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.

A003586 3-smooth numbers: numbers of the form 2^i*3^j with i, j >= 0.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162, 192, 216, 243, 256, 288, 324, 384, 432, 486, 512, 576, 648, 729, 768, 864, 972, 1024, 1152, 1296, 1458, 1536, 1728, 1944, 2048, 2187, 2304, 2592, 2916, 3072, 3456, 3888
Offset: 1

Views

Author

Paul Zimmermann, Dec 11 1996

Keywords

Comments

This sequence is easily confused with A033845, which gives numbers of the form 2^i*3^j with i, j >= 1. Don't simply say "numbers of the form 2^i*3^j", but specify which sequence you mean. - N. J. A. Sloane, May 26 2024
These numbers were once called "harmonic numbers", see Lenstra links. - N. J. A. Sloane, Jul 03 2015
Successive numbers k such that phi(6k) = 2k. - Artur Jasinski, Nov 05 2008
Where record values greater than 1 occur in A088468: A160519(n) = A088468(a(n)). - Reinhard Zumkeller, May 16 2009
Also numbers that are divisible by neither 6k - 1 nor 6k + 1, for all k > 0. - Robert G. Wilson v, Oct 26 2010
Also numbers m such that the rooted tree with Matula-Goebel number m has m antichains. The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T. The vertices of a rooted tree can be regarded as a partially ordered set, where u<=v holds for two vertices u and v if and only if u lies on the unique path between v and the root. An antichain is a nonempty set of mutually incomparable vertices. Example: m=4 is in the sequence because the corresponding rooted tree is \/=ARB (R is the root) having 4 antichains (A, R, B, AB). - Emeric Deutsch, Jan 30 2012
A204455(3*a(n)) = 3, and only for these numbers. - Wolfdieter Lang, Feb 04 2012
The number of terms less than or equal to n is Sum_{i=0..floor(log_2(n))} floor(log_3(n/2^i) + 1), or Sum_{i=0..floor(log_3(n))} floor(log_2(n/3^i) + 1), which requires fewer terms to compute. - Robert G. Wilson v, Aug 17 2012
Named 3-friables in French. - Michel Marcus, Jul 17 2013
In the 14th century Levi Ben Gerson proved that the only pairs of terms which differ by 1 are (1,2), (2,3), (3,4), and (8,9); see A235365, A235366, A236210. - Jonathan Sondow, Jan 20 2014
Range of values of A000005(n) (and also A181819(n)) for cubefree numbers n. - Matthew Vandermast, May 14 2014
A036561 is a permutation of this sequence. - L. Edson Jeffery, Sep 22 2014
Also the sorted union of A000244 and A007694. - Lei Zhou, Apr 19 2017
The sum of the reciprocals of the 3-smooth numbers is equal to 3. Brief proof: 1 + 1/2 + 1/3 + 1/4 + 1/6 + 1/8 + 1/9 + ... = (Sum_{k>=0} 1/2^k) * (Sum_{m>=0} 1/3^m) = (1/(1-1/2)) * (1/(1-1/3)) = (2/(2-1)) * (3/(3-1)) = 3. - Bernard Schott, Feb 19 2019
Also those integers k for which, for every prime p > 3, p^(2k) - 1 == 0 (mod 24k). - Federico Provvedi, May 23 2022
For n>1, the exponents’ parity {parity(i), parity(j)} of one out of four consecutive terms is {odd, odd}. Therefore, for n>1, at least one out of every four consecutive terms is a Zumkeller number (A083207). If for the term whose parity is {even, odd}, even also means nonzero, then this term is also a Zumkeller number (as is the case with the last of the four consecutive terms 1296, 1458, 1536, 1728). - Ivan N. Ianakiev, Jul 10 2022
Except the initial terms 2, 3, 4, 8, 9 and 16, these are numbers k such that k^6 divides 6^k. Except the initial terms 2, 3, 4, 6, 8, 9, 16, 18 and 27, these are numbers k such that k^12 divides 12^k. - Mohammed Yaseen, Jul 21 2022
In music theory, a comma is a ratio, close to 1 (typically less than 1.04), between two natural numbers divisible by only small primes (typically single digit). In this sequence, a(131) / a(130) = 531441 / 524288 ~ 1.013643 is the Pythagorean comma (A221363), the difference between 12 perfect fifths and 7 octaves. - Hal M. Switkay, Mar 23 2025

References

  • J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 654 pp. 85, 287-8, Ellipses Paris 2004.
  • S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, p. xxiv.
  • R. Tijdeman, Some applications of Diophantine approximation, pp. 261-284 of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett et al., Peters, 2003.

