A003586 -id:A003586 - cp's OEIS frontend

A062051 Number of partitions of n into powers of 3.

1, 1, 1, 2, 2, 2, 3, 3, 3, 5, 5, 5, 7, 7, 7, 9, 9, 9, 12, 12, 12, 15, 15, 15, 18, 18, 18, 23, 23, 23, 28, 28, 28, 33, 33, 33, 40, 40, 40, 47, 47, 47, 54, 54, 54, 63, 63, 63, 72, 72, 72, 81, 81, 81, 93, 93, 93, 105, 105, 105, 117, 117, 117, 132, 132, 132, 147, 147, 147, 162

Offset: 0

Amarnath Murthy, Jun 06 2001

nonn

Number of different partial sums of 1+[1,*3]+[1,*3]+..., where [1,*3] means we can either add 1 or multiply by 3. E.g., a(6)=3 because we have 6=1+1+1+1+1+1=(1+1)*3=1*3+1+1+1. - Jon Perry, Jan 01 2004
Also number of partitions of n into distinct 3-smooth parts. E.g., a(10) = #{9+1, 8+2, 6+4, 6+3+1, 4+3+2+1} = #{9+1, 3+3+3+1, 3+3+1+1+1+1, 3+1+1+1+1+1+1+1, 1+1+1+1+1+1+1+1+1+1} = 5. - Reinhard Zumkeller, Apr 07 2005
Starts to differ from A008650 at a(81). - R. J. Mathar, Jul 31 2010
If m=ceiling(log_3(2k)) and n=(3^m+1)/2-k for k in the range (3^(m-1)+1)/2+(3^(m-2))<=k<=(3^m-1)/2, this sequence gives the number of "feasible" partitions described in the sequence A254296. For instance, the terms starting at 121st term of A254296 backwards to 68th term of A254296 provide the first 54 terms of this sequence. - Md. Towhidul Islam, Mar 01 2015
From Gary W. Adamson, Sep 03 2016: (Start)
Let M =
  1, 0, 0, 0, 0, ...
  1, 0, 0, 0, 0, ...
  1, 0, 0, 0, 0, ...
  1, 1, 0, 0, 0, ...
  1, 1, 0, 0, 0, ...
  1, 1, 0, 0, 0, ...
  1, 1, 1, 0, 0, ...
  1, 1, 1, 0, 0, ...
  ..., where the leftmost column is all 1's, and all other columns are 1's shifted down thrice. Lim_{k=1..inf} M^k has a single nonzero column, which gives the sequence. (End)

			a(4) = 2 and the partitions are 3+1, 1+1+1+1;
a(9) = 5 and the partitions are 9; 3+3+3; 3+3+1+1+1; 3+1+1+1+1+1+1; 1+1+1+1+1+1+1+1+1.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Vassil Dimitrov, Laurent Imbert, and Andrew Zakaluzny, Multiplication by a Constant is Sublinear, 18th IEEE Symposium on Computer Arithmetic (2007). See Theorem 1.
Md Towhidul Islam and Md Shahidul Islam, Number of Partitions of an n-kilogram Stone into Minimum Number of Weights to Weigh All Integral Weights from 1 to n kg(s) on a Two-pan Balance, arXiv:1502.07730 [math.CO], 2015.
M. Latapy, Partitions of an integer into powers, DMTCS Proceedings AA (DM-CCG), 2001, 215-228.
M. Latapy, Partitions of an integer into powers, DMTCS Proceedings AA (DM-CCG), 2001, 215-228. [Cached copy, with permission]

A005704 with terms repeated 3 times.

Cf. A000123, A018819, A000009, A003586, A105420, A039966, A023893, A105420, A106244, A131995, A179051, A254296.

nn=70;a=Product[1/(1-x^(3^i)),{i,0,4}];CoefficientList[Series[a,{x,0,nn}],x] (* Geoffrey Critzer, Oct 30 2012 *)

