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 1-6 of 6 results.

A179480 Let m>k>0 be odd numbers and denote by the symbol "m<->k" the value A000265(m-k). Then the sequence m<->k, m<->(m<->k), m<->(m<->(m<->k)), ... is periodic; a(n) is the smallest period in the case m=2*n-1, k=1.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 2, 1, 5, 2, 6, 5, 5, 7, 2, 1, 6, 9, 6, 3, 3, 6, 12, 10, 4, 13, 10, 3, 15, 15, 2, 1, 17, 10, 18, 2, 10, 14, 20, 13, 21, 2, 14, 4, 6, 4, 18, 11, 9, 25, 26, 4, 27, 9, 18, 5, 22, 4, 12, 27, 10, 25, 2, 1, 33, 6, 18, 15, 35, 22, 30, 3, 22, 37, 6, 12, 10, 13, 26
Offset: 2

Views

Author

Vladimir Shevelev, Jul 16 2010

Keywords

Comments

A dual sequence to A179382.
Let b = (2*n-1) and k = A003558(n-1). If a(n) is odd, b divides (2^k + 1); but if a(n) is even, b divides (2^k - 1). Examples: a(14) = 5, odd; with b = 27 and A003558(13) = 9. Then 27 divides (2^9 + 1) or 513 = 27 * 19. a(18) = 6, even. b = 35, with k= A003558(17) = 12. Then 35 divides (2^12 - 1). - Gary W. Adamson, Aug 20 2012.
Iff a(n) = n/2 or (n-1)/2, then 2*n - 1 is a prime with one coach and is in A216371. Examples: a(19) = 9, so 37 is in A216371. a(12) = 6, so 23 is in A216371. - _Gary W. Adamson, Sep 08 2012.

Examples

			If n=14, then m=27 and we have 27<->1=13, 27<->13=7, 27<->7=5, 27<->5=11, 27<->11=1. Thus a(14)=5.

Crossrefs

Cf. A179382, A179383, A000265.
Cf. A003558.
Cf. A216371.

Programs

  • Maple
    Contribution from R. J. Mathar, Nov 04 2010: (Start)
A179480aux := proc(x,y) local xtrack,xitr,xpos ; xtrack := [y] ; while true do xitr := A000265(x-op(-1,xtrack)) ; if not member(xitr, xtrack,'xpos') then xtrack := [op(xtrack),xitr] ; else return 1+nops(xtrack)-xpos ; end if; end do: end proc:
A179480 := proc(n) A179480aux(2*n-1,1) ; end proc: seq(A179480(n),n=2..80) ; (End)
  • Mathematica
    oddres[n_] := n/2^IntegerExponent[n, 2];
b[x_, y_] := Module[{xtrack = {y}, xitr}, While[True, xitr = oddres[x - Last@ xtrack]; If[FreeQ[xtrack, xitr], AppendTo[xtrack, xitr], Return[ Length[xtrack]]]]];
a[n_] := b[2n-1, 1];
a /@ Range[2, 80] (* Jean-François Alcover, Apr 13 2020, after R. J. Mathar *)

Extensions

Edited by N. J. A. Sloane, Jul 18 2010
More terms from R. J. Mathar, Nov 04 2010

A179481 a(n) = 2*t(n)-1 where t(n) is the sequence of records positions of A179480.

Original entry on oeis.org

3, 7, 11, 19, 23, 29, 37, 47, 53, 59, 67, 71, 79, 83, 101, 103, 107, 131, 139, 149, 163, 167, 173, 179, 191, 197, 199, 211, 227, 239, 263, 269, 271, 293, 311, 317, 347, 359, 367, 373, 379, 383, 389, 419, 443, 461, 463, 467, 479, 487, 491, 503, 509, 523, 541
Offset: 2

Views

Author

Vladimir Shevelev, Jul 16 2010

Keywords

Comments

Question. Is every term of this sequence prime?
From Gary W. Adamson, Sep 04 2012: (Start)
In answer to the primality question and pursuant to the Coach Theorem of Hilton and Pedersen: phi(b) = 2 * k * c, with b an odd integer and k in A003558, and c (the numbers of coaches) in A135303; iff phi(b) = (b-1) then b = p, prime. This implies that if b has one coach and k = (b-1)/2, b must be prime since phi(b) = 2 * k * c = 2 * (b-1)/2 * 1 = (b-1). Conjectures: all terms in A179481 have one coach with k = (b-1)/2 and are therefore primes. Next, if A179480(n) is a new record high value, then so is A003558(n-1); but not necessarily the converse (e.g. 13), and the corresponding value of k for b is (b-1)/2. Examples: b = 13 has one coach with k (sum of bottom row terms ) = 6 = A003558(6); and r (number of entries in each row) = 3:
13: [1, 3, 5]
......2, 1, 3. This example satisfies the primality requirements since phi(13) = 12 = 2 * k * c = 2 * 6 * 1; but not the new record requirement for r = 3 since A179480(6) = 3, corresponding to 11, not 13. As shown in the coach for 11:
11: [1, 3, 3]
......1, 1, 3; k = (b-1)/2 with r = 3 and c = 1. Therefore, 11 is in A179481 but not 13. (End)