Links

Crossrefs

Cf. A051037, A002473, A051038, A080197, A080681, A080682, A117221, A105420, A062051, A117222, A117220, A090184, A131096, A131097, A186711, A186712, A186771, A088468, A061987, A080683 (p-smooth numbers with other values of p), A025613 (a subsequence).
Cf. also A235365, A235366, A236210, A036561.
Cf. also A000244, A007694. - Lei Zhou, Apr 19 2017
Cf. A191475 (successive values of i), A191476 (successive values of j), A022330 (indices of the pure terms 2^i), A022331 (indices of the pure terms 3^j). - N. J. A. Sloane, May 26 2024
Cf. A221363.

Programs

  • Haskell
    import Data.Set (Set, singleton, insert, deleteFindMin)
smooth :: Set Integer -> [Integer]
smooth s = x : smooth (insert (3*x) $ insert (2*x) s')
  where (x, s') = deleteFindMin s
a003586_list = smooth (singleton 1)
a003586 n = a003586_list !! (n-1)
-- Reinhard Zumkeller, Dec 16 2010
  • Magma
    [n: n in [1..4000] | PrimeDivisors(n) subset [2,3]]; // Bruno Berselli, Sep 24 2012
  • Maple
    A003586 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do numtheory[factorset](a) minus {2,3} ; if % = {} then return a; end if; end do: end if; end proc: # R. J. Mathar, Feb 28 2011
with(numtheory): for i from 1 to 23328 do if(i/phi(i)=3)then print(i/6) fi od; # Gary Detlefs, Jun 28 2011
  • Mathematica
    a[1] = 1; j = 1; k = 1; n = 100; For[k = 2, k <= n, k++, If[2*a[k - j] < 3^j, a[k] = 2*a[k - j], {a[k] = 3^j, j++}]]; Table[a[i], {i, 1, n}] (* Hai He (hai(AT)mathteach.net) and Gilbert Traub, Dec 28 2004 *)
aa = {}; Do[If[EulerPhi[6 n] == 2 n, AppendTo[aa, n]], {n, 1, 1000}]; aa (* Artur Jasinski, Nov 05 2008 *)
fQ[n_] := Union[ MemberQ[{1, 5}, # ] & /@ Union@ Mod[ Rest@ Divisors@ n, 6]] == {False}; fQ[1] = True; Select[ Range@ 4000, fQ] (* Robert G. Wilson v, Oct 26 2010 *)
powerOfTwo = 12; Select[Nest[Union@Join[#, 2*#, 3*#] &, {1}, powerOfTwo-1], # < 2^powerOfTwo &] (* Robert G. Wilson v and T. D. Noe, Mar 03 2011 *)
fQ[n_] := n == 3 EulerPhi@ n; Select[6 Range@ 4000, fQ]/6 (* Robert G. Wilson v, Jul 08 2011 *)
mx = 4000; Sort@ Flatten@ Table[2^i*3^j, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx/2^i]}] (* Robert G. Wilson v, Aug 17 2012 *)
f[n_] := Block[{p2, p3 = 3^Range[0, Floor@ Log[3, n] + 1]}, p2 = 2^Floor[Log[2, n/p3] + 1]; Min[ Select[ p2*p3, IntegerQ]]]; NestList[f, 1, 54] (* Robert G. Wilson v, Aug 22 2012 *)
Select[Range@4000, Last@Map[First, FactorInteger@#] <= 3 &] (* Vincenzo Librandi, Aug 25 2016 *)
Select[Range[4000],Max[FactorInteger[#][[All,1]]]<4&] (* Harvey P. Dale, Jan 11 2017 *)
  • PARI
    test(n)=for(p=2,3, while(n%p==0, n/=p)); n==1;
for(n=1,4000,if(test(n),print1(n",")))
  • PARI
    list(lim)=my(v=List(),N);for(n=0,log(lim\1+.5)\log(3),N=3^n;while(N<=lim,listput(v,N);N<<=1));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011
  • PARI
    is_A003586(n)=n<5||vecmax(factor(n,5)[, 1])<5 \\ M. F. Hasler, Jan 16 2015
  • PARI
    list(lim)=my(v=List(), N); for(n=0, logint(lim\=1,3), N=3^n; while(N<=lim, listput(v, N); N<<=1)); Set(v) \\ Charles R Greathouse IV, Jan 10 2018
  • Python
    from itertools import count, takewhile
def aupto(lim):
    pows2 = list(takewhile(lambda x: xMichael S. Branicky, Jul 08 2022
  • Python
    from sympy import integer_log
def A003586(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Sep 15 2024
  • Python
    # faster for initial segment of sequence
import heapq
from itertools import islice
def A003586gen(): # generator of terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], [2, 3]
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield v
            oldv = v
            for p in psmooth_primes:
                heapq.heappush(h, v*p)
print(list(islice(A003586gen(), 65))) # Michael S. Branicky, Sep 17 2024
(C++) // Returns A003586 <= threshold without approximations nor sorting
#include 
std::forward_list A003586(const int threshold) {
    std::forward_list sequence;
    auto start_it = sequence.before_begin();
    for (int i = 1; i <= threshold; i *= 2) {
        for (int inc = 1; std::next(start_it) != sequence.end() && inc <= i; inc *= 3)
            ++start_it;
        auto it = start_it;
        for (int j = 1; i * j <= threshold; j *= 3) {
            sequence.emplace_after(it, i * j);
            for (int inc = 1; std::next(it) != sequence.end() && inc <= i; inc *= 2)
                ++it;
        }
    }
    return sequence;
} // Eben Gino Lester, Apr 17 2025
  • Sage
    def isA003586(n) :
    return not any(d != 2 and d != 3 for d in prime_divisors(n))
@CachedFunction
def A003586(n) :
    if n == 1 : return 1
    k = A003586(n-1) + 1
    while not isA003586(k) : k += 1
    return k
[A003586(n) for n in (1..55)] # Peter Luschny, Jul 20 2012

