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.

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A014659 Odd numbers that do not divide 2^k + 1 for any k >= 1.

Original entry on oeis.org

7, 15, 21, 23, 31, 35, 39, 45, 47, 49, 51, 55, 63, 69, 71, 73, 75, 77, 79, 85, 87, 89, 91, 93, 95, 103, 105, 111, 115, 117, 119, 123, 127, 133, 135, 141, 143, 147, 151, 153, 155, 159, 161, 165, 167, 175, 183, 187, 189, 191, 195, 199, 203, 207, 213, 215, 217, 219
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

This is the subset of odd integers > 1 as (2*n - 1) in A179480 such that A179480(n) is even. Example: A179480(18) = 6, even; corresponding to (2*18 - 1), 35. Then 35 is in A014659. A014657 is the subset of odd terms > 1 corresponding to odd terms in A179480. - Gary W. Adamson, Aug 20 2012
From Wolfdieter Lang, Aug 22 2020: (Start)
These odd numbers are the moduli named 2*n+1 in the definition of A003558(n), for n >= 1, for which the + sign applies. The signs in the definition of A003558 are given in A332433.
These are the odd numbers N >= 3 for which A003558((N-1)/2) = A002326((N+1)/2), the period length P(N) of the cycles {2^k (mod N)}_{k=0}^(P(N)-1). Compare the periods given in A201908((N+1)/2, k). (End)

Links

Crossrefs

Cf. A179480, A002326, A003558, A201908, A332433.
Cf. A014657, numbers that divide 2^k + 1 for some k.

Extensions

More terms from Don Reble, Nov 03 2001

A179738 a(n) = length of (eventual) period of the sequence defined by s(0) = 1, s(n+1) = odd_part(2n-1 + (s(n) if n odd else s(n)*3)), where odd_part = A000265.

Original entry on oeis.org

1, 2, 4, 1, 4, 8, 2, 4, 8, 4, 2, 2, 1, 4, 12, 4, 4, 26, 2, 12, 6, 2, 4, 4, 4, 16, 4, 8, 10, 6, 4, 8, 4, 2, 6, 8, 12, 12, 28, 1, 22, 16, 2, 16, 12, 12, 16, 6, 4, 40, 24, 4, 32, 6, 26, 16, 6, 8, 4, 8, 12, 2, 36, 6, 46, 12, 2, 36, 60, 6, 16, 8, 4, 18, 20, 4, 60, 36
Offset: 2

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Author

Vladimir Shevelev, Jul 25 2010

Keywords

Comments

Original definition (edited): Let x, y be odd numbers and the operation x <+> y := A000265(x+y). Consider sequence s(0) = x <+> y, s(2*k+1) = x <+> 3*s(2*k), s(2*k+2) = x <+> s(2*k+1); a(n) is the smallest period in case x = 2*n-1, y = 1.
The operation x <+> y = A000265(x+y) = odd part of x+y is also considered in A179382.
Record values are: a(2) = 1, a(3) = 2, a(4) = 4, a(7) = 8, a(16) = 12, a(19) = 26, a(40) = 28, a(51) = 40, a(66) = 46, a(70) = 60, a(111) = 64, a(126) = 70, a(147) = 80, a(162) = 96, a(225) = 120, a(379) = 170, a(619) = 184, a(640) = 228, a(727) = 248, a(759) = 256, a(916) = 348, ... - M. F. Hasler, Feb 16 2025

Examples

			For n = 4, 2*n-1 = 7, we get: 7 <+> 1 = 1, 7 <+> 3*1 = 5, 7 <+> 5 = 3, 7 <+> 3*3 = 1, and from here on it starts over with 7 <+> 1 = 1, etc., so the period is [1, 5, 3, 1], of length 4, whence a(4) = 4.
For n = 6, 2*n-1 = 11, we get:
  11 <+> 1 = 3, 11 <+> 3*3 =  5, 11 <+>  5  = 1, 11 <+> 3*1 = 7,
  11 <+> 7 = 9, 11 <+> 3*9 = 19, 11 <+> 19 = 15, 11 <+> 3*15 = 7, 11 <+> 7 = 9, ...
Thus we have an eventually periodic sequence with the smallest period 4 (with elements 7, 9, 19, 15). Thus a(6) = 4.

