1,2

Sometimes called the Hamming sequence, since Hamming asked for an efficient algorithm to generate the list, in ascending order, of all numbers of the form 2^i*3^j*5^k for i,j,k >= 0. The problem was popularized by Edsger Dijkstra.

Numbers k such that 8*k = EulerPhi(30*k). -Artur Jasinski,Nov 05 2008

Where record values greater than 1 occur inA165704:A165705(n) =A165704(a(n)). -Reinhard Zumkeller,Sep 26 2009

A051916is a subsequence. -Reinhard Zumkeller,Mar 20 2010

Also called "harmonic whole numbers", see Howard and Longair, 1982, Table I, page 121. -Hugo Pfoertner,Jul 16 2020

Also called ugly numbers, although it is not clear why. -Gus Wiseman,May 21 2021

Some woody bamboo species have extraordinarily long and stable flowering intervals that belong to this sequence. The model by Veller, Nowak & Davis justifies this observation from the evolutionary point of view. -Andrey Zabolotskiy,Jun 27 2021

Also those integers k for which, for every prime p > 5, p^(4*k) - 1 == 0 (mod 240*k). -Federico Provvedi,May 23 2022

As noted in the comments toA085152,Størmer's theorem implies that the only pairs of consecutive integers that appear as consecutive terms of this sequence are (1,2), (2,3), (3,4), (4,5), (5,6), (8,9), (9,10), (15,16), (24,25), and (80,81). These all represent significant musical intervals. -Hal M. Switkay,Dec 05 2022

Reinhard Zumkeller,Table of n, a(n) for n = 1..10000(first 1000 terms from T. D. Noe)

Benoit Cloitre,Plot of abs(f(n)-s(n)) vs its mean values (blue) and vs loglog(n) (red).

M. J. Dominus,Infinite Lists in Perl.

Deborah Howard and Malcolm Longair,Harmonic Proportion and Palladio's "Quattro Libri",Journal of the Society of Architectural Historians (1982) 41 (2): 116-143.

Vaclav Kotesovec,Plot of a(n) / (exp((6*log(2)*log(3)*log(5)*n)^(1/3))/sqrt(30)) for n = 1..1200000

Rosetta Code,A collection of computer codes to compute 5-smooth numbers.

Raphael Schumacher,The Formula for the Distribution of the 3-Smooth Numbers, 5-Smooth, 7-Smooth and all other Smooth Numbers,arXiv:1608.06928 [math.NT], 2016.

Sci.math,Ugly numbers.

Carl Veller, Martin A. Nowak and Charles C. Davis,Extended flowering intervals of bamboos evolved by discrete multiplication,Ecology Letters, 18 (2015), 653-659.

Eric Weisstein's World of Mathematics,Smooth Number.

Wikipedia,Regular number.

Wikipedia,Talk:Regular number.Includes a discussion of the name.

Wikipedia,Størmer's theorem.

Let s(n) = Card(k | a(k)<n) and f(n) = log(n*sqrt(30))^3/(6*log(2)*log(3)*log(5)). Then s(n) = f(n) + O(log(n)). Conjecture: s(n)=f(n) + O(log log n). For example, s(10000000) = 768 is well approximated by f(10000000) = 769.3... (see graphic given as link). -Benoit Cloitre,Dec 30 2001

The characteristic function of this sequence is given by:

Sum_{n>=1} x^a(n) = Sum_{n>=1} -Möbius(30*n)*x^n/(1-x^n). -Paul D. Hanna,Sep 18 2011

a(n) =A143207(n) / 30. -Reinhard Zumkeller,Sep 13 2011

A204455(15*a(n)) = 15, and only for these numbers. -Wolfdieter Lang,Feb 04 2012

A006530(a(n)) <= 5. -Reinhard Zumkeller,May 16 2015

Sum_{n>=1} 1/a(n) = Product_{primes p <= 5} p/(p-1) = (2*3*5)/(1*2*4) = 15/4. -Amiram Eldar,Sep 22 2020

FromGus Wiseman,May 21 2021: (Start)

The sequence of terms together with their prime indices begins:

1: {} 25: {3,3}

2: {1} 27: {2,2,2}

3: {2} 30: {1,2,3}

4: {1,1} 32: {1,1,1,1,1}

5: {3} 36: {1,1,2,2}

6: {1,2} 40: {1,1,1,3}

8: {1,1,1} 45: {2,2,3}

9: {2,2} 48: {1,1,1,1,2}

10: {1,3} 50: {1,3,3}

12: {1,1,2} 54: {1,2,2,2}

15: {2,3} 60: {1,1,2,3}

16: {1,1,1,1} 64: {1,1,1,1,1,1}

18: {1,2,2} 72: {1,1,1,2,2}

20: {1,1,3} 75: {2,3,3}

24: {1,1,1,2} 80: {1,1,1,1,3}

(End)