Formula

An asymptotic formula for a(n) is roughly a(n) ~ 1/sqrt(6)*exp(sqrt(2*log(2)*log(3)*n)). - Benoit Cloitre, Nov 20 2001
A061987(n) = a(n + 1) - a(n), a(A084791(n)) = A084789(n), a(A084791(n) + 1) = A084790(n). - Reinhard Zumkeller, Jun 03 2003
Union of powers of 2 and 3 with n such that psi(n) = 2*n, where psi(n) = n*Product_(1 + 1/p) over all prime factors p of n = A001615(n). - Lekraj Beedassy, Sep 07 2004; corrected by Franklin T. Adams-Watters, Mar 19 2009
a(n) = 2^A022328(n)*3^A022329(n). - N. J. A. Sloane, Mar 19 2009
The characteristic function of this sequence is given by Sum_{n >= 1} x^a(n) = Sum_{n >= 1} moebius(6*n)*x^n/(1 - x^n). - Paul D. Hanna, Sep 18 2011
a(n) = A007694(n+1)/2. - Lei Zhou, Apr 19 2017

Extensions

Deleted claim that this sequence is union of 2^n (A000079) and 3^n (A000244) sequences -- this does not include the terms which are not pure powers. - Walter Roscello (wroscello(AT)comcast.net), Nov 16 2008

A229943 Decimal expansion of 256/243, the Pythagorean semitone.