{ n=15; v=vector(n); for (i=1,n,v[i]=vector(2^(i-1))); v[1][1]=1; for (i=2,n, k=length(v[i-1]); for (j=1,k, v[i][j]=v[i-1][j]+1; v[i][j+k]=v[i-1][j]*3)); c=vector(n); for (i=1,n, for (j=1,2^(i-1), if (v[i][j]<=n, c[v[i][j]]++))); c } \\ Jon Perry

from functools import lru_cache
@lru_cache(maxsize=None)
def A062051(n): return A062051(n-1)+(0 if n%3 else A062051(n//3)) if n>2 else 1 # Chai Wah Wu, Sep 21 2022

a(n) = A005704([n/3]).
G.f.: Product_{k>=0} 1/(1-x^(3^k)). - R. J. Mathar, Jul 31 2010
If m = ceiling(log_3(2k)), define n = (3^m + 1)/2 - k for k in the range (3^(m-1)+1)/2 + (3^(m-2)) <= k <= (3^m-1)/2. Then, a(n) = Sum_{s=ceiling((k-1)/3)..(3^(m-1)-1)/2} a(s). This gives the first 2(3^(m-1))/3 terms. - Md. Towhidul Islam, Mar 01 2015
G.f.: 1 + Sum_{i>=0} x^(3^i) / Product_{j=0..i} (1 - x^(3^j)). - Ilya Gutkovskiy, May 07 2017

More terms from Larry Reeves (larryr(AT)acm.org), Jun 11 2001

A065333 Characteristic function of 3-smooth numbers, i.e., numbers of the form 2^i*3^j (i, j >= 0).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Oct 29 2001

Dirichlet inverse of b(n) where b(n) = 0 except for: b(1) = b(6) = -b(2) = -b(3) = 1. - Alexander Adam, Dec 26 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
A. Pakapongpun and T. Ward, Functorial Orbit counting, JIS 12 (2009) 09.2.4, example 9.
Index entries for characteristic functions

Characteristic function of A003586.

Cf. A000265, A007814, A007949, A038502, A065330, A065332, A071521 (partial sums), A072078 (inverse Möbius transform).

a065333 = fromEnum . (== 1) . a038502 . a000265
-- Reinhard Zumkeller, Jan 08 2013, Apr 12 2012

a[n_] := Boole[ 2^IntegerExponent[n, 2] * 3^IntegerExponent[n, 3] == n]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, May 16 2013, after Charles R Greathouse IV *)

a(n)=sumdiv(n,d,moebius(6*d)) \\ Benoit Cloitre, Oct 18 2009

a(n)=3^valuation(n,3)<Charles R Greathouse IV, Aug 21 2011

from sympy import multiplicity
def A065333(n): return int(3**(multiplicity(3,m:=n>>(~n&n-1).bit_length()))==m) # Chai Wah Wu, Dec 20 2024

a(n) = if n = A003586(k) for some k then 1 else 0.
a(n) = signum(A065332(n)), where signum = A057427.
a(n) = if A065330(n) = 1 then 1 else 0 = 1 - signum(A065330(n) - 1).
a(n) = Product_{p prime and p|n} 0^floor(p/4). - Reinhard Zumkeller, Nov 19 2004
Multiplicative with a(2^e) = a(3^e) = 1, a(p^e) = 0 for prime p > 3. Dirichlet g.f. 1/(1-2^-s)/(1-3^-s). - Franklin T. Adams-Watters, Sep 01 2006
a(n) = 0^(A038502(A000265(n)) - 1). - Reinhard Zumkeller, Sep 28 2008
a(n) = Sum_{d|n} mu(6*d). - Benoit Cloitre, Oct 18 2009

A002072 a(n) = smallest number m such that for all k > m, either k or k+1 has a prime factor > prime(n).

Original entry on oeis.org

1, 8, 80, 4374, 9800, 123200, 336140, 11859210, 11859210, 177182720, 1611308699, 3463199999, 63927525375, 421138799639, 1109496723125, 1453579866024, 20628591204480, 31887350832896, 31887350832896, 119089041053696, 2286831727304144, 9591468737351909375, 9591468737351909375, 9591468737351909375, 9591468737351909375, 9591468737351909375, 19316158377073923834000

Offset: 1

N. J. A. Sloane

An effective abc conjecture (c < rad(abc)^2) would imply that a(27) = a(28) = ... = a(32), and a(33) = 124225935845233319439173. - Lucas A. Brown, Sep 20 2020

			a(1) = 1 since for any number k greater than 1, it is impossible that k and k+1 both are powers of 2, so at least one of them has a prime factor > 2. (For m = 0 this would not hold for k = 1, k+1 = 2.)
a(2) = 8 since for any larger k, we cannot have k and k+1 both 3-smooth (cf. A003586).
31887350832897 = 3^9*7*37*41^2*61^2, 31887350832896 = 2^8*13*19*23*29^4*31, this number appears twice because there is no pair of numbers with max. factor = 67 that is larger than this number.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Lucas A. Brown, Python program.
E. F. Ecklund and R. B. Eggleton, Prime factors of consecutive integers, Amer. Math. Monthly, 79 (1972), 1082-1089.
D. H. Lehmer, On a problem of Størmer, Ill. J. Math., 8 (1964), 57-79.
Don Reble, Python program
Jim White, Results to P = 127
Wikipedia, Størmer's theorem

Cf. A002071, A003032, A003033, A122463, A145606, A175607.

