A305800 -id:A305800 - cp's OEIS frontend

A340365 a(n) = A005940(n) / gcd(A005940(n), A324106(n)), where A324106(n) is multiplicative with a(p^e) = A005940(p^e).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 35, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 13, 1, 11, 1, 1, 1, 21, 1, 1, 35, 1, 1, 5, 1, 1, 1, 1, 1, 49, 1, 1, 1, 3, 1, 49, 1, 1, 9, 1, 1, 27, 1, 17, 13, 1, 1, 13, 11, 1, 1, 1, 1, 55, 1, 55, 21, 1, 1, 1, 1, 1, 35, 7, 1, 21, 1, 1, 5, 7, 1, 875, 1, 27, 1, 1, 1, 121

Offset: 1

Antti Karttunen, Jan 06 2021

nonn

It is conjectured that A070776 gives the positions of all ones after the initial one. If that holds, then for all i, j: a(i) = a(j) => A340363(i) = A340363(j).

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A005940, A070776, A324106, A340362, A340364, A340366.

Cf. also A305800, A340363.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A324106(n) = { my(f=factor(n)); prod(i=1, #f~, A005940(f[i,1]^f[i,2])); };
A340365(n) = { my(t=A005940(n)); t / gcd(t, A324106(n)); };

a(n) = A005940(n) / A340364(n) = A005940(n) / gcd(A005940(n), A324106(n)).

A355831 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A354347(i) = A354347(j) for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 20 2022

nonn

Restricted growth sequence transform of the ordered pair [A046523(n), A354347(n)].

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

Cf. A305800, A355830, A355832.

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; };
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A345000(n) = gcd(A003415(n),A003415(A276086(n)));
v354347 = DirInverseCorrect(vector(up_to,n,A345000(n)));
A354347(n) = v354347[n];
Aux355831(n) = [A046523(n), A354347(n)];
v355831 = rgs_transform(vector(up_to,n,Aux355831(n)));
A355831(n) = v355831[n];

A355834 Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(j) and A355931(i) = A355931(j) for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 20 2022

nonn

Restricted growth sequence transform of the ordered pair [A348717(n), A355931(n)], where A355931(n) = A000265(A009194(i)).

			a(450) = a(3675) [= 274 as allotted by rgs-transform] because A003961(450) = 3675, therefore 450 and 3675 are in the same column of the prime shift array A246278, and because A355931(450) = A355931(3675) = 3.
a(3185) = a(14399) [= 2020 as allotted by rgs-transform] because A003961(3185) = 14399 and A355931(3185) = A355931(14399) = 7.
a(5005) = a(17017) [= 3184 as allotted by rgs-transform] because A003961(5005) = 17017 and A355931(5005) = A355931(17017) = 7.

Cf. A000203, A000265, A009194, A246278, A348717, A355931.

Cf. also A300230, A305800, A305897, A355833.

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; };
A000265(n) = (n>>valuation(n,2));
A009194(n) = gcd(n, sigma(n));
A348717(n) = if(1==n, 1, my(f = factor(n), k = primepi(f[1, 1])-1); for (i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])-k)); factorback(f));
Aux355834(n) = [A000265(A009194(n)), A348717(n)];
v355834 = rgs_transform(vector(up_to,n,Aux355834(n)));
A355834(n) = v355834[n];

A326192 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => A009195(i) = A009195(j) and f(i) = f(j), where f(n) = gcd(n,sigma(n)) * (-1)^[gcd(n,sigma(n))==n] and A009195(n) = gcd(n, phi(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 24 2019

nonn

Restricted growth sequence transform of the ordered pair [A009195(n), A326193(n)].
For all i, j:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A300242(i) = A300242(j),
  a(i) = a(j) => A326196(i) = A326196(j).

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

Cf. A009194, A009195, A300242, A305800, A326193, A326194, A326196.

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; };
Aux326192(n) = { my(u=gcd(n,sigma(n))); [gcd(n,eulerphi(n)), u*((-1)^(u==n))]; };
v326192 = rgs_transform(vector(up_to, n, Aux326192(n)));
A326192(n) = v326192[n];

A329353 Lexicographically earliest infinite sequence such that a(i) = a(j) => A329352(i) = A329352(j) for all i, j.