A051037:= proc(n)

option remember;

local a;

if n = 1 then

1;

else

for a from procname(n-1)+1 do

numtheory[factorset](a) minus {2, 3, 5 };

if % = {} then

return a;

end if;

end do:

end if;

end proc:

seq(A051037(n), n=1..100); #R. J. Mathar,Nov 05 2017

mx = 405; Sort@ Flatten@ Table[ 2^a*3^b*5^c, {a, 0, Log[2, mx]}, {b, 0, Log[3, mx/2^a]}, {c, 0, Log[5, mx/(2^a*3^b)]}] (* Or *)

Select[ Range@ 405, Last@ Map[First, FactorInteger@ #] < 7 &] (*Robert G. Wilson v*)

With[{nn=10}, Select[Union[Times@@@Flatten[Table[Tuples[{2, 3, 5}, n], {n, 0, nn}], 1]], #<=2^nn&]] (*Harvey P. Dale,Feb 28 2022 *)

(PARI) test(n)= {m=n; forprime(p=2, 5, while(m%p==0, m=m/p)); return(m==1)}

for(n=1, 500, if(test(n), print1(n "," )))

(PARI) a(n)=local(m); if(n<1, 0, n=a(n-1); until(if(m=n, forprime(p=2, 5, while(m%p==0, m/=p)); m==1), n++); n)

(PARI) list(lim)=my(v=List(), s, t); for(i=0, logint(lim\=1, 5), t=5^i; for(j=0, logint(lim\t, 3), s=t*3^j; while(s<=lim, listput(v, s); s<<=1))); Set(v) \\Charles R Greathouse IV,Sep 21 2011; updated Sep 19 2016

(PARI) smooth(P:vec, lim)={ my(v=List([1]), nxt=vector(#P, i, 1), indx, t);

while(1, t=vecmin(vector(#P, i, v[nxt[i]]*P[i]), &indx);

if(t>lim, break); if(t>v[#v], listput(v, t)); nxt[indx]++);

Vec(v)

};

smooth([2, 3, 5], 1e4) \\Charles R Greathouse IV,Dec 03 2013

(PARI) is_A051037(n)=n<7||vecmax(factor(n, 6)[, 1])<7 \\M. F. Hasler,Jan 16 2015

(Magma) [n: n in [1..500] | PrimeDivisors(n) subset [2, 3, 5]]; //Bruno Berselli,Sep 24 2012

(Haskell)

import Data.Set (singleton, deleteFindMin, insert)

a051037 n = a051037_list!! (n-1)

a051037_list = f $ singleton 1 where

f s = y: f (insert (5 * y) $ insert (3 * y) $ insert (2 * y) s')

where (y, s') = deleteFindMin s

--Reinhard Zumkeller,May 16 2015

(Python)

def isok(n):

while n & 1 == 0: n >>= 1

while n % 3 == 0: n //= 3

while n % 5 == 0: n //= 5

return n == 1 #Darío Clavijo,Dec 30 2022

(Python)

from sympy import integer_log

defA051037(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):

c = n+x

for i in range(integer_log(x, 5)[0]+1):

for j in range(integer_log(y:=x//5**i, 3)[0]+1):

c -= (y//3**j).bit_length()

return c

return bisection(f, n, n) #Chai Wah Wu,Sep 16 2024

(Python) # faster for initial segment of sequence

import heapq

from itertools import islice

def A051037gen(): # generator of terms

v, oldv, h, psmooth_primes, = 1, 0, [1], [2, 3, 5]

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(A051037gen(), 65))) #Michael S. Branicky,Sep 17 2024

For p-smooth numbers with other values of p, seeA003586,A002473,A051038,A080197,A080681,A080682,A080683.

Cf.A006530,A112757,A112758,A112759,A112763,A112764,A291719.

Subsequences:A003592,A003593,A257997.

The partitions with these Heinz numbers are counted byA001399.

The conjugate opposite isA033942,counted byA004250.

The opposite isA059485,counted byA004250.

The non-3-smooth case isA080193,counted byA069905.

The conjugate isA037144,counted byA001399.

The complement isA279622,counted byA035300.

Requiring the sum of prime indices to be even givesA344297.

Cf.A000244,A002182,A002183,A035301,A056239,A112798,A261144,A344293.

Sequence in context:A097752A014866A051661*A250089A257997A070023

Adjacent sequences:A051034A051035A051036*A051038A051039A051040

easy,nonn

approved