Equals A117581(n) - 1.

smoothNumbers[p_?PrimeQ, max_Integer] := Module[{a, aa, k, pp, iter}, k = PrimePi[p]; aa = Array[a, k]; pp = Prime[Range[k]]; iter = Table[{a[j], 0, PowerExpand[Log[pp[[j]], max/Times @@ (Take[pp, j-1]^Take[aa, j-1])]] }, {j, 1, k}]; Sort[Flatten[Table[Times @@ (pp^aa), Evaluate[ Sequence @@ iter]]]]]; a[n_] := Module[{sn = smoothNumbers[Prime[n], Ceiling[2000 + 10^n/n]], pos}, pos = Position[Differences[sn], 1][[-1, 1]]; sn[[pos]]]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 12}] (* Jean-François Alcover, Nov 17 2016, after M. F. Hasler's observation *)

A002072(n, a=[1, 8, 80, 4374, 9800, 123200, 336140, 11859210, 11859210, 177182720, 1611308699, 3463199999, 63927525375, 421138799639, 1109496723125, 1453579866024])=a[n] \\ "practical" solution for use in other sequences, easily extended to more values. - M. F. Hasler, Jan 16 2015

A2072=List(1); A002072(n)={while(#A2072 best && isSmooth(sol, P) && isSmooth(sol+1, P) && best=sol, p=primes([1, P])); for(i=1, 2^#p, i==2 && next; my(qq = 2*vecprod(vecextract(p,i-1)), qn = [qq, sqrtint(qq), 0, 1], cf = [1,0,0,1], xi, aa, x0, x1, y0, y1); until(x0, aa = (qn[2]+qn[3])\qn[4]; qn[3] = aa*qn[4] - qn[3]; qn[4] = (qn[1] - qn[3]^2) \ qn[4]; cf = [aa*cf[1]+cf[3], aa*cf[2]+cf[4], cf[1], cf[2]]; cf[1]^2 - qq*cf[2]^2 == 1 && [x0,x1, y0,y1] = [x1, cf[1], y1, cf[2]] ); isSmooth(y0, P) || next; check(xi = x0); check(x1); for (i=3, max(P\/2, 3), [x0, x1] = [x1, x1 * xi * 2 - x0]; check(x1)))/*for i*/; listput(A2072, best) } \\ Following Don Reble's Python program. - M. F. Hasler, Mar 01 2025

a(n) < 10^n/n except for n=4. (Conjectured, from experimental data.) - M. F. Hasler, Jan 16 2015

More terms from Don Reble, Jan 11 2005
a(18)-a(26) from Fred Schneider, Sep 09 2006
Corrected and extended by Andrey V. Kulsha, Aug 10 2011, according to Jim White's computations.

A003594 Numbers of the form 3^i*7^j with i, j >= 0.

Original entry on oeis.org

1, 3, 7, 9, 21, 27, 49, 63, 81, 147, 189, 243, 343, 441, 567, 729, 1029, 1323, 1701, 2187, 2401, 3087, 3969, 5103, 6561, 7203, 9261, 11907, 15309, 16807, 19683, 21609, 27783, 35721, 45927, 50421, 59049, 64827, 83349, 107163, 117649

Offset: 1

N. J. A. Sloane

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 70 terms from Vincenzo Librandi)
Vaclav Kotesovec, Graph - the asymptotic ratio (600000 terms).

Cf. A003586, A003591, A003592, A003593, A003595.

Cf. A025642, A025665.