Original entry on oeis.org

1, 0, 5, 3, 4, 9, 7, 9, 4, 2, 3, 8, 6, 8, 3, 1, 2, 7, 5, 7, 2, 0, 1, 6, 4, 6, 0, 9, 0, 5, 3, 4, 9, 7, 9, 4, 2, 3, 8, 6, 8, 3, 1, 2, 7, 5, 7, 2, 0, 1, 6, 4, 6, 0, 9, 0, 5, 3, 4, 9, 7, 9, 4, 2, 3, 8, 6, 8, 3, 1, 2, 7, 5, 7, 2, 0, 1, 6, 4, 6, 0, 9, 0, 5, 3, 4, 9, 7, 9, 4, 2, 3, 8, 6, 8, 3, 1, 2, 7, 5, 7, 2, 0, 1, 6, 4, 6, 0, 9
Offset: 1

Views

Author

Omar E. Pol, Oct 25 2013

Keywords

Comments

The Pythagorean diatonic semitone is one of the musical intervals. Has a ratio of 256/243, and is often called the Pythagorean limma. It is also sometimes called the Pythagorean minor semitone.
After the initial term the sequence has period 27, repeat: 0, 5, 3, 4, 9, 7, 9, 4, 2, 3, 8, 6, 8, 3, 1, 2, 7, 5, 7, 2, 0, 1, 6, 4, 6, 0, 9.

Examples

			1.053497942386831275720164609...

References

  • J. M. Merino de la Fuente, Las vibraciones de la música, Editorial Club Universitario (2006), 133.
  • Alberto Rojo, La física en la vida cotidiana, Siglo Veintiuno Editores (2011), 137.

Links

Crossrefs

Cf. A010774, A221363, A229948, A230437.

Programs

  • Mathematica
    RealDigits[256/243,10,120][[1]] (* Harvey P. Dale, Jul 17 2019 *)

Formula

A229948/A221363 = (3^7/2^11)/(3^12/2^19) = 2^8/3^5 = 256/243.

A229948 Decimal expansion of 2187/2048, the Pythagorean apotome.

Original entry on oeis.org

1, 0, 6, 7, 8, 7, 1, 0, 9, 3, 7, 5
Offset: 1

Views

Author

Omar E. Pol, Oct 25 2013

Keywords

Comments

The Pythagorean apotome 2187/2048 (also called the apotome Pythagorica) is one of the musical intervals.

Examples

			1.06787109375

Links

Crossrefs

Cf. A010774, A221363, A229943, A230437.

Formula

A221363 * A229943 = (3^12/2^19)*(2^8/3^5) = 3^7/2^11 = 2187/2048.

A258054 Circle of fifths cycle (counterclockwise).

Original entry on oeis.org

1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8, 1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8, 1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8, 1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8, 1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8, 1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8, 1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8, 1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8, 1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8
Offset: 1

Views

Author

Peter Woodward, May 17 2015

Keywords

Comments

The twelve notes dividing the octave are numbered 1 through 12 sequentially. This sequence begins at a certain note, travels down a perfect fifth twelve times (seven semitones), and arrives back at the same note. If justly tuned fifths are used, the final note will be flat by the Pythagorean comma (roughly 23.46 cents or about a quarter of a semitone).
Period 12: repeat [1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8]. - Omar E. Pol, May 18 2015
The string [1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8] is also in both A023127 and A054073. - Omar E. Pol, May 19 2015

Examples

			For a(3), 1+5+5 = 11 (mod 12).
For a(4), 1+5+5+5 = 4 (mod 12).

Links

Crossrefs

Cf. A221363 (Pythagorean comma), A257811 (clockwise circle of fifths cycle).

Programs

  • Magma
    [1+5*(n-1) mod 12: n in [1..80]]; // Vincenzo Librandi, May 19 2015
  • Maple
    A258054:=n->1+(5*(n-1) mod 12): seq(A258054(n), n=1..100); # Wesley Ivan Hurt, May 22 2015
  • Mathematica
    PadRight[{}, 100, {1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8}] (* Vincenzo Librandi, May 19 2015 *)
LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1},{1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8},108] (* Ray Chandler, Aug 27 2015 *)
  • PARI
    a(n)=1+5*(n-1) \\ Charles R Greathouse IV, May 22 2015
  • PARI
    Vec(x*(1 + 6*x + 11*x^2 + 4*x^3 + 9*x^4 + 2*x^5 + 7*x^6 + 12*x^7 + 5*x^8 + 10*x^9 + 3*x^10 + 8*x^11) / (1 - x^12) + O(x^80)) \\ Colin Barker, Nov 15 2019

Formula

Periodic with period 12: a(n) = 1 + (5(n-1) mod 12).
From Colin Barker, Nov 15 2019: (Start)
G.f.: x*(1 + 6*x + 11*x^2 + 4*x^3 + 9*x^4 + 2*x^5 + 7*x^6 + 12*x^7 + 5*x^8 + 10*x^9 + 3*x^10 + 8*x^11) / (1 - x^12).
a(n) = a(n-12) for n>12.
(End)

Extensions

Extended by Ray Chandler, Aug 27 2015

A230437 Decimal expansion of (2/(3 - 2^(1/2)))^(1/4).

