A324400 -id:A324400 - cp's OEIS frontend

A331747 Lexicographically earliest infinite sequence such that a(i) = a(j) => A009194(i) = A009194(j) and A278222(i) = A278222(j) for all i, j.

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

Offset: 1

Antti Karttunen, Feb 04 2020

nonn

Restricted growth sequence transform of the ordered pair [A009194(n), A278222(n)].
For all i, j:
   A331746(i) = A331746(j) => a(i) = a(j).

Cf. A009194, A278222.

Cf. also A324400, A286622, A318310, A318311, A324389, A331744, A331745, A331746.

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));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1)));
t };
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));
Aux331747(n) = [A009194(n),A278222(n)];
v331747 = rgs_transform(vector(up_to, n, Aux331747(n)));
A331747(n) = v331747[n];

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

A336155 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(1+i) = A007814(1+j) and A335915(i) = A335915(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 11 2020

nonn

Restricted growth sequence transform of the ordered pair [A007814(1+n), A335915(n)]. Note that A007814(1+n) gives the number of trailing 1-bits in the binary expansion of n.

For all i, j: A324400(i) = A324400(j) => a(i) = a(j).

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

Cf. A000265, A007814, A335915.

Cf. also A324400, A336154, A336156, A336162.

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; };
A007814(n) = valuation(n,2);
A000265(n) = (n>>valuation(n,2));
A335915(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265(f[k,1]-1)*A000265(f[k,1]+1))^f[k,2])); };
Aux336155(n) = [A007814(1+n), A335915(n)];
v336155 = rgs_transform(vector(up_to, n, Aux336155(n)));
A336155(n) = v336155[n];

A336162 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(i) = A278222(j) and A335915(i) = A335915(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 11 2020

nonn

Restricted growth sequence transform of the ordered pair [A278222(n), A335915(n)].

For all i, j: A324400(i) = A324400(j) => a(i) = a(j).

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

Cf. A000265, A005940, A046523, A278222, A335915.

Cf. also A286622, A324400, A336155, A336159, A336160.

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));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
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));
A335915(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265(f[k,1]-1)*A000265(f[k,1]+1))^f[k,2])); };
Aux336162(n) = [A278222(n), A335915(n)];
v336162 = rgs_transform(vector(up_to, n, Aux336162(n)));
A336162(n) = v336162[n];

A336935 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007733(i) = A007733(j) and A278222(i) = A278222(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 10 2020

nonn

Restricted growth sequence transform of the ordered pair [A007733(n), A278222(n)].

For all i, j: A324400(i) = A324400(j) => A003602(i) = A003602(j) => a(i) = a(j).

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

Cf. A000265, A003602, A007733, A278222, A286622, A324400, A336933, A336934.

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; };
A007733(n) = znorder(Mod(2, n/2^valuation(n, 2))); \\ From A007733
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
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));
Aux336935(n) = [A007733(n), A278222(n)];
v336935 = rgs_transform(vector(up_to, n, Aux336935(n)));
A336935(n) = v336935[n];

A324531 Lexicographically earliest sequence such that for all i, j >= 1, a(i) = a(j) => f(i) = f(j), where f(n) = [A278222(n), A318458(n)] for all other numbers, except f(1) = 0.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Mar 05 2019

nonn

For all i, j:

a(i) = a(j) => A324532(i) = A324532(j).

Cf. A278222, A318458, A324400, A324530, A324532.

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; };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of M. F. Hasler
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A278222(n) = A046523(A005940(1+n));
A318458(n) = bitand(n,sigma(n)-n);
Aux324531(n) = if(1==n,0,[A278222(n), A318458(n)]);
v324531 = rgs_transform(vector(up_to,n,Aux324531(n)));
A324531(n) = v324531[n];

For n >= 1, a(2^n) = 2.

A331300 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = min(n, A057889(n)), and A057889 is a bijective base-2 reverse.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 19, 22, 25, 26, 23, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 34, 38, 39, 40, 41, 42, 43, 44, 32, 35, 45, 40, 39, 46, 47, 48, 36, 42, 47, 49, 43, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 57, 62, 69, 70, 71, 72, 73, 74, 65, 75, 76, 77, 78, 79, 80, 81, 55, 58, 82, 64, 69, 83, 84, 74, 63

Offset: 0

Antti Karttunen, Jan 18 2020

Restricted growth sequence transform of A331166. See comments in that sequence.

Antti Karttunen, Table of n, a(n) for n = 0..100000
Index entries for sequences related to binary expansion of n

Cf. A057889, A331166.

Cf. also A324400, A331303, A305801, A305801, A305900, A295300 for other "top level" filtering sequences.

up_to = 100000;
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; };
A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
A057889(n) = if(!n,n,A030101(n/(2^valuation(n,2))) * (2^valuation(n, 2)));
A331166(n) = min(n, A057889(n));
v331300 = rgs_transform(vector(1+up_to,n,A331166(n-1)));
A331300(n) = v331300[1+n];
for(n=0,up_to,write("b331300.txt", n, " ", A331300(n)));

A336148 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278221(i) = A278221(j) and A336158(i) = A336158(j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 2, 7, 8, 9, 5, 10, 11, 12, 2, 13, 14, 15, 8, 16, 17, 18, 5, 19, 20, 21, 11, 22, 23, 24, 2, 25, 26, 27, 14, 28, 29, 30, 8, 31, 32, 33, 17, 34, 35, 36, 5, 37, 38, 39, 20, 40, 41, 25, 11, 42, 43, 44, 23, 45, 46, 47, 2, 48, 49, 50, 26, 51, 32, 52, 14, 53, 54, 34, 29, 55, 56, 57, 8, 58, 59, 60, 32, 61, 62, 63, 17, 64, 65, 30, 35, 66, 67, 68, 5, 69, 70, 71, 38, 72, 73, 74, 20, 75

Offset: 1

Antti Karttunen, Jul 12 2020

nonn

Restricted growth sequence transform of the ordered pair [A278221(n), A336158(n)], i.e., of the ordered pair [A046523(A122111(n)), A046523(A000265(n))].

