A323897 -id:A323897 - cp's OEIS frontend

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.

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.

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.

cp's OEIS Frontend

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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