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, 12, 18, 19, 2, 20, 2, 21, 22, 23, 24, 25, 2, 26, 27, 28, 2, 29, 2, 30, 31, 32, 2, 33, 7, 34, 35, 25, 2, 36, 27, 37, 38, 39, 2, 40, 2, 41, 42, 43, 44, 45, 2, 46, 47, 48, 2, 49, 2, 50, 51, 52, 53, 54, 2, 55, 56, 57, 2, 58, 59, 60, 61, 62, 2, 63, 64, 65, 66, 67, 68, 69, 2, 70, 71, 72, 2, 73, 2, 49, 74

Offset: 1

Antti Karttunen, Nov 12 2019

nonn

Restricted growth sequence transform of A329352.
For all i, j:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A069359(i) = A069359(j).

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

Cf. A010051, A019565, A069359, A305800, A329352.
Cf. also A329351.
Differs from A319682 for the first time at n=254, where a(254)=123, while A319682(254)=184.

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; };
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A329352(n) = { my(m=1); fordiv(n,d,if(isprime(n/d), m *= A019565(d))); (m); };
v329353 = rgs_transform(vector(up_to, n, A329352(n)));
A329353(n) = v329353[n];

A335424 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(A335423(i)) = A046523(A335423(j)) for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 13 2020

nonn

For all i, j: A305800(i) = A305800(j) => a(i) = a(j) => A162642(i) = A162642(j).

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

Cf. A046523, A162642, A248663, A335423, A335425.

Cf. also A286622, A305800.

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; };
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A248663(n) = A048675(core(n));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A335423(n) = A005940(1+A248663(n));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
v335424 = rgs_transform(vector(up_to,n,A046523(A335423(n))));
A335424(n) = v335424[n];

A335425 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000188(i) = A000188(j) and A335424(i) = A335424(j) for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 13 2020

nonn

Restricted growth sequence transform of the ordered pair [A000188(n), A046523(A335423(n))].

For all i, j: A305800(i) = A305800(j) => a(i) = a(j) => A001222(i) = A001222(j).

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

Cf. A000188, A001222, A046523, A162642, A248663, A305800, A335423, A335424.

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; };
A000188(n) = core(n, 1)[2]; \\ From A000188
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A248663(n) = A048675(core(n));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A335423(n) = A005940(1+A248663(n));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
Aux335425(n) = [A000188(n),A046523(A335423(n))];
v335425 = rgs_transform(vector(up_to,n,Aux335425(n)));
A335425(n) = v335425[n];

A336311 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A336120(i)) = A278222(A336120(j)) and A278222(A336125(i)) = A278222(A336125(j)) for all i, j >= 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 5, 3, 2, 4, 2, 3, 5, 6, 2, 7, 2, 4, 5, 3, 2, 6, 3, 3, 8, 4, 2, 7, 2, 9, 5, 3, 3, 10, 2, 3, 5, 6, 2, 7, 2, 4, 8, 3, 2, 9, 5, 4, 5, 4, 2, 11, 3, 6, 5, 3, 2, 10, 2, 3, 8, 12, 3, 7, 2, 4, 5, 4, 2, 13, 2, 3, 4, 4, 5, 7, 2, 9, 14, 3, 2, 10, 3, 3, 5, 6, 2, 11, 5, 4, 5, 3, 3, 12, 2, 7, 8, 6, 2, 7, 2, 6, 4

Offset: 1

Antti Karttunen, Jul 17 2020

nonn

Restricted growth sequence transform of the ordered pair [A336312(n), A336313(n)].
For all i, j:
  A305800(i) = A305800(j) => a(i) = a(j) => A001222(i) = A001222(j).

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

Cf. A001222, A278222, A336120, A336124, A336125.

Cf. also A292584, A305800, A336312, A336313.