Filtered([1..120000],n->PowerMod(21,n,n)=0); # Muniru A Asiru, Mar 19 2019

import Data.Set (singleton, deleteFindMin, insert)
a003594 n = a003594_list !! (n-1)
a003594_list = f $ singleton 1 where
   f s = y : f (insert (3 * y) $ insert (7 * y) s')
               where (y, s') = deleteFindMin s
-- Reinhard Zumkeller, May 16 2015

[n: n in [1..120000] | PrimeDivisors(n) subset [3,7]]; // Bruno Berselli, Sep 24 2012

f[upto_]:=Sort[Select[Flatten[3^First[#] 7^Last[#] & /@ Tuples[{Range[0, Floor[Log[3, upto]]], Range[0, Floor[Log[7, upto]]]}]], # <= upto &]]; f[120000]  (* Harvey P. Dale, Mar 04 2011 *)
fQ[n_] := PowerMod[21, n, n] == 0; Select[Range[120000], fQ] (* Bruno Berselli, Sep 24 2012 *)

list(lim)=my(v=List(),N);for(n=0,log(lim)\log(7),N=7^n;while(N<=lim,listput(v,N);N*=3));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

from sympy import integer_log
def A003594(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(integer_log(x//7**i,3)[0]+1 for i in range(integer_log(x,7)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

The characteristic function of this sequence is given by Sum_{n >= 1} x^a(n) = Sum_{n >= 1} mu(21*n)*x^n/(1 - x^n), where mu(n) is the Möbius function A008683. Cf. with the formula of Hanna in A051037. - Peter Bala, Mar 18 2019
Sum_{n>=1} 1/a(n) = (3*7)/((3-1)*(7-1)) = 7/4. - Amiram Eldar, Sep 22 2020
a(n) ~ exp(sqrt(2*log(3)*log(7)*n)) / sqrt(21). - Vaclav Kotesovec, Sep 22 2020
a(n) = 3^A025642(n) * 7^A025665(n). - R. J. Mathar, Jul 06 2025

A003595 Numbers of the form 5^i*7^j with i, j >= 0.

Original entry on oeis.org

1, 5, 7, 25, 35, 49, 125, 175, 245, 343, 625, 875, 1225, 1715, 2401, 3125, 4375, 6125, 8575, 12005, 15625, 16807, 21875, 30625, 42875, 60025, 78125, 84035, 109375, 117649, 153125, 214375, 300125, 390625, 420175, 546875, 588245, 765625, 823543, 1071875, 1500625

Offset: 1

N. J. A. Sloane

nonn

Successive k such that phi(35*k) = 24*k: 35*a(n) = A033851(n). - Artur Jasinski, Nov 09 2008

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
Vaclav Kotesovec, Graph - the asymptotic ratio (400000 terms).
David Ryan, An algorithm to assign musical prime commas to every prime number and construct a universal and compact free Just Intonation musical notation, Preprint, arXiv:1612.01860 [cs.SD], 2016-2017.

Cf. A033851, A143207, A147571-A147580.
Cf. A003586, A003592, A003593, A003594.
Cf. A025652, A025667.

import Data.Set (singleton, deleteFindMin, insert)
a003595 n = a003595_list !! (n-1)
a003595_list = f $ singleton 1 where
   f s = y : f (insert (5 * y) $ insert (7 * y) s')
               where (y, s') = deleteFindMin s
-- Reinhard Zumkeller, May 16 2015

[n: n in [1..600000] | PrimeDivisors(n) subset [5,7]]; // Bruno Berselli, Sep 24 2012

a = {}; Do[If[EulerPhi[35 k] == 24 k, AppendTo[a, k]], {k, 1, 10000}]; a (* Artur Jasinski, Nov 09 2008 *)
fQ[n_] := PowerMod[35, n, n] == 0; Select[Range[600000], fQ] (* Bruno Berselli, Sep 24 2012 *)

list(lim)=my(v=List(),N);for(n=0,log(lim)\log(7),N=7^n;while(N<=lim,listput(v,N);N*=5));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

from sympy import integer_log
def A003595(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(integer_log(x//7**i,5)[0]+1 for i in range(integer_log(x,7)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

Sum_{n>=1} 1/a(n) = (5*7)/((5-1)*(7-1)) = 35/24. - Amiram Eldar, Sep 22 2020
a(n) ~ exp(sqrt(2*log(5)*log(7)*n)) / sqrt(35). - Vaclav Kotesovec, Sep 22 2020
a(n) = 5^A025652(n) * 7^A025667(n). - R. J. Mathar, Jul 06 2025

A065119 Numbers k such that the k-th cyclotomic polynomial is a trinomial.

