A323892 -id:A323892 - cp's OEIS frontend

A318310 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000120(i) = A000120(j) and A033879(i) = A033879(j), for all i, j >= 0.

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

Offset: 1

Antti Karttunen, Aug 25 2018

Restricted growth sequence transform of ordered pair [A000120(n), A033879(n)], or equally, of ordered pair [A000120(n), A294898(n)].
For all i, j:
  A318311(i) = A318311(j) => a(i) = a(j),
  a(i) = a(j) => A286449(i) = A286449(j),
  a(i) = a(j) => A294898(i) = A294898(j).
In the scatter plot one can see the effects of both base-2 related A000120 (binary weight of n) and prime factorization related A033879 (deficiency of n) graphically mixed: from the former, a square grid pattern, and from the latter the black rays that emanate from the origin. The same is true for A323898, while in the ordinal transform of this sequence, A331184, such effects are harder to visually discern. - Antti Karttunen, Jan 13 2020

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

Cf. A000120, A033879, A286449, A295881, A295882, A294898.
Cf. A318311, A323889, A323892, A323898, A324344, A324380, A324390 for similar constructions.
Cf. A331184 (ordinal transform).

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; };
A318310aux(n) = [hammingweight(n), (2*n) - sigma(n)];
v318310 = rgs_transform(vector(up_to,n,A318310aux(n)));
A318310(n) = v318310[n];

Name changed by Antti Karttunen, Jan 13 2020

A323889 Lexicographically earliest positive sequence such that a(i) = a(j) => A002487(i) = A002487(j) and A278222(i) = A278222(j), for all i, j >= 0.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Feb 09 2019

Restricted growth sequence transform of the ordered pair [A002487(n), A278222(n)].

Cf. A002487, A278222, A286622, A324400.

Cf. also A103391, A278243, A286378, A318311, A323892, A323897 and A324533 for a "deformed variant".

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; };
A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
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));
Aux323889(n) = [A002487(n), A278222(n)];
v323889 = rgs_transform(vector(1+up_to,n,Aux323889(n-1)));
A323889(n) = v323889[1+n];

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

A323898 Lexicographically earliest sequence such that a(i) = a(j) => A000120(i) = A000120(j) and A083254(i) = A083254(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 09 2019

nonn

Restricted growth sequence transform of the ordered pair [A000120(n), A083254(n)].

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

Cf. A000120, A083254.

Cf. also A318310, A323892, A323898.

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; };
A083254(n) = (2*eulerphi(n)-n);
A323898aux(n) = [hammingweight(n), A083254(n)];
v323898 = rgs_transform(vector(up_to,n,A323898aux(n)));
A323898(n) = v323898[n];

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

A323897 Lexicographically earliest sequence such that a(i) = a(j) => A002487(i) = A002487(j) and A083254(i) = A083254(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 09 2019

nonn

Restricted growth sequence transform of the ordered pair [A002487(n), A083254(n)].

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to Stern's sequences

Cf. A000010, A002487, A083254.

Cf. also A323892, A323898.

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; };
A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A083254(n) = (2*eulerphi(n)-n);
A323897aux(n) = [A002487(n), A083254(n)];
v323897 = rgs_transform(vector(up_to,n,A323897aux(n)));
A323897(n) = v323897[n];

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

A323904 Lexicographically earliest sequence such that a(i) = a(j) => A033879(i) = A033879(j) and A083254(i) = A083254(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 10 2019

nonn

Restricted growth sequence transform of the ordered pair [A033879(n), A083254(n)], the deficiency of n, and its Möbius transform.

For all i, j: a(i) = a(j) => A297159(i) = A297159(j).

			For n=39, we have A033879(39) = 2*39 - A000203(39) = 22, and A083254(39) = 2*A000010(39)-39 = 9. For n=63 the results are same, with A033879(63) = 22 and A083254(63) = 9, thus a(39) and a(63) are allotted the same number by the restricted growth sequence transform, which in this case is 35, thus a(39) = a(63) = 35.
For n=42 and 54, we have A033879(42) = -12, A083254(42) = -18 and A033879(54) = -12, A083254(54) = -18, thus a(42) = a(54) (= 38).

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

Cf. A000010, A000203, A033879, A083254.

Cf. also A318310, A323892, A323897, A323898.

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; };
A033879(n) = (2*n-sigma(n));
A083254(n) = (2*eulerphi(n)-n);
A323904aux(n) = [A033879(n), A083254(n)];
v323904 = rgs_transform(vector(up_to,n,A323904aux(n)));
A323904(n) = v323904[n];

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

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A318310 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000120(i) = A000120(j) and A033879(i) = A033879(j), for all i, j >= 0.

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A323889 Lexicographically earliest positive sequence such that a(i) = a(j) => A002487(i) = A002487(j) and A278222(i) = A278222(j), for all i, j >= 0.

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A323898 Lexicographically earliest sequence such that a(i) = a(j) => A000120(i) = A000120(j) and A083254(i) = A083254(j), for all i, j >= 1.

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A323897 Lexicographically earliest sequence such that a(i) = a(j) => A002487(i) = A002487(j) and A083254(i) = A083254(j), for all i, j >= 1.

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A323904 Lexicographically earliest sequence such that a(i) = a(j) => A033879(i) = A033879(j) and A083254(i) = A083254(j), for all i, j >= 1.

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