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.
Differences[Table[NestWhile[#+1&,Prime[n],SquareFreeQ[#]&],{n,100}]]
Union[Table[NestWhile[#+1&,Prime[n],SquareFreeQ],{n,100}]]
lns[p_]:=Module[{k=p+1},While[SquareFreeQ[k],k++];k]; Table[lns[p],{p,Prime[Range[70]]}]//Union (* Harvey P. Dale, Jun 12 2025 *)
The terms together with their prime indices begin:
9: {2,2}
25: {3,3}
27: {2,2,2}
28: {1,1,4}
36: {1,1,2,2}
45: {2,2,3}
49: {4,4}
50: {1,3,3}
52: {1,1,6}
56: {1,1,1,4}
64: {1,1,1,1,1,1}
76: {1,1,8}
81: {2,2,2,2}
88: {1,1,1,5}
92: {1,1,9}
96: {1,1,1,1,1,2}
nn=100;
y=Table[NestWhile[#+1&,Prime[n],SquareFreeQ[#]&],{n,nn}];
Complement[Select[Range[Prime[nn]],!SquareFreeQ[#]&],y]
The terms together with their prime indices begin:
12: {1,1,2}
16: {1,1,1,1}
18: {1,2,2}
20: {1,1,3}
24: {1,1,1,2}
40: {1,1,1,3}
48: {1,1,1,1,2}
54: {1,2,2,2}
60: {1,1,2,3}
63: {2,2,4}
68: {1,1,7}
72: {1,1,1,2,2}
75: {2,3,3}
80: {1,1,1,1,3}
84: {1,1,2,4}
90: {1,2,2,3}
98: {1,4,4}
108: {1,1,2,2,2}
112: {1,1,1,1,4}
116: {1,1,10}
128: {1,1,1,1,1,1,1}
132: {1,1,2,5}
q:= 3: R:= NULL: flag:= false: count:= 0:
while count < 100 do
p:= q; q:= nextprime(q);
for k from p+1 to q-1 do
found:= false;
if not numtheory:-issqrfree(k) then
if flag then
count:= count+1; R:= R,k
fi;
found:= true; break
fi;
od;
flag:= found;
od:
R; # Robert Israel, Nov 20 2024
y=Table[NestWhile[#+1&,Prime[n],SquareFreeQ],{n,100}];
Select[Most[Union[y]],Count[y,#]==1&]
The terms together with their prime indices begin:
4: {1,1}
8: {1,1,1}
32: {1,1,1,1,1}
44: {1,1,5}
104: {1,1,1,6}
140: {1,1,3,4}
284: {1,1,20}
464: {1,1,1,1,10}
572: {1,1,5,6}
620: {1,1,3,11}
644: {1,1,4,9}
824: {1,1,1,27}
860: {1,1,3,14}
1232: {1,1,1,1,4,5}
y=Table[NestWhile[#+1&,Prime[n],SquareFreeQ[#]&],{n,1000}];
Select[Union[y],Count[y,#]==2&]
6 is the largest squarefree less than the 4th prime, 7. So a(4) = 6.