Original entry on oeis.org

3, 6, 9, 12, 18, 24, 27, 36, 48, 54, 72, 81, 96, 108, 144, 162, 192, 216, 243, 288, 324, 384, 432, 486, 576, 648, 729, 768, 864, 972, 1152, 1296, 1458, 1536, 1728, 1944, 2187, 2304, 2592, 2916, 3072, 3456, 3888, 4374, 4608, 5184, 5832, 6144, 6561, 6912, 7776, 8748, 9216

Offset: 1

Len Smiley, Nov 12 2001

nonn

Appears to be numbers of form 2^a * 3^b, a >= 0, b > 0. - Lekraj Beedassy, Sep 10 2004
This is true: see link "Cyclotomic trinomials". - Robert Israel, Jul 14 2015
3-smooth numbers (A003586) which are not powers of 2 (A000079). - Amiram Eldar, Nov 10 2020
These are the conjugates of semiprimes, where conjugation is A122111; or Heinz numbers of conjugates of length-2 partitions. - Gus Wiseman, Nov 09 2023
A multiplicative semigroup: if m and n are in the sequence, then so is m*n. - Antti Karttunen, Jul 13 2024

			The 54th cyclotomic polynomial is x^18 - x^9 + 1 which is trinomial, so 54 is in the sequence.
From _Gus Wiseman_, Nov 09 2023: (Start)
The terms and conjugate semiprimes, showing their respective Heinz partitions, begin:
    3: (2)              4: (1,1)
    6: (2,1)            6: (2,1)
    9: (2,2)            9: (2,2)
   12: (2,1,1)         10: (3,1)
   18: (2,2,1)         15: (3,2)
   24: (2,1,1,1)       14: (4,1)
   27: (2,2,2)         25: (3,3)
   36: (2,2,1,1)       21: (4,2)
   48: (2,1,1,1,1)     22: (5,1)
   54: (2,2,2,1)       35: (4,3)
   72: (2,2,1,1,1)     33: (5,2)
   81: (2,2,2,2)       49: (4,4)
   96: (2,1,1,1,1,1)   26: (6,1)
(End)

Jean-Marie De Koninck and Armel Mercier, 1001 Problèmes en Théorie Classique Des Nombres, Problem 733, pp. 74 and 310, Ellipses Paris, 2004.

Robert Israel, Table of n, a(n) for n = 1..10000
Robert Israel, Cyclotomic trinomials

Differs at the 18th term from A063996.
Cf. A003586, A033845, A206787.
For primes (A008578) we have conjugates A000079.
For triprimes (A014612) we have conjugates A080193.
A001358 lists semiprimes, squarefree A006881, complement A100959.
Cf. A000040, A000720, A001248, A046682, A056239, A086971, A122111, A220264.

with(numtheory): a := []; for m from 1 to 3000 do if nops([coeffs(cyclotomic(m,x))])=3 then a := [op(a),m] fi od; print(a);

max = 5000; Sort[Flatten[Table[2^a 3^b, {a, 0, Floor[Log[2, max]]}, {b, Floor[Log[3, max/2^a]]}]]] (* Alonso del Arte, May 19 2016 *)

isok(n)=my(vp = Vec(polcyclo(n))); sum(k=1, #vp, vp[k] != 0) == 3; \\ Michel Marcus, Jul 11 2015

list(lim)=my(v=List(),N); for(n=1,logint(lim\1,3), N=3^n; while(N<=lim, listput(v,N); N<<=1)); Set(v) \\ Charles R Greathouse IV, Aug 07 2015

A206787(a(n)) = 4. - Reinhard Zumkeller, Feb 12 2012
a(n) = A033845(n)/2 = 3 * A003586(n). - Robert Israel, Jul 14 2015
Sum_{n>=1} 1/a(n) = 1. - Amiram Eldar, Nov 10 2020

Offset set to 1 and more terms from Michel Marcus, Jul 11 2015

A071521 Number of 3-smooth numbers <= n.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 5, 6, 7, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18

Offset: 1

Benoit Cloitre, Jun 02 2002

A 3-smooth number is a number of the form 2^x * 3^y where x >= 0 and y >= 0.