References

  • P. Hilton and J. Pedersen, A Mathematics Tapestry, Demonstrating the Beautiful Unity of Mathematics, 2010, Cambridge University Press, pp. 260-264.

Crossrefs

Cf. A179480, A179460, A179382, A179383.
Cf. A003558, A000040.

Extensions

Edited by N. J. A. Sloane, Jul 18 2010
More terms from R. J. Mathar, Jul 18 2010

A179687 a(n) = 2*q(n)-1 where q(n) is the sequence of records positions of A179686.

Original entry on oeis.org

3, 5, 7, 19, 27, 37, 47, 53, 67, 95, 101, 103, 107, 131, 149, 173, 181, 191, 227, 239, 263, 293, 311, 317, 349, 359, 367, 373, 383, 419, 461, 463, 479, 509, 547, 557, 587, 613, 619, 647, 653, 677, 701, 743, 751, 773, 787, 821, 841, 863, 887, 947
Offset: 1

Views

Author

Vladimir Shevelev, Jul 24 2010

Keywords

Comments

The record positions are q(n) = 2, 3, 4, 10, 14, 19, 24, 27, 34,...
Sequences A179382, A179383, A179480, A179481, A179686 and this sequence show that their terms depend on prime power factorization of 2*n-1. Nevertheless, this question yet is waiting its research. Most likely that almost all terms of this sequence are primes (27, 95, 841,... are not).

Crossrefs

Cf. A179382, A179383, A179480, A179481, A179686

Extensions

Extended beyond a(7) by R. J. Mathar, Dec 04 2011

A179538 Numbers 2n+1 for which A003558(n), n>=1, are record values of A003558.

Original entry on oeis.org

3, 5, 7, 11, 13, 19, 23, 29, 37, 47, 53, 59, 61, 67, 71, 79, 83, 101, 103, 107, 121, 131, 139, 149, 163, 167, 173, 179, 181, 191, 197, 199, 211, 227, 239, 263, 269, 271, 293, 311, 317, 347, 349, 359, 367, 373, 379, 383, 389, 419, 421, 443, 461, 463, 467, 479
Offset: 1

Views

Author

Vladimir Shevelev, Jul 18 2010

Keywords

Comments

Conjecture. All terms are primes except for a finite set of squares of primes.
All terms from a(1) to a(5000) are primes except for a(21) = 121 = 11^2, supporting V. Shevelev's conjecture. [John W. Layman, Jul 22 2010]

Links

Crossrefs

Cf. A003558, A139099, A179383, A179481.

Programs

  • Mathematica
    s = {}; am = 0; Do[a = Min[MultiplicativeOrder[2, n, {-1, 1}]]; If[a > am, am = a; AppendTo[s, n]], {n, 3, 480, 2}]; s (* Amiram Eldar, Sep 13 2019 *)

Extensions

a(31)-a(56) from John W. Layman, Jul 22 2010

A179739 a(n) = 2*h(n)-1 where h(n) is the sequence of records positions of A179738.

Original entry on oeis.org

3, 5, 7, 11, 13, 31, 37
Offset: 2

Views

Author

Vladimir Shevelev, Jul 25 2010

Keywords

Crossrefs

Cf. A179738 A179382, A179383, A179480, A179481, A179686 A179687

A179788 2*r(n)-1 where r(n) is the sequence of records positions of A179787.

Original entry on oeis.org

1, 7, 9, 23, 25, 33, 39, 41, 57, 71, 105, 119, 135, 151, 169, 183, 185, 199, 217, 231, 263, 265, 281, 297, 311, 343, 359, 375, 377, 391, 423, 441, 471, 505, 519, 535
Offset: 1

Views

Author

Vladimir Shevelev, Jul 27 2010

Keywords

Comments

r(n) = 1, 4, 5, 12, 13, 17, 20, 21, 29, 36, 53, 60, 68, 76, 85, 92, 93,...

Crossrefs

Cf. A179787, A179383, A179480, A179481, A179686, A179687 A179738, A179739

Extensions

Values a(2)-a(9) corrected, a(10) etc added by R. J. Mathar, Nov 04 2010
Showing 1-6 of 6 results.

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