A351037 -id:A351037 - cp's OEIS frontend

A351461 Lexicographically earliest infinite sequence such that a(i) = a(j) => A206787(i) = A206787(j) and A336651(i) = A336651(j) for all i, j >= 1.

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

Offset: 1

Antti Karttunen, Feb 11 2022

nonn

Restricted growth sequence transform of the ordered pair [A206787(n), A336651(n)], or equally, of sequence b(n) = A291750(A000265(n)).
For all i, j >= 1:
  A003602(i) = A003602(j) => A351040(i) = A351040(j) => a(i) = a(j),
  A324400(i) = A324400(j) => A351460(i) = A351460(j) => a(i) = a(j),
  a(i) = a(j) => A000593(i) = A000593(j),
  a(i) = a(j) => A347385(i) = A347385(j),
  a(i) = a(j) => A351037(i) = A351037(j) => A347240(i) = A347240(j).
From Antti Karttunen, Nov 23 2023: (Start)
Conjectured to be equal to the lexicographically earliest infinite sequence such that b(i) = b(j) => A000593(i) = A000593(j) and A336467(i) = A336467(j) for all i, j >= 1. In any case, a(i) = a(j) => b(i) = b(j) for all i, j >= 1 [because both A000593(n) and A336467(n) can be computed from the values of A206787(n) and A336651(n)], but whether the implication holds to the opposite direction is still open. Empirically this has been checked up to n = 2^22. See also comment in A351040.
(End)

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

Cf. A206787, A336651.
Cf. also A000593, A003602, A291750, A291751, A324400, A336467, A347240, A347385, A351040, A351460.
Differs from A351037 for the first time at n=103, where a(103) = 42 while A351037(103) = 27.

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; };
A206787(n) = sumdiv(n, d, d*(d % 2)*issquarefree(d)); \\ From A206787
A336651(n) = { my(f=factor(n)); prod(i=1, #f~, if(2==f[i,1],1,f[i,1]^(f[i,2]-1))); };
Aux351461(n) = [A206787(n), A336651(n)];
v351461 = rgs_transform(vector(up_to, n, Aux351461(n)));
A351461(n) = v351461[n];

A351090 Lexicographically earliest infinite sequence such that a(i) = a(j) => A351091(i) = A351091(j) and A351092(i) = A351092(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, 18, 5, 19, 10, 20, 3, 21, 11, 22, 6, 23, 12, 24, 2, 25, 13, 26, 7, 27, 14, 28, 4, 29, 15, 30, 8, 31, 16, 32, 1, 33, 17, 34, 9, 35, 18, 36, 5, 37, 19, 38, 10, 39, 20, 40, 3, 41, 21, 42, 11, 43, 22, 44, 6, 45, 23

Offset: 1

Antti Karttunen, Jan 31 2022

Restricted growth sequence transform of the ordered pair [A351091(n), A351092(n)], or equally, of the ordered pair [A351093(n), A351094(n)].

For all i, j: A003602(i) = A003602(j) => a(i) = a(j) => A000593(i) = A000593(j).

			Consider two odd semiprimes, 689 and 697. The divisors of 689 are 1, 13, 53, 689, and the divisors of 697 are 1, 17, 41, 697. Applying A019565(A289813(x)) to the former gives [2, 30, 7, 105], while with the latter it gives [2, 5, 105, 42], and the product of both sequences is 44100. Applying A019565(A289814(x)) to the former gives [1, 1, 30, 286], while with the latter it gives [1, 6, 2, 715]. Product of both sequences is 8580. Therefore, because A351091(689) = A351091(697) and A351092(689) = A351092(697), also a(689) = a(697).

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

Cf. A000593, A351037, A351091, A351092, A351093, A351094.
Differs from A003602 for the first time at n=697, where a(697) = 345 while A003602(697) = 349.
Cf. also A293226, A351030.