Bruce C. Berndt and Robert A. Rankin, "Ramanujan : letters and commentary", History of Mathematics Volume 9, AMS-LMS, p. 23, p. 35.
G. H. Hardy, Ramanujan: Twelve lectures on subjects suggested by his life and work, AMS Chelsea Pub., 1999, pages 67-82.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Thierry Bousch, La Tour de Stockmeyer, Séminaire Lotharingien de Combinatoire 77 (2017), Article B77d.
M. Haussman and H. N. Shapiro, On Ramanujan right triangle conjecture, Comm. Pure Appl. Math. 42 (1989), 885-889.
A. M. Ostrowski, Bemerkungen zur Theorie der Diophantischen Approximationen, Abh. Math. Sem. Univ. Hamburg 1 (1922), 77-98; 250-251.
Raphael Schumacher, The Formulas for the Distribution of the 3-Smooth, 5-Smooth, 7-Smooth and all other Smooth Numbers, arXiv preprint arXiv:1608.06928 [math.NT], 2016.

Cf. A003586.

Cf. A000079, A000244, A022330, A022331, A065331, A322026.

a071521 n = length $ takeWhile (<= n) a003586_list
-- Reinhard Zumkeller, Aug 14 2011

N:= 10000: # to get a(1) to a(N)
V:= Vector(N):
for y from 0 to floor(log[3](N)) do
  for x from 0 to ilog2(N/3^y) do
    V[2^x*3^y]:= 1
od od:
convert(map(round,Statistics:-CumulativeSum(V)),list); # Robert Israel, Dec 16 2014

a[n_] := Sum[ MoebiusMu[6k]*Floor[n/k], {k, 1, n}]; Table[a[n], {n, 1, 75}] (* Jean-François Alcover, Oct 11 2011, after Benoit Cloitre *)
f[n_] := Sum[Floor@Log[3, n/2^i] + 1, {i, 0, Log[2, n]}]; Array[f, 75] (* faster, or *)
f[n_] := Sum[Floor@Log[2, n/3^i] + 1, {i, 0, Log[3, n]}]; Array[f, 75] (* Robert G. Wilson v, Aug 18 2012 *)
Accumulate[Table[If[Max[FactorInteger[n][[All,1]]]<4,1,0],{n,80}]] (* Harvey P. Dale, Jan 11 2017 *)

for(n=1,100,print1(sum(k=1,n,if(sum(i=3,n,if(k%prime(i),0,1)),0,1)),","))

a(n)=sum(k=1,n,moebius(2*3*k)*floor(n/k)) \\ Benoit Cloitre, Jun 14 2007

a(n)=my(t=1/3); sum(k=0,logint(n,3), t*=3; logint(n\t,2)+1) \\ Charles R Greathouse IV, Jan 08 2018

from sympy import integer_log
def A071521(n): return sum((n//3**i).bit_length() for i in range(integer_log(n,3)[0]+1)) # Chai Wah Wu, Sep 15 2024

a(n) = Card{ k | A003586(k) <= n }. Asymptotically: let a=1/(2*log(2)*log(3)), b=sqrt(6), then from Ramanujan a(n) ~ a*log(2*n)*log(3*n) or equivalently a(n) ~ a*log(b*n)^2.
A022331(n) = a(A000079(n)); A022330(n) = a(A000244(n)). - Reinhard Zumkeller, May 09 2006
a(n) = Sum_{k=1..n} mu(6k)*floor(n/k). - Benoit Cloitre, Jun 14 2007
a(n) = Sum_{k=1..n} (floor(6^k/k)-floor((6^k-1)/k)). - Anthony Browne, May 19 2016
From Ridouane Oudra, Jul 17 2020: (Start)
a(n) = Sum_{i=0..floor(log_2(n))} (floor(log_3(n/2^i)) + 1).
a(n) = Sum_{i=0..floor(log_3(n))} (floor(log_2(n/3^i)) + 1). (End)
A322026(n) = a(A065331(n)). - Antti Karttunen, Sep 08 2024

A064553 a(1) = 1, a(prime(i)) = i + 1 for i > 0 and a(u * v) = a(u) * a(v) for u, v > 0.

