cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A322317 Ordinal transform of A322316.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Dec 04 2018

Keywords

Links

Crossrefs

Cf. A007814, A007949, A122841, A244417, A322316.
Cf. also A126760.

Programs

  • PARI
    up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
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);
A122841(n) = min(A007814(n), A007949(n));
A244417(n) = max(valuation(n,2), valuation(n,3));
v322316 = rgs_transform(vector(up_to, n, [A122841(n), A244417(n)]));
v322317 = ordinal_transform(v322316);
A322317(n) = v322317[n];

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.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Dec 03 2018

Keywords

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

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)

A244417 Exponents of 6 in appearing in the 6-adic value of 1/n, n>=1 (A244416).

Original entry on oeis.org

0, 1, 1, 2, 0, 1, 0, 3, 2, 1, 0, 2, 0, 1, 1, 4, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 3, 2, 0, 1, 0, 5, 1, 1, 0, 2, 0, 1, 1, 3, 0, 1, 0, 2, 2, 1, 0, 4, 0, 1, 1, 2, 0, 3, 0, 3, 1, 1, 0, 2, 0, 1, 2, 6, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 1, 2, 0, 1, 0, 4, 4, 1, 0, 2, 0, 1, 1, 3, 0, 2, 0, 2, 1, 1, 0, 5, 0, 1, 2, 2
Offset: 1

Views

Author

Wolfdieter Lang, Jul 02 2014

Keywords

Comments

For the definition of 'g-dic value of 1/n' see a comment on A244416. In the Mahler reference, p. 7, the present exponent of 6 is there called f = f(1/n) for g = 6.

Examples

			See A244416.

References

  • Kurt Mahler, p-adic numbers and their functions, second ed., Cambridge University Press, 1981.

Links

Crossrefs

Cf. A122841, A244416, A007814 (g=2), A007949 (g=3), A244415 (g=4), A112765 (g=5), A051903, A065331.
Cf. also A322026, A322316.

Programs

  • Mathematica
    a[n_] := Max[IntegerExponent[n, {2, 3}]]; Array[a, 100] (* Amiram Eldar, Aug 19 2024 *)
  • PARI
    A244417(n) = max(valuation(n,2), valuation(n,3)); \\ Antti Karttunen, Dec 04 2018

Formula

a(n) = 0 if n is congruent 1 or 5 (mod 6). a(n) = max(A007814(n), A007949(n)) if n == 0 (mod 6). a(n) = A007814(n) if n == 2 or 4 (mod 6) and a(n) = A007949(n) if n == 3 (mod 6).
a(n) = max(A007814(n), A007949(n)), in all cases. - Antti Karttunen, Dec 04 2018
From Amiram Eldar, Aug 19 2024: (Start)
a(n) = A051903(A065331(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 13/10. (End)
Showing 1-3 of 3 results.

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