A324380 -id:A324380 - 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

A324386 a(n) = A324383(A006068(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 4, 4, 4, 2, 2, 6, 6, 1, 2, 4, 8, 4, 4, 6, 12, 2, 8, 6, 10, 6, 22, 10, 8, 1, 4, 4, 6, 2, 8, 6, 8, 4, 6, 12, 14, 2, 16, 10, 16, 2, 8, 16, 4, 6, 14, 8, 24, 6, 30, 18, 20, 6, 26, 18, 26, 1, 6, 8, 8, 4, 12, 12, 6, 8, 12, 14, 18, 4, 20, 20, 20, 4, 16, 16, 8, 12, 28, 16, 10, 12, 22, 26, 14, 12, 34, 20, 22, 2, 12

Offset: 0

Antti Karttunen, Feb 27 2019

nonn

This is most likely equal to A276150(A086141(n)), apart from the different offset used in A086141.

The same comments about the parity of terms as in A324383 and A324387 apply also here, except here 1's occur at positions given by 2^k - 1.

Cf. A002110, A006068, A025487, A086141, A276150, A322827, A324342, A324382.

Cf. also A324383, A324387 (permutations of this sequence) and A324380, A324390.

A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ From A006068
A276150(n) = { my(s=0,m); forprime(p=2, , if(!n, return(s)); m = n%p; s += m; n = (n-m)/p); };
A322827(n) = if(!n,1,my(bits = Vecrev(binary(n)), rl=1, o = List([])); for(i=2,#bits,if(bits[i]==bits[i-1], rl++, listput(o,rl))); listput(o,rl); my(es=Vecrev(Vec(o)), m=1); for(i=1,#es,m *= prime(i)^es[i]); (m));
A324383(n) = A276150(A322827(n));
A324386(n) = A324383(A006068(n));

a(n) = A324383(A006068(n)) = A276150(A322827(A006068(n))).

a(A000225(n)) = 1 for all n.

A324390 Lexicographically earliest positive sequence such that a(i) = a(j) => A278219(i) = A278219(j) and A324386(i) = A324386(j), for all i, j >= 0.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Feb 27 2019

nonn

Restricted growth sequence transform of the ordered pair [A278219(n), A324386(n)].

Cf. A278219, A324386.

Cf. also A286619, A324343, A324344, A324380 (compare the scatter-plots).

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 }; \\ 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));
A003188(n) = bitxor(n, n>>1);
A278219(n) = A278222(A003188(n));
Aux324390(n) = [A278219(n), A324386(n)]; \\ See code for A324386 in that entry.
v324390 = rgs_transform(vector(1+up_to,n,Aux324390(n-1)));
A324390(n) = v324390[1+n];

a(A000225(n)) = 2 for all n >= 1.

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.

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A324386 a(n) = A324383(A006068(n)).

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A324390 Lexicographically earliest positive sequence such that a(i) = a(j) => A278219(i) = A278219(j) and A324386(i) = A324386(j), for all i, j >= 0.

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