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

1,2

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.

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)

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; };
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)<<valuation(n, 2)); \\ From 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

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.

Sequence in context: A319826 A278108 A302786 * A363247 A087165 A238326

Adjacent sequences: A322023 A322024 A322025 * A322027 A322028 A322029

Keyword

nonn

Author

Antti Karttunen, Dec 03 2018

Status

approved