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 5, 8, 9, 8, 6, 12, 7, 10, 12, 16, 8, 18, 9, 16, 15, 12, 10, 24, 16, 14, 27, 20, 11, 24, 12, 32, 18, 16, 20, 36, 13, 18, 21, 32, 14, 30, 15, 24, 36, 20, 16, 48, 25, 32, 24, 28, 17, 54, 24, 40, 27, 22, 18, 48, 19, 24, 45, 64, 28, 36, 20, 32, 30, 40, 21, 72, 22, 26

Offset: 1

Reinhard Zumkeller, Sep 21 2001

a(n) <= n for all n and a(x) = x iff x = 2^i * 3^j for i, j >= 0: a(A003586(n)) = A003586(n) for n > 0. By definition a is completely multiplicative and also surjective. a(p) < a(q) for primes p < q.
Completely multiplicative with a(prime(i)) = i + 1. - Charles R Greathouse IV, Sep 07 2012
a(A080688(n,k)) = A080444(n,k) = n for k=1..A001055(n). - Reinhard Zumkeller, Oct 01 2012

			a(5) = a(prime(3)) = 3 + 1 = 4; a(14) = a(2*7) = a(prime(1)* prime(4)) = (1+1)*(4+1) = 10.

T. D. Noe, Table of n, a(n) for n = 1..8000
T. D. Noe, Plot of A064553
Index to divisibility sequences
Index entries for sequences computed from indices in prime factorization

Cf. A000005, A000040, A003961, A003963, A049084, A020639, A064554, A064555, A001055, A003586, A064557, A064558, A027746, A027748, A124010, A181819.

A left inverse of A290641.

a064553 1 = 1
a064553 n = product $ map ((+ 1) . a049084) $ a027746_row n
-- Reinhard Zumkeller, Apr 09 2012, Feb 17 2012, Jan 28 2011

A064553 := proc(n)
    local a,f,p,e ;
    a := 1 ;
    for f in ifactors(n)[2] do
        p :=op(1,f) ;
        e :=op(2,f) ;
        a := a*(numtheory[pi](p)+1)^e ;
    end do:
    a ;
end proc: # R. J. Mathar, Sep 07 2012

nn=100; a=Table[0, {nn}]; a[[1]]=1; Do[If[PrimeQ[i], a[[i]]=PrimePi[i]+1, p=FactorInteger[i][[1,1]]; a[[i]] = a[[p]]*a[[i/p]]], {i, 2, nn}]; a (* T. D. Noe, Dec 12 2004, revised Sep 27 2011 *)
Array[Apply[Times, Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger[ #]] /. p_ /; PrimeQ@ p :> PrimePi@ p + 1] &, 74] (* Michael De Vlieger, Aug 22 2017 *)

A064553(n)={n=factor(n);n[,1]=apply(f->1+primepi(f),n[,1]);factorback(n)} \\ M. F. Hasler, Aug 28 2012

(define (A064553 n) (if (= 1 n) n (* (+ 1 (A055396 n)) (A064553 (A032742 n))))) ;; Antti Karttunen, Aug 22 2017

a(A000040(n)) = n+1.
Let the prime factorization of n be p1^e1...pk^ek, then a(n) = (pi(p1)+1)^e1...(pi(pk)+1)^ek, where pi(p) is the index of prime p. - T. D. Noe, Dec 12 2004
From Antti Karttunen, Aug 22 2017: (Start)
a(n) = A003963(A003961(n)).
a(A181819(n)) = A000005(n).
a(A290641(n)) = n. (End)

Displayed values double-checked with new PARI code by M. F. Hasler, Aug 28 2012

A065338 a(1) = 1, a(p) = p mod 4 for p prime and a(u * v) = a(u) * a(v) for u, v > 0.

Original entry on oeis.org

1, 2, 3, 4, 1, 6, 3, 8, 9, 2, 3, 12, 1, 6, 3, 16, 1, 18, 3, 4, 9, 6, 3, 24, 1, 2, 27, 12, 1, 6, 3, 32, 9, 2, 3, 36, 1, 6, 3, 8, 1, 18, 3, 12, 9, 6, 3, 48, 9, 2, 3, 4, 1, 54, 3, 24, 9, 2, 3, 12, 1, 6, 27, 64, 1, 18, 3, 4, 9, 6, 3, 72, 1, 2, 3, 12, 9, 6, 3, 16, 81, 2, 3, 36, 1, 6, 3, 24, 1, 18, 3

Offset: 1

Reinhard Zumkeller, Oct 29 2001

			a(120) = a(2*2*2*3*5) = a(2)*a(2)*a(2)*a(3)*a(5) = 2*2*2*3*1 = 24.
a(150) = a(2*3*5*5) = a(2)*a(3)*a(5)*a(5) = 2*3*1*1 = 6.
a(210) = a(2*3*5*7) = a(2)*a(3)*a(5)*a(7) = 2*3*1*3 = 18.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A039702, A000040, A003586, A007814, A065339, A065331.