For all i, j: A324400(i) = A324400(j) => A336146(i) = A336146(j) => a(i) = a(j).

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

Cf. A000265, A046523, A064989, A122111, A278221, A336158.

Cf. also A286621, A318890, A324400, A336146, A336149, A336159.

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; };
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)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A278221(n) = A046523(A122111(n));
A000265(n) = (n>>valuation(n,2));
A336158(n) = A046523(A000265(n));
Aux336148(n) = [A278221(n),A336158(n)];
v336148 = rgs_transform(vector(up_to, n, Aux336148(n)));
A336148(n) = v336148[n];

A336150 Lexicographically earliest infinite sequence such that a(i) = a(j) => A001221(i) = A001221(j) and A020639(i) = A020639(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 11 2020

nonn

Restricted growth sequence transform of the ordered pair [A001221(n), A020639(n)]. The first member of pair gives the number of distinct prime divisors of n, and the second member gives its smallest prime factor.

For all i, j: A324400(i) = A324400(j) => a(i) = a(j).

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

Cf. A001221, A020639.

Cf. also A324400, A336151, A336152.

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; };
A020639(n) = if(1==n, n, factor(n)[1, 1]);
Aux336150(n) = [omega(n), A020639(n)];
v336150 = rgs_transform(vector(up_to, n, Aux336150(n)));
A336150(n) = v336150[n];

A336152 Lexicographically earliest infinite sequence such that a(i) = a(j) => A001221(i) = A001221(j) and A007814(1+i) = A007814(1+j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 2, 4, 5, 3, 5, 4, 5, 7, 2, 4, 5, 3, 5, 8, 5, 6, 5, 4, 5, 3, 5, 4, 9, 10, 2, 8, 5, 11, 5, 4, 5, 12, 5, 4, 9, 3, 5, 8, 5, 13, 5, 4, 5, 11, 5, 4, 5, 12, 5, 8, 5, 3, 9, 4, 5, 14, 2, 8, 9, 3, 5, 8, 9, 6, 5, 4, 5, 11, 5, 8, 9, 13, 5, 4, 5, 3, 9, 8, 5, 12, 5, 4, 9, 11, 5, 8, 5, 15, 5, 4, 5, 11, 5, 4, 9, 6, 5, 16

Offset: 1

Antti Karttunen, Jul 11 2020

nonn

Restricted growth sequence transform of the ordered pair [A001221(n), A007814(1+n)]. The first member of pair gives the number of distinct prime divisors of n, and the second member gives the number of trailing 1-bits in its binary expansion.

For all i, j: A324400(i) = A324400(j) => a(i) = a(j).

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

Cf. A001221, A007814.

Cf. also A324400, A336150, A336151, A336156.

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; };
A007814(n) = valuation(n,2);
Aux336152(n) = [omega(n), A007814(1+n)];
v336152 = rgs_transform(vector(up_to, n, Aux336152(n)));
A336152(n) = v336152[n];

A336154 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(1+i) = A007814(1+j) and A278222(i) = A278222(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 11 2020

nonn

Restricted growth sequence transform of the ordered pair [A007814(1+n), A278222(n)]. Note that A007814(1+n) gives the number of trailing 1-bits in the binary expansion of n.

For all i, j: A324400(i) = A324400(j) => a(i) = a(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A007814, A278222.

Cf. also A286622, A324400, A336153, A336155, A336156.

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; };
A007814(n) = valuation(n,2);
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
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));
Aux336154(n) = [A007814(1+n), A278222(n)];
v336154 = rgs_transform(vector(up_to, n, Aux336154(n)));
A336154(n) = v336154[n];

cp's OEIS Frontend

A331747 Lexicographically earliest infinite sequence such that a(i) = a(j) => A009194(i) = A009194(j) and A278222(i) = A278222(j) for all i, j.

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A336155 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(1+i) = A007814(1+j) and A335915(i) = A335915(j), for all i, j >= 1.

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A336162 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(i) = A278222(j) and A335915(i) = A335915(j), for all i, j >= 1.

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A336935 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007733(i) = A007733(j) and A278222(i) = A278222(j), for all i, j >= 1.

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A324531 Lexicographically earliest sequence such that for all i, j >= 1, a(i) = a(j) => f(i) = f(j), where f(n) = [A278222(n), A318458(n)] for all other numbers, except f(1) = 0.

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A331300 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = min(n, A057889(n)), and A057889 is a bijective base-2 reverse.

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PARI

A336148 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278221(i) = A278221(j) and A336158(i) = A336158(j), for all i, j >= 1.

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PARI

A336150 Lexicographically earliest infinite sequence such that a(i) = a(j) => A001221(i) = A001221(j) and A020639(i) = A020639(j), for all i, j >= 1.

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PARI

A336152 Lexicographically earliest infinite sequence such that a(i) = a(j) => A001221(i) = A001221(j) and A007814(1+i) = A007814(1+j), for all i, j >= 1.

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