Original entry on oeis.org

1, 0, 5, 9, 7, 3, 2, 6, 7, 2, 2, 0, 2, 1, 3, 9, 8, 0, 8, 7, 4, 6, 9, 6, 9, 3, 7, 6, 9, 9, 2, 5, 8, 3, 2, 5, 1, 2, 1, 1, 6, 7, 7, 0, 5, 7, 3, 6, 2, 0, 3, 6, 5, 1, 5, 3, 0, 6, 8, 4, 1, 6, 4, 8, 2, 6, 6, 1, 8, 9, 1, 1, 4, 6, 4, 1, 1, 8, 3, 9, 2, 9, 8, 3, 2, 8, 3, 2, 7, 0, 5, 3, 4, 9, 5, 8, 3, 7, 3, 7, 9, 6, 2, 0
Offset: 1

Views

Author

Omar E. Pol, Oct 25 2013

Keywords

Comments

Approximation to the frequency ratio of a semitone in equal temperament, suggested by Marin Mersenne.
This constant is close to A010774, the frequency ratio of a semitone in equal temperament. The difference is 0.0002695778428445...

Examples

			1.059732672202139808746969376992583251211677...

Links

Crossrefs

Cf. A010774, A221363, A229943, A229948.

Programs

  • Mathematica
    RealDigits[Surd[2/(3 - 2^(1/2)), 4], 10, 120][[1]] (* Amiram Eldar, May 16 2023 *)

A358029 Decimal expansion of the ratio between step sizes of the diatonic and chromatic semitones produced by a circle of 12 perfect fifths in Pythagorean tuning.

Original entry on oeis.org

1, 2, 6, 0, 0, 1, 6, 7, 5, 2, 6, 7, 0, 8, 2, 4, 5, 3, 5, 9, 3, 1, 2, 7, 6, 1, 2, 2, 6, 0, 3, 9, 2, 4, 2, 3, 3, 7, 1, 8, 1, 1, 5, 5, 7, 9, 2, 3, 2, 7, 6, 7, 8, 3, 3, 4, 1, 0, 6, 5, 2, 0, 1, 6, 1, 6, 2, 0, 8, 7, 4, 8, 0, 0, 8, 3, 1, 2, 2, 7, 8, 4, 6, 8, 8, 1, 4
Offset: 1

Views

Author

Eliora Ben-Gurion, Oct 25 2022

Keywords

Comments

Pythagorean tuning is a form of tuning produced by repeated stacking of the perfect fifth, which has the frequency ratio of 3:2. A circle of 12 perfect fifths is approximately equal to the tuning system predominantly in use in the world today. If the perfect fifth is stacked 12 times and the resulting sequence is octave-reduced, then this divides the octave into 5 chromatic semitones which are equal to 2187/2048 (A229948), and 7 diatonic semitones which are equal to 256/243 (A229943). Diatonic semitones are those which are derived from a circle of 7 perfect fifths, the diatonic scale, and 5 chromatic semitones are a byproduct of an addition of 5 more perfect fifths, that is, another rotation, to the scale.

Examples

			1.2600167526708245359312761226039242337181155792327678334106520161...

Links

Crossrefs

Cf. A131071, A221363, A258054, A257811.
Cf. A229943, A229948.

Programs

  • Mathematica
    RealDigits[(7*Log[3] - 11*Log[2])/(8*Log[2] - 5*Log[3]), 10, 120][[1]] (* Amiram Eldar, Jun 21 2023 *)

Formula

Equals log(2187/2048) / log(256/243).
Equals log(A229948) / log(A229943).
Equals (7*log(3) - 11*log(2))/(8*log(2) - 5*log(3)).
Showing 1-6 of 6 results.

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