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&,n,#>1&&!SquareFreeQ[#]&],{n,100}]]
The prime indices of 39 are {2,6}, so a(39) = 2.
The prime indices of 70 are {1,3,4}, so a(70) = 2.
The prime indices of 98 are {1,4,4}, so a(98) = 1.
The prime indices of 294 are {1,2,4,4}, a(294) = 2.
The prime indices of 1911 are {2,4,4,6}, so a(1911) = 2.
The prime indices of 2548 are {1,1,4,4,6}, so a(2548) = 3.
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[prix[n],SquareFreeQ]],{n,100}]
For n = 9, there are numbers 2, 3, 5, 6 and 7 which are not divisible by the square of their largest prime factor, while 1 is excluded (no prime factors) and 4 and 8 are divisible both by 2^2 and 9 is divisible by 3^2. Thus a(9) = 5.
ndsQ[n_]:=Mod[n,Max[Select[Divisors[n],!CompositeQ[#]&]]^2]!=0; Accumulate[Table[If[ ndsQ[n],1,0],{n,80}]] (* Harvey P. Dale, Oct 14 2023 *)
from sympy import primefactors def a243285(n): return 0 if n==1 else sum([1 for k in range(2, n + 1) if k%(primefactors(k)[-1]**2)!=0]) # Indranil Ghosh, Jun 15 2017
(define (A243285 n) (- n (A243283 n)))
f[p_, e_] := Count[Range[e], ?SquareFreeQ] + 1; a[1] = 1; a[n] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
s(n) = sum(k=1, n, issquarefree(k)) + 1; a(n) = vecprod(apply(x -> s(x), factor(n)[, 2]));
a(4)=2; there are two partitions of 4 into two parts: (3,1) and (2,2). Both of the larger parts are squarefree, thus a(4)=2. a(5)=1; there are two partitions of 5 into two parts: (4,1) and (3,2). Among the larger parts, only 3 is squarefree, thus a(5)=1.
with(numtheory): A262868:=n->add(mobius(n-i)^2, i=1..floor(n/2)): seq(A262868(n), n=1..100);
Table[Sum[MoebiusMu[n - i]^2, {i, Floor[n/2]}], {n, 100}]
Table[Count[IntegerPartitions[n,{2}][[All,1]],?SquareFreeQ],{n,80}] (* _Harvey P. Dale, Jan 03 2022 *)
a(n) = sum(i=1, n\2, moebius(n-i)^2); \\ Michel Marcus, Oct 04 2015
f(n)=my(s); forfactored(k=1,sqrtint(n),s+=n\k[1]^2*moebius(k)); s a(n)=n--; f(n) - f(n\2) \\ Charles R Greathouse IV, Nov 04 2017
a(5)=2; there are two partitions of 5 into two parts: (4,1) and (3,2). Both of the smaller parts are squarefree, thus a(5)=2. a(6)=3; there are three partitions of 6 into two parts: (5,1), (4,2) and (3,3). Among the three smaller parts, all are squarefree, thus a(6)=3.
with(numtheory): A262869:=n->add(mobius(i)^2, i=1..floor(n/2)): seq(A262869(n), n=1..100);
Table[Sum[MoebiusMu[i]^2, {i, Floor[n/2]}], {n, 100}]
Table[Count[IntegerPartitions[n,{2}][[All,2]],?SquareFreeQ],{n,80}] (* _Harvey P. Dale, Oct 17 2021 *)
a(n) = sum(i=1, n\2, moebius(i)^2); \\ Michel Marcus, Oct 04 2015
a(n)=my(s); n\=2; forsquarefree(k=1, sqrtint(n), s += n\k[1]^2*moebius(k)); s \\ Charles R Greathouse IV, Jan 08 2018
a(7)=6, because the set {1,2,3,4,5,7} has no three distinct elements multiplying to a square, but {1,2,3,4,5,6,7} has 2*3*6 = 6^2.
\\ See PARI link
from math import isqrt
def is_square(n):
return isqrt(n) ** 2 == n
def valid_subset(A):
length = len(A)
for i in range(length):
for j in range(i + 1, length):
for k in range(j + 1, length):
if is_square(A[i] * A[j] * A[k]):
return False
return True
def largest_subset_size(N):
from itertools import combinations
max_size = 0
for size in range(1, N + 1):
for subset