A322026 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(i) = A007814(j) and A007949(i) = A007949(j), for all i, j, where A007814 and A007949 give the 2- and 3-adic valuations of n.

1, 2, 3, 4, 1, 5, 1, 6, 7, 2, 1, 8, 1, 2, 3, 9, 1, 10, 1, 4, 3, 2, 1, 11, 1, 2, 12, 4, 1, 5, 1, 13, 3, 2, 1, 14, 1, 2, 3, 6, 1, 5, 1, 4, 7, 2, 1, 15, 1, 2, 3, 4, 1, 16, 1, 6, 3, 2, 1, 8, 1, 2, 7, 17, 1, 5, 1, 4, 3, 2, 1, 18, 1, 2, 3, 4, 1, 5, 1, 9, 19, 2, 1, 8, 1, 2, 3, 6, 1, 10, 1, 4, 3, 2, 1, 20, 1, 2, 7, 4, 1, 5, 1, 6, 3

Offset: 1

Author

Antti Karttunen, Dec 03 2018

Keywords

• nonn

Comments

Restricted growth sequence transform of the ordered pair [A007814(n), A007949(n)].

For all i, j:

A305900(i) = A305900(j) => a(i) = a(j),

a(i) = a(j) => A122841(i) = A122841(j),

a(i) = a(j) => A244417(i) = A244417(j),

a(i) = a(j) => A322316(i) = A322316(j) => A072078(i) = A072078(j).

If and only if a(k) > a(i) for all k > i then k is in A003586, - David A. Corneth, Dec 03 2018

That is, A003586 gives the positions of records (1, 2, 3, 4, 5, ...) in this sequence.

Sequence A126760 (without its initial zero) and this sequence are ordinal transforms of each other.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Jon Maiga, Computer-generated formulas for A322026, Sequence Machine.

Crossrefs

Cf. A003586 (positions of records, the first occurrence of n), A007814, A007949, A065331, A071521, A072078, A087465, A122841, A126760 (ordinal transform), A322316, A323883, A323884.

Cf. also A247714 and A255975.

Programs

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; };
A007814(n) = valuation(n,2);
A007949(n) = valuation(n,3);
v322026 = rgs_transform(vector(up_to, n, [A007814(n), A007949(n)]));
A322026(n) = v322026[n];

PARI

A065331(n) = (3^valuation(n, 3)<A065331
A071521(n) = { my(t=1/3); sum(k=0, logint(n, 3), t*=3; logint(n\t, 2)+1); }; \\ From A071521.
A322026(n) = A071521(A065331(n)); \\ Antti Karttunen, Sep 08 2024

Formula

For s = A003586(n), a(s) = n = a((6k+1)*s) = a((6k-1)*s), where s is the n-th 3-smooth number and k > 0. - David A. Corneth, Dec 03 2018

A065331(n) = A003586(a(n)). - David A. Corneth, Dec 04 2018

From Antti Karttunen, Sep 08 2024: (Start)

a(n) = Sum{k=1..n} [A126760(k)==A126760(n)], where [ ] is the Iverson bracket.

a(n) = A071521(A065331(n)). [Found by Sequence Machine and also by LODA miner]

a(n) = A323884(25*n). [Conjectured by Sequence Machine]

(End)