with(numtheory): a:=proc(n) local p,B,j: p:=ithprime(n): B:={}: for j from 1 to p-1 do if abs(mobius(j))>0 then B:=B union {j} else B:=B fi od: B[nops(B)] end: seq(a(m),m=1..75); # Emeric Deutsch, Oct 14 2005
With[{k = 120}, Table[SelectFirst[Range[Prime@ n - 1, Prime@ n - Min[Prime@ n - 1, k], -1], SquareFreeQ], {n, 60}]] (* Michael De Vlieger, Aug 16 2017 *)
a(n,p=prime(n))=while(!issquarefree(p--),); p \\ Charles R Greathouse IV, Aug 16 2017
10 is the smallest squarefree number greater than the 4th prime, 7. So a(4) = 10. From _Gus Wiseman_, Dec 07 2024: (Start) The first number line below shows the squarefree numbers. The second shows the primes: --1--2--3-----5--6--7-------10-11----13-14-15----17----19----21-22-23-------26-- =====2==3=====5=====7==========11====13==========17====19==========23=========== (End)
with(numtheory): a:=proc(n) local p,B,j: p:=ithprime(n): B:={}: for j from p+1 to p+20 do if abs(mobius(j))>0 then B:=B union {j} else B:=B fi od: B[1] end: seq(a(m),m=1..75); # Emeric Deutsch, Oct 10 2005
Do[k = Prime[n] + 1; While[ !SquareFreeQ[k], k++ ]; Print[k], {n, 1, 100}] (* Ryan Propper, Oct 10 2005 *)
With[{k = 120}, Table[SelectFirst[Range[Prime@ n + 1, Prime@ n + k], SquareFreeQ], {n, 58}]] (* Michael De Vlieger, Aug 16 2017 *)
a(n,p=prime(n))=while(!issquarefree(p++),); p \\ Charles R Greathouse IV, Aug 16 2017
Between 113 and 127 the 7 numbers which are not squarefree are {116,117,120,121,124,125,126}, so a(30)=7.
From _Gus Wiseman_, Dec 07 2024: (Start)
The a(n) nonsquarefree numbers for n = 1..15:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
----------------------------------------------------------
. 4 . 8 12 16 18 20 24 . 32 40 . 44 48
9 25 36 45 49
27 50
28 52
(End)
Count[Range[#[[1]]+1,#[[2]]-1],?(!SquareFreeQ[#]&)]&/@Partition[Prime[Range[120]],2,1] (* _Harvey P. Dale, Mar 31 2024 *)
{ n=0; q=2; forprime (p=3, prime(1001), a=0; for (i=q+1, p-1, a+=!issquarefree(i)); write("b061399.txt", n++, " ", a); q=p ) } \\ Harry J. Smith, Jul 22 2009
a(n) = my(p=prime(n)); sum(k=p, nextprime(p+1), ! issquarefree(k)); \\ Michel Marcus, Dec 09 2024
from sympy import mobius, prime def A061399(n): return sum(not mobius(m) for m in range(prime(n)+1,prime(n+1))) # Chai Wah Wu, Jul 20 2024
From _Gus Wiseman_, Dec 10 2024: (Start)
The squarefree numbers <= n are the following columns, with maxima a(n):
1 2 3 3 5 6 7 7 7 10 11 11 13 14 15 15
1 2 2 3 5 6 6 6 7 10 10 11 13 14 14
1 1 2 3 5 5 5 6 7 7 10 11 13 13
1 2 3 3 3 5 6 6 7 10 11 11
1 2 2 2 3 5 5 6 7 10 10
1 1 1 2 3 3 5 6 7 7
1 2 2 3 5 6 6
1 1 2 3 5 5
1 2 3 3
1 2 2
1 1
(End)
A070321 := proc(n) local a; for a from n by -1 do if issqrfree(a) then return a; end if; end do: end proc: seq(A070321(n),n=1..100) ; # R. J. Mathar, May 25 2023
a[n_] :=For[ k = n, True, k--, If[ SquareFreeQ[k], Return[k]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Mar 27 2013 *)
gsfn[n_]:=Module[{k=n},While[!SquareFreeQ[k],k--];k]; Array[gsfn,80] (* Harvey P. Dale, Mar 27 2013 *)
a(n) = while (! issquarefree(n), n--); n; \\ Michel Marcus, Mar 18 2017
from itertools import count from sympy import factorint def A070321(n): return next(m for m in count(n,-1) if max(factorint(m).values(),default=0)<=1) # Chai Wah Wu, Dec 04 2024
lpp[n_]:=Module[{k=n-1},While[!PrimePowerQ[k],k--];k]; Join[{1},Table[ lpp[ n],{n,Prime[Range[2,60]]}]] (* Harvey P. Dale, Nov 24 2018 *)
from sympy import factorint, prime def A065514(n): return next(filter(lambda m:len(factorint(m))<=1, range(prime(n)-1,0,-1))) # Chai Wah Wu, Oct 25 2024
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