A324390 -id:A324390

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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.

+10
15

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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

LINKS

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

PROG

(PARI)

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];

CROSSREFS

Cf. A000120, A033879, A286449, A295881, A295882, A294898.

Cf. A318311, A323889, A323892, A323898, A324344, A324380, A324390 for similar constructions.

Cf. A331184 (ordinal transform).

KEYWORD

nonn,look

AUTHOR

Antti Karttunen, Aug 25 2018

EXTENSIONS

Name changed by Antti Karttunen, Jan 13 2020

STATUS

approved

A324386

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

+10
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

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.

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..16383

Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537

Index entries for sequences related to binary expansion of n

Index entries for sequences related to primorial base

Index entries for sequences related to primorial numbers

FORMULA

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

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

PROG

(PARI)

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));

CROSSREFS

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

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

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 27 2019

STATUS

approved

A324389

Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A009194(n), A318458(n)] for all other numbers, except f(1) = -1.

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

For all i, j:

A324401(i) = A324401(j) => a(i) = a(j).

Regarding the scatter plot of this sequence, see also comments in A318310. - Antti Karttunen, Feb 04 2020

LINKS

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

Index entries for sequences related to binary expansion of n

Index entries for sequences related to sigma(n)

PROG

(PARI)

up_to = 65537;

A009194(n) = gcd(n, sigma(n));

A318458(n) = bitand(n, sigma(n)-n);

Aux324389(n) = if(1==n, -1, [A009194(n), A318458(n)]);

v324389 = rgs_transform(vector(up_to, n, Aux324389(n)));

A324389(n) = v324389[n];

CROSSREFS

Cf. A009194, A318310, A318458, A324390, A324401, A324530, A331744, A331746, A331747.

KEYWORD

nonn,look

AUTHOR

Antti Karttunen, Mar 05 2019

STATUS

approved

A324380

Lexicographically earliest positive sequence such that a(i) = a(j) => A069010(i) = A069010(j) and A324386(i) = A324386(j), for all i, j >= 0.

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Restricted growth sequence transform of the ordered pair [A069010(n), A324386(n)].

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..65537

Index entries for sequences related to binary expansion of n

Index entries for sequences related to primorial base

FORMULA

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

PROG