\\ Needs also code from A336120, A336124, A336125, etc:
up_to = 1024;
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; };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
Aux336311(n) = [A278222(A336120(n)),A278222(A336125(n))];
v336311 = rgs_transform(vector(up_to,n,Aux336311(n)));
A336311(n) = v336311[n];

A340362 a(n) = A005940(n) - A324106(n), where A324106(n) is multiplicative with a(p^e) = A005940(p^e).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 24, 0, 0, -32, 0, -12, 0, 0, 0, -12, 0, 0, 16, 0, 0, 140, 0, 0, 0, 0, 0, 114, 0, 0, 0, 150, 0, 280, 0, 0, 48, 0, 0, 180, 0, -108, -64, 0, 0, -70, -24, 0, 0, 0, 0, 18, 0, 140, -24, 0, 0, 0, 0, 0, 32, 330, 0, -60, 0, 0, 280, 300, 0, 632, 0, 300

Offset: 1

Antti Karttunen, Jan 06 2021

sign

It is conjectured that A070776 gives the positions of all zeros after the initial a(1) = 0. If that holds, then for all i, j: a(i) = a(j) => A340363(i) = A340363(j).

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A005940, A070776, A340364, A340365, A340366.

Cf. also A305800, A340363.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A324106(n) = { my(f=factor(n)); prod(i=1, #f~, A005940(f[i,1]^f[i,2])); };
A340362(n) = (A005940(n)-A324106(n));

A353523 Lexicographically earliest infinite sequence such that a(i) = a(j) => A349905(i) = A349905(j) and A003557(i) = A003557(j), for all i, j >= 1.

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, 14, 2, 15, 16, 17, 18, 19, 2, 20, 2, 21, 22, 23, 22, 24, 2, 25, 23, 26, 2, 27, 2, 28, 29, 30, 2, 31, 32, 33, 34, 35, 2, 36, 17, 37, 38, 39, 2, 40, 2, 41, 42, 43, 34, 44, 2, 45, 39, 46, 2, 47, 2, 48, 49, 50, 34, 51, 2, 52, 53, 54, 2, 55, 25, 56, 57, 58, 2, 59, 38, 60

Offset: 1

Antti Karttunen, Apr 27 2022

nonn

Restricted growth sequence transform of the ordered pair [A003557(n), A349905(n)], or equally, of the ordered pair [A003415(A003961(n)), A003557(A003961(n))].
This is a prime-shifted variant of A344025, as this is the restricted growth sequence transform of A344025(A003961(n)).
For all i, j:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A349905(i) = A349905(j) => A008836(i) = A008836(j),
  a(i) = a(j) => A353571(i) = A353571(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A003415, A003557, A003961, A008836, A305800, A344025, A349905, A353571.

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; };
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
Aux353523(n) = { my(s=A003961(n)); [A003415(s), A003557(s)]; };
v353523 = rgs_transform(vector(up_to, n, Aux353523(n)));
A353523(n) = v353523[n];

cp's OEIS Frontend

A340365 a(n) = A005940(n) / gcd(A005940(n), A324106(n)), where A324106(n) is multiplicative with a(p^e) = A005940(p^e).

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A355831 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A354347(i) = A354347(j) for all i, j >= 1.

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A355834 Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(j) and A355931(i) = A355931(j) for all i, j >= 1.

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A326192 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => A009195(i) = A009195(j) and f(i) = f(j), where f(n) = gcd(n,sigma(n)) * (-1)^[gcd(n,sigma(n))==n] and A009195(n) = gcd(n, phi(n)).

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A329353 Lexicographically earliest infinite sequence such that a(i) = a(j) => A329352(i) = A329352(j) for all i, j.

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A335424 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(A335423(i)) = A046523(A335423(j)) for all i, j >= 1.

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A335425 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000188(i) = A000188(j) and A335424(i) = A335424(j) for all i, j >= 1.

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A336311 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A336120(i)) = A278222(A336120(j)) and A278222(A336125(i)) = A278222(A336125(j)) for all i, j >= 1.

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PARI

A340362 a(n) = A005940(n) - A324106(n), where A324106(n) is multiplicative with a(p^e) = A005940(p^e).

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