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 1-4 of 4 results.

A228957 Numbers n such that n/rad(n) is greater than the greatest prime dividing n.

Original entry on oeis.org

8, 16, 24, 27, 32, 36, 48, 54, 64, 72, 80, 81, 96, 100, 108, 112, 125, 128, 135, 144, 160, 162, 180, 189, 192, 196, 200, 216, 224, 225, 240, 243, 250, 256, 270, 288, 300, 320, 324, 336, 343, 352, 360, 375, 378, 384, 392, 400, 405, 416, 432, 441, 448, 450, 480
Offset: 1

Views

Author

Michel Lagneau, Sep 09 2013

Keywords

Comments

n such that n/rad(n)> gpf(n); numbers n such that n/A007947(n) > A006530(n) where A007947 is the product of the distinct prime factors of n and A006530 is the greatest prime dividing n.
The sequence A137845 (logarithmically smooth numbers)is included in this sequence.
It appears that there exists consecutive numbers such that (80,81), (224,225), (675,676), (1088,1089), (1215,1216), (2375,2376), (2400,2401), (2600, 2601), (3024,3025), (3249,3250), (3968,3969), (4224,4225), (4374,4375), (5831,5832),...
But it appears also that (2400,2401) and (4374,4375) are the only consecutive numbers in the sequence A137845.

Examples

			24 is in the sequence because the prime divisors of 24 are 2 and 3 and 24/2*3 > 3.

Links

Crossrefs

Cf. A006530, A007947, A137845.
A366250 is a subsequence.

Programs

  • Maple
    with(numtheory) :for n from 1 to 400 do:x:=factorset(n):n1:=nops(x): p:= product('x[i]', 'i'=1..n1):m:=n/p:if m> x[n1]then printf(`%d, `,n):else fi:od:
  • Mathematica
    rad[n_]:=Times@@(First@#&/@FactorInteger@n);Select[Range[2,1000],FactorInteger[#][[-1,1]]<#/rad[#]&]
nrQ[n_]:=Module[{x=FactorInteger[n][[All,1]]},n/Times@@x>Last[x]]; Select[Range[ 500],nrQ] (* Harvey P. Dale, Jun 15 2022 *)
  • PARI
    is(n)=my(f=factor(n)); prod(i=1,#f~,f[i,1]^(f[i,2]-1)) > f[#f~,1] \\ Charles R Greathouse IV, Sep 09 2013

A230400 Numbers n such that n = abc = 2(ab+ac+bc) for some positive integers a,b,c.

Original entry on oeis.org

216, 250, 256, 288, 400, 432, 450, 486, 576, 882
Offset: 1

Views

Author

M. F. Hasler, Oct 18 2013

Keywords

Comments

Otherwise said: Volumes of integer-sided cubes equal to their surface area (assuming dimensionless unit of length).
The sequence is a finite subsequence of A055744, A069167, A073539, A090779 and A137845.

Examples

			The triples (a,b,c) ordered by largest member(s) are (6,6,6), (8,8,4), (10,5,5), (12,6,4), (12,12,3), (15,10,3), (18,9,3), (20,5,4), (24,8,3), (42,7,3).

Links

Crossrefs

Cf. A229941.

Programs

  • PARI
    L=[];for(a=1,99,for(b=1,a,for(c=1,b,a*b*c==2*(a*b+b*c+a*c)&&!printf("(%d,%d,%d), ",a,b,c)&&L=concat(L,a*b*c))));vecsort(L)

A333534 a(n) is the number of log(n)-smooth numbers <= n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 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, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19
Offset: 2

Views

Author

N. J. A. Sloane, Apr 08 2020

Keywords

Comments

Number of k <= n such that the greatest prime factor of k is <= log(n).

Links

Crossrefs

Cf. A001671, A006530, A051102, A137845, A295084.

Programs

  • Maple
    A333534 := n -> nops(select(k -> A006530(k) <= ilog(n), [$1..n])):
seq(A333534(n), n=2..86); # Peter Luschny, Apr 09 2020
# second Maple program:
b:= proc(n) option remember; max(1, map(i-> i[1], ifactors(n)[2])) end:
a:= n-> (t-> add(`if`(b(i)<= t, 1, 0), i=1..n))(ilog(n)):
seq(a(n), n=2..100);  # Alois P. Heinz, Apr 09 2020
  • Mathematica
    a[n_] := Select[Range[n], FactorInteger[#][[-1, 1]] <= Log[n]&] // Length;
a /@ Range[2, 100] (* Jean-François Alcover, May 17 2020 *)
  • PARI
    gpf(j)={if(j==1,1,my(f=factor(j));f[#f[,2],1])};
for(n=2,80,my(L=log(n));print1(sum(k=1,n,gpf(k)<=L),", ")) \\ Hugo Pfoertner, Apr 09 2020
  • PARI
    sm(lim, p)=if(p==2, return(logint(lim\1, 2)+1)); my(s=0, q=precprime(p-1), t=1); for(e=0, logint(lim\=1, p), s+=sm(lim\t, q); t*=p); s
a(n)=if(n<8,return(n>2)); sm(n, precprime(log(n))) \\ Charles R Greathouse IV, Apr 16 2020

Formula

a(n) = A096300(n), n>2. - R. J. Mathar, Apr 27 2020

A333961 Numbers k with property that k is the least logarithmically smooth numbers (meaning largest prime factor of k is less than log(k)) having squarefree kernel equal to squarefree kernel of k.

Original entry on oeis.org

8, 24, 27, 150, 160, 225, 625, 1120, 1134, 1225, 1260, 1323, 1372, 1575, 2401, 59895, 60368, 60500, 60984, 61600, 61952, 62370, 63360, 63525, 63888, 65219, 67375, 68607, 72171, 73205, 161051, 443625, 445900, 446160, 446875, 447174, 448448, 449280, 449878, 450450
Offset: 1

Views

Author

David A. Corneth, Apr 12 2020

Keywords

Comments

Primitive sequence underlying A137845. If k is in this sequence then any number larger than k having the same squarefree kernel is in the sequence.

Examples

			8 is in the sequence as the largest prime factor of 8 is 2 and log(8) > 2 and 8 is the least positive integer with this property and squarefree kernel 2.

Crossrefs

Cf. A007947, A137845.

Programs

  • Mathematica
    rads = seq = {}; Do[rad = Times @@ (p = First /@ FactorInteger[n]); pmax = Max[p]; If[pmax < Log[n] && ! MemberQ[rads, rad], AppendTo[rads, rad]; AppendTo[seq, n]], {n, 1, 5*10^5}]; seq (* Amiram Eldar, Apr 12 2020 *)
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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