a065338 1 = 1
a065338 n = (spf `mod` 4) * a065338 (n `div` spf) where spf = a020639 n
-- Reinhard Zumkeller, Nov 18 2011

a[1] = 1; a[n_] := a[n] = Mod[p = FactorInteger[n][[1, 1]], 4]*a[n/p]; Table[ a[n], {n, 1, 100} ] (* Jean-François Alcover, Jan 20 2012 *)

a(n)=my(f=factor(n)); prod(i=1,#f~, (f[i,1]%4)^f[i,2]) \\ Charles R Greathouse IV, Feb 07 2017

a(n) = 1 if n = 1, otherwise (A020639(n) mod 4) * n / A020639(n).
a(n) = (2^A007814(n)) * (3^A065339(n)).
a(n) <= n.
a(a(n)) = a(n).
a(x) = x iff x = 2^i * 3^j for i, j >= 0.
a(A003586(n)) = A003586(n).
a(A065331(n)) = A065331(n).
a(A004613(n)) = 1; A065333(a(n)) = 1. - Reinhard Zumkeller, Jul 10 2010
Dirichlet g.f.: (1/(1-2^(-s+1))) * Product_{prime p=4k+1} (1/(1-p^(-s))) * Product_{prime p=4k+3} 1/(1-3*p^(-s)). - Ralf Stephan, Mar 28 2015

A108090 Numbers of the form (11^i)*(13^j).

Original entry on oeis.org

1, 11, 13, 121, 143, 169, 1331, 1573, 1859, 2197, 14641, 17303, 20449, 24167, 28561, 161051, 190333, 224939, 265837, 314171, 371293, 1771561, 2093663, 2474329, 2924207, 3455881, 4084223, 4826809, 19487171, 23030293, 27217619

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Jun 03 2005

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A003586, A003592, A003593, A003591, A003594, A003595, A003596, A003597, A003598, A003599, A107326, A107364, A107466, A108056.

import Data.Set (singleton, deleteFindMin, insert)
a108090 n = a108090_list !! (n-1)
a108090_list = f $ singleton (1,0,0) where
   f s = y : f (insert (11 * y, i + 1, j) $ insert (13 * y, i, j + 1) s')
         where ((y, i, j), s') = deleteFindMin s
-- Reinhard Zumkeller, May 15 2015

[n: n in [1..10^7] | PrimeDivisors(n) subset [11, 13]]; // Vincenzo Librandi, Jun 27 2016

mx = 3*10^7; Sort@ Flatten@ Table[ 11^i*13^j, {i, 0, Log[11, mx]}, {j, 0, Log[13, mx/11^i]}] (* Robert G. Wilson v, Aug 17 2012 *)
fQ[n_]:=PowerMod[143, n, n] == 0; Select[Range[2 10^7], fQ] (* Vincenzo Librandi, Jun 27 2016 *)

list(lim)=my(v=List(),t); for(j=0,logint(lim\=1,13), t=13^j; while(t<=lim, listput(v,t); t*=11)); Set(v) \\ Charles R Greathouse IV, Aug 29 2016

from sympy import integer_log
def A108090(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = 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(integer_log(x//13**i,11)[0]+1 for i in range(integer_log(x,13)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Mar 25 2025

Sum_{n>=1} 1/a(n) = (11*13)/((11-1)*(13-1)) = 143/120. - Amiram Eldar, Sep 23 2020

a(n) ~ exp(sqrt(2*log(11)*log(13)*n)) / sqrt(143). - Vaclav Kotesovec, Sep 23 2020

cp's OEIS Frontend

A062051 Number of partitions of n into powers of 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A065333 Characteristic function of 3-smooth numbers, i.e., numbers of the form 2^i*3^j (i, j >= 0).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

Python

Formula

A002072 a(n) = smallest number m such that for all k > m, either k or k+1 has a prime factor > prime(n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A003594 Numbers of the form 3^i*7^j with i, j >= 0.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Haskell

Magma

Mathematica

PARI

Python

Formula

A003595 Numbers of the form 5^i*7^j with i, j >= 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

Python

Formula

A065119 Numbers k such that the k-th cyclotomic polynomial is a trinomial.

Views

Author

Keywords

Comments

Examples

References

Links