Links

Crossrefs

Cf. A179382, A179480, A179686.

Programs

  • PARI
    apply( {A179738(n, y=1, T(y, x=2*n-1)=(x+y)>>valuation(x+y,2))=my(s=[], P); until(, s=concat(s, y=T(3^(#s%2)*y)); for(L=1, #s\3, P=[vecextract(s,Str(-L-t,"..-",1+t)) | t<-[0,L,2*L]]; P[1]==P[2] && P[1]==P[3] && return(#P[1])))}, [2..90])
  • Python
    def A179738(n):
    s = [1]; x = 2*n-1; odd = lambda z: all(z&1 or(z:=z>>1)for _ in range(z))and z
    while not(p := next((p for p in range(1, len(s)//3+1) if
        s[-p:]==s[-2*p:-p]==s[-3*p:-2*p]), 0)): s.append(odd(x+3**(len(s)&1)*s[-1]))
    return p
print([A179738(n)for n in range(2,99)]) # M. F. Hasler, Feb 16 2025

Extensions

Name edited and corrections proposed by Jason Yuen, Feb 09 2025
Edited, a(4) and a(18) corrected, and extended by M. F. Hasler, Feb 15 2025

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

A179787 Let the operation <+> be defined by x<+>y = A038502(x+y). a(n) is the period in the track of the iterated application x<+>(x<+>...(x<+>1)) for x = A001651(n-1).

Original entry on oeis.org

2, 1, 2, 4, 6, 1, 4, 4, 2, 6, 3, 16, 18, 2, 3, 8, 20, 1, 6, 28, 30, 7, 16, 10, 18, 18, 2, 8, 42, 8, 11, 18, 42, 20, 4, 52, 20, 3, 28, 26, 10, 30, 15, 10, 22, 12, 8, 28, 12, 18, 18, 28, 78, 1, 8, 38, 14, 42, 9, 88, 4, 22, 23, 28, 48, 42, 18, 100, 34, 3, 52, 50, 22, 20, 9, 112, 38, 22, 23, 38
Offset: 1

Views

Author

Vladimir Shevelev, Jul 27 2010

Keywords

Comments

The symbol <+> removes powers of three of the sum of the two operands.
The process of starting with 1, adding some constant number x = A001651(n-1) and reducing it iteratively with this operation defines a track 1, x<+>1, x<+>(x<+>1), ... which enters a cycle.
The period of this cycle specifies a(n).
Similar iterated reductions can be defined for power bases m other than 3.

Examples

			For n=5 we take x=A001651(4)=7. The iteration yields 1, 7<+>1=8, 7<+>8=5, 7<+>5=4, 7<+>4=11, 7<+>11=2, 7<+>2=1.
We have reached the 1 of the beginning and therefore a cycle of length a(5)=6.

Crossrefs

Cf. A179382, A179480, A179686, A179738.

Programs

  • Maple
    A038502 := proc(n) a := 1; for p in ifactors(n)[2] do if op(1,p) <> 3 then a := a*op(1,p)^op(2,p) ; end if; end do; a ; end proc:
A179787aux := proc(x,y) local xtrack,xitr,xpos ; xtrack := [y] ; while true do xitr := A038502(op(-1,xtrack)+x) ;
if not member(xitr, xtrack,'xpos') then xtrack := [op(xtrack),xitr] ; else return 1+nops(xtrack)-xpos ; end if; end do: end proc:
A001651 := proc(n) option remember; if n <=2 then n; else procname(n-2)+3 ; end if; end proc:
A179787 := proc(n) A179787aux(A001651(n),1) ; end proc: seq(A179787(n),n=1..80) ; # R. J. Mathar, Nov 04 2010

Extensions

a(22) corrected, definition tightened removing new terminology, sequence extended beyond a(55) by R. J. Mathar, Nov 04 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
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