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.
a(6)=10: A136743(10) = A001221(A136742(10)) = A001221(39270) = A001221(2*3*5*7*11*17) = 6.
a005117 n = a005117_list !! (n-1) a005117_list = filter ((== 1) . a008966) [1..] -- Reinhard Zumkeller, Aug 15 2011, May 10 2011
[ n : n in [1..1000] | IsSquarefree(n) ];
with(numtheory); a := [ ]; for n from 1 to 200 do if issqrfree(n) then a := [ op(a), n ]; fi; od: t:= n-> product(ithprime(k),k=1..n): for n from 1 to 113 do if(t(n) mod n = 0) then print(n) fi od; # Gary Detlefs, Dec 07 2011 A005117 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do if numtheory[issqrfree](a) then return a; end if; end do: end if; end proc: # R. J. Mathar, Jan 09 2013
Select[ Range[ 113], SquareFreeQ] (* Robert G. Wilson v, Jan 31 2005 *) Select[Range[150], Max[Last /@ FactorInteger[ # ]] < 2 &] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 26 2006 *) NextSquareFree[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sf = n + sgn; While[c < Abs[k], While[ ! SquareFreeQ@ sf, If[sgn < 0, sf--, sf++]]; If[ sgn < 0, sf--, sf++]; c++]; sf + If[ sgn < 0, 1, -1]]; NestList[ NextSquareFree, 1, 70] (* Robert G. Wilson v, Apr 18 2014 *) Select[Range[250], MoebiusMu[#] != 0 &] (* Robert D. Rosales, May 20 2024 *)
bnd = 1000; L = vector(bnd); j = 1; for (i=1,bnd, if(issquarefree(i),L[j]=i; j=j+1)); L
{a(n)= local(m,c); if(n<=1,n==1, c=1; m=1; while( cMichael Somos, Apr 29 2005 */
list(n)=my(v=vectorsmall(n,i,1),u,j); forprime(p=2,sqrtint(n), forstep(i=p^2, n, p^2, v[i]=0)); u=vector(sum(i=1,n,v[i])); for(i=1,n,if(v[i],u[j++]=i)); u \\ Charles R Greathouse IV, Jun 08 2012
for(n=1, 113, if(core(n)==n, print1(n, ", "))); \\ Arkadiusz Wesolowski, Aug 02 2016
S(n) = my(s); forsquarefree(k=1,sqrtint(n),s+=n\k[1]^2*moebius(k)); s; a(n) = my(min=1, max=231, k=0, sc=0); if(n >= 144, min=floor(zeta(2)*n - 5*sqrt(n)); max=ceil(zeta(2)*n + 5*sqrt(n))); while(min <= max, k=(min+max)\2; sc=S(k); if(abs(sc-n) <= sqrtint(n), break); if(sc > n, max=k-1, if(sc < n, min=k+1, break))); while(!issquarefree(k), k-=1); while(sc != n, my(j=1); if(sc > n, j = -1); k += j; sc += j; while(!issquarefree(k), k += j)); k; \\ Daniel Suteu, Jul 07 2022
first(n)=my(v=vector(n),i); forsquarefree(k=1,if(n<268293,(33*n+30)\20,(n*Pi^2/6+0.058377*sqrt(n))\1), if(i++>n, return(v)); v[i]=k[1]); v \\ Charles R Greathouse IV, Jan 10 2023
A5117=[1..3]; A005117(n)={if(n>#A5117, my(N=#A5117); A5117=Vec(A5117, max(n+999, N*5\4)); iferr(forsquarefree(k=A5117[N]+1, #A5117*Pi^2\6+sqrtint(#A5117)\17+11, A5117[N++]=k[1]),E,)); A5117[n]} \\ M. F. Hasler, Aug 08 2025
from sympy.ntheory.factor_ import core def ok(n): return core(n, 2) == n print(list(filter(ok, range(1, 114)))) # Michael S. Branicky, Jul 31 2021
from itertools import count, islice from sympy import factorint def A005117_gen(startvalue=1): # generator of terms >= startvalue return filter(lambda n:all(x == 1 for x in factorint(n).values()),count(max(startvalue,1))) A005117_list = list(islice(A005117_gen(),20)) # Chai Wah Wu, May 09 2022
from math import isqrt from sympy import mobius def A005117(n): def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)) m, k = n, f(n) while m != k: m, k = k, f(k) return m # Chai Wah Wu, Jul 22 2024
1 begins a new chain 1, 2, 3 of squarefree integers. 4 is not squarefree. Then 5 begins a new chain 5, 6, 7 of squarefree integers. Hence 1 and 5 are terms of the sequence.
Select[Range[100], MoebiusMu[# - 1] == 0 && Abs[MoebiusMu[#]] == 1 &] (* Amiram Eldar, Feb 14 2021 *) SequencePosition[Table[If[SquareFreeQ[n],1,0],{n,0,250}],{0,1}][[All,2]]-1 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 24 2021 *)
n=1; for(k=1,100, while(!issquarefree(n),n=n+1); print1(n","); while(issquarefree(n),n=n+1))
The runs of squarefree integers are as follows: (1,2,3), (5,6,7), (10,11), (13,14,15), (17), (19), (21,22,23),...
with(numtheory): a:=proc(n) if mobius(n)=0 then n else fi end: A:=[0,seq(a(n),n=1..500)]: b:=proc(n) if A[n]-A[n-1]>1 then A[n]-A[n-1]-1 else fi end: seq(b(n),n=2..nops(A)); # Emeric Deutsch, Jul 24 2006
t = {}; cnt = 0; Do[If[SquareFreeQ[n], cnt++, If[cnt > 0, AppendTo[t, cnt]; cnt = 0]], {n, 500}]; t (* T. D. Noe, Mar 19 2013 *)
n=1; while(n<1000, c=0; while(issquarefree(n), n++; c++); print1(c,", "); while(!issquarefree(n), n++)) \\ Rick L. Shepherd, Jul 25 2006
;; With Antti Karttunen's IntSeq-library. (define (A120992 n) (if (= n 1) (Aincr_points_of_A243348 n) (- (Aincr_points_of_A243348 n) (Aincr_points_of_A243348 (- n 1))))) ;; Using these two auxiliary functions, not submitted separately: (define Aincr_points_of_A243348 (COMPOSE -1+ (NONZERO-POS 1 1 Afirst_diffs_of_A243348))) (define (Afirst_diffs_of_A243348 n) (if (< n 2) (- n 1) (- (A243348 n) (A243348 (- n 1)))))
n=10: A072284(10)=33, A120992(10)=3: a(10)=33*34*35=39270.
R:= NULL:
state:= false:
count:= 0:
for n from 1 while count < 100 do
if numtheory:-issqrfree(n) then
if state then P:= P*n
else P:= n; state:= true
fi
elif state then
R:= R, P;
count:= count+1;
state:= false
fi
od:
R; # Robert Israel, Jan 06 2020
Times @@ # & /@ DeleteCases[#, ?(! SquareFreeQ@ First@ # &)] &@ SplitBy[Range[140], SquareFreeQ] (* _Michael De Vlieger, Jan 06 2020 *)
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