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(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A289813(n) = { my(d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); }; \\ From A289813
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); }; \\ From A289814
A351091(n) = { my(m=1); fordiv(n>>valuation(n,2),d,m *= A019565(A289813(d))); (m); };
A351092(n) = { my(m=1); fordiv(n>>valuation(n,2),d,m *= A019565(A289814(d))); (m); };
Aux351090(n) = [A351091(n),A351092(n)];
v351090 = rgs_transform(vector(up_to, n, Aux351090(n)));
A351090(n) = v351090[n];

A351040 Lexicographically earliest infinite sequence such that a(i) = a(j) => A336158(i) = A336158(j), A206787(i) = A206787(j) and A336651(i) = A336651(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, 17, 5, 18, 10, 19, 3, 20, 11, 21, 6, 22, 12, 23, 2, 24, 13, 25, 7, 26, 14, 25, 4, 27, 15, 28, 8, 29, 16, 30, 1, 31, 17, 32, 9, 33, 17, 34, 5, 35, 18, 36, 10, 33, 19, 37, 3, 38, 20, 39, 11, 40, 21, 41, 6, 42

Offset: 1

Antti Karttunen, Jan 31 2022

Restricted growth sequence transform of the ordered triplet [A336158(n), A206787(n), A336651(n)].
For all i, j >= 1:
  A003602(i) = A003602(j) => a(i) = a(j),
  a(i) = a(j) => A336390(i) = A336390(j) => A336391(i) = A336391(j),
  a(i) = a(j) => A347374(i) = A347374(j),
  a(i) = a(j) => A351036(i) = A351036(j) => A113415(i) = A113415(j),
  a(i) = a(j) => A351461(i) = A351461(j).
From Antti Karttunen, Nov 23 2023: (Start)
Conjectured to be equal to the lexicographically earliest infinite sequence such that b(i) = b(j) => A000593(i) = A000593(j), A336158(i) = A336158(j) and A336467(i) = A336467(j), for all i, j >= 1 (this was the original definition). In any case it holds that a(i) = a(j) => b(i) = b(j) for all i, j >= 1. See comment in A351461.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..65539

Cf. A000265, A000593, A336158, A336467.
Cf. also A003602, A336390, A336391, A351037, A366891.
Differs from A347374 for the first time at n=103, where a(103) = 48, while A347374(103) = 30.
Differs from A351035 for the first time at n=175, where a(175) = 80, while A351035(175) = 78.
Differs from A351036 for the first time at n=637, where a(637) = 272, while A351036(637) = 261.

up_to = 65539;
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));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
A336158(n) = A046523(A000265(n));
A206787(n) = sumdiv(n, d, d*(d % 2)*issquarefree(d));
A336651(n) = { my(f=factor(n)); prod(i=1, #f~, if(2==f[i,1],1,f[i,1]^(f[i,2]-1))); };
Aux351040(n) = [A336158(n), A206787(n), A336651(n)];
v351040 = rgs_transform(vector(up_to, n, Aux351040(n)));
A351040(n) = v351040[n];

Original definition moved to the comment section and replaced with a definition that is at least as encompassing, and conjectured to be equal to the original one. - Antti Karttunen, Nov 23 2023

A351036 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000593(i) = A000593(j) and A336158(i) = A336158(j), for all i, j >= 1.

Original entry on oeis.org

Offset: 1

Antti Karttunen, Jan 30 2022

Restricted growth sequence transform of the ordered pair [A000593(n), A336158(n)], where A000593(n) is the sum of odd divisors of n, and A336158(n) is the least representative of the prime signature of the odd part of n.
For all i, j:
  A003602(i) = A003602(j) => A351040(i) = A351040(j) => a(i) = a(j),
  a(i) = a(j) => A113415(i) = A113415(j).

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

Cf. A000265, A000593, A003602, A046523, A113415, A336158.
Cf. also A351037.
Differs from A347374 for the first time at n=103, where a(103) = 48, while A347374(103) = 30.
Differs from A351035 for the first time at n=175, where a(175) = 80, while A351035(175) = 78.
Differs from A351040 for the first time at n=637, where a(637) = 261, while A351040(637) = 272.

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));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A336158(n) = A046523(A000265(n));
A000593(n) = sigma(A000265(n));
Aux351036(n) = [A000593(n), A336158(n)];
v351036 = rgs_transform(vector(up_to, n, Aux351036(n)));
A351036(n) = v351036[n];

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A351461 Lexicographically earliest infinite sequence such that a(i) = a(j) => A206787(i) = A206787(j) and A336651(i) = A336651(j) for all i, j >= 1.

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A351090 Lexicographically earliest infinite sequence such that a(i) = a(j) => A351091(i) = A351091(j) and A351092(i) = A351092(j), for all i, j >= 1.

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A351040 Lexicographically earliest infinite sequence such that a(i) = a(j) => A336158(i) = A336158(j), A206787(i) = A206787(j) and A336651(i) = A336651(j) for all i, j >= 1.

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A351036 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000593(i) = A000593(j) and A336158(i) = A336158(j), for all i, j >= 1.

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