7, 15, 38, 39, 40, 41, 42, 14, 43, 41, 44, 22, 1, 45, 46, 47, 24, 48, 49, 50, 13, 51, 52, 53, 54, 55, 56, 57, 7, 58, 59, 60, 61, 62, 63, 64, 33, 65, 66, 67, 68, 69, 70, 71, 18, 69, 30, 72, 19

Offset: 1

Antti Karttunen, Feb 04 2020

nonn

Restricted growth sequence transform of the ordered pair [A009194(n), A323901(n)].

Cf. A002487, A009194, A163511, A323901.

Cf. also A324400, A318310, A324389, A331745, A331746, A331747.

\\ Needs also code from A323901.
up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A009194(n) = gcd(n, sigma(n));
Aux331744(n) = [A009194(n),A323901(n)];
v331744 = rgs_transform(vector(up_to, n, Aux331744(n)));
A331744(n) = v331744[n];

a(2^n) = 1 for all n >= 0.

A324389 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A009194(n), A318458(n)] for all other numbers, except f(1) = -1.

Original entry on oeis.org

1, 2, 3, 2, 3, 4, 3, 2, 2, 5, 3, 6, 3, 7, 8, 2, 3, 9, 3, 10, 3, 11, 3, 12, 2, 13, 14, 15, 3, 16, 3, 2, 17, 18, 3, 19, 3, 11, 3, 20, 3, 21, 3, 22, 23, 7, 3, 6, 2, 24, 25, 26, 3, 27, 28, 29, 28, 30, 3, 31, 3, 32, 33, 2, 3, 34, 3, 18, 17, 35, 3, 36, 3, 5, 3, 37, 3, 38, 3, 39, 2, 18, 3, 40, 41, 11, 17, 42, 3, 43, 44, 45, 3, 46, 47, 12, 3, 48, 23, 49, 3, 50, 3

Offset: 1

Antti Karttunen, Mar 05 2019

For all i, j:
  A324401(i) = A324401(j) => a(i) = a(j).
Regarding the scatter plot of this sequence, see also comments in A318310. - Antti Karttunen, Feb 04 2020

Cf. A009194, A318310, A318458, A324390, A324401, A324530, A331744, A331746, A331747.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A009194(n) = gcd(n,sigma(n));
A318458(n) = bitand(n,sigma(n)-n);
Aux324389(n) = if(1==n,-1,[A009194(n), A318458(n)]);
v324389 = rgs_transform(vector(up_to,n,Aux324389(n)));
A324389(n) = v324389[n];

A300242 Filter sequence combining gcd(n,sigma(n)) and gcd(n,phi(n)), (A009194 and A009195).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 5, 6, 1, 7, 1, 6, 8, 9, 1, 10, 1, 11, 5, 6, 1, 12, 13, 6, 14, 15, 1, 3, 1, 16, 8, 6, 1, 17, 1, 6, 5, 18, 1, 19, 1, 7, 20, 6, 1, 21, 22, 23, 8, 11, 1, 24, 13, 25, 5, 6, 1, 26, 1, 6, 14, 27, 1, 3, 1, 11, 8, 6, 1, 28, 1, 6, 13, 7, 1, 19, 1, 29, 30, 6, 1, 31, 1, 6, 8, 32, 1, 33, 34, 7, 5, 6, 35, 36, 1, 37, 20, 38, 1, 3, 1, 39, 20

Offset: 1

Antti Karttunen, Mar 02 2018

nonn

Restricted growth sequence transform of P(A009194(n), A009195(n)), where P(a,b) is a two-argument form of A000027 used as a Cantor pairing function N x N -> N.

			a(15) = a(33) (= 8) because A009194(15) = A009194(33) = 3 and A009195(15) = A009195(33) = 1.
a(20) = a(52) (= 11) because A009194(20) = A009194(52) = 2 and A009195(20) = A009195(52) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A009194, A009195.

Cf. also A286591, A300225, A300240, A300241, A300243.

rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A009194(n) = gcd(n, sigma(n));
A009195(n) = gcd(n, eulerphi(n));
Aux300242(n) = (1/2)*(2 + ((A009194(n)+A009195(n))^2) - A009194(n) - 3*A009195(n));
write_to_bfile(1,rgs_transform(vector(65537,n,Aux300242(n))),"b300242.txt");

A324394 a(n) = A009194(A005940(1+n)), where A005940 is the Doudna sequence and A009194(n) = gcd(n,sigma(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 3, 4, 1, 3, 1, 1, 1, 2, 1, 2, 1, 6, 3, 12, 1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 3, 28, 1, 6, 1, 10, 1, 2, 3, 12, 1, 18, 15, 4, 1, 1, 3, 1, 1, 6, 1, 3, 1, 2, 3, 4, 1, 3, 1, 1, 1, 2, 1, 4, 1, 6, 3, 8, 7, 2, 3, 28, 1, 6, 1, 2, 1, 2, 3, 28, 1, 6, 3, 120, 1, 2, 1, 6, 1, 90, 3, 12, 1, 1, 1, 7, 1, 6, 3, 5, 7, 2

Offset: 0

Antti Karttunen, Mar 05 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to sigma(n)

Cf. A009194, A324054, A324055, A324058, A324349, A324395.

A324394(n) = { my(m1=1,m2=1,p=2,mp=p*p); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, m1 *= p; if(3==(n%4),mp *= p,m2 *= (mp-1)/(p-1))); n>>=1); gcd(m1,m2); };

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A009194(n) = gcd(n, sigma(n));
A324394(n) = A009194(A005940(1+n));

a(n) = A009194(A005940(1+n)) = gcd(A005940(1+n), A324054(n)).

cp's OEIS Frontend

A300230 Restricted growth sequence transform of A286570, combining A009194(n) and A046523(n), i.e., gcd(n,sigma(n)) and the prime signature of n.

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A355925 Square array A(n, k) = A009194(A246278(n, k)), read by falling antidiagonals.

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A300231 Filter sequence combining A001065(n) and A009194(n), the sum of proper divisors of n and gcd(n,sigma(n)).

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A300233 Filter sequence combining A051953(n) and A009194(n), cototient of n and gcd(n,sigma(n)).

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A324396 a(1) = 0; for n > 1, a(n) = A009194(A156552(n)).

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A326194 Number of iterations of x -> A009194(x) needed to reach a fixed point when starting from x = n, where A009194(x) = gcd(x, sigma(x)).

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A331744 Lexicographically earliest infinite sequence such that a(i) = a(j) => A009194(i) = A009194(j) and A323901(i) = A323901(j) for all i, j.

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A324389 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A009194(n), A318458(n)] for all other numbers, except f(1) = -1.

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A300242 Filter sequence combining gcd(n,sigma(n)) and gcd(n,phi(n)), (A009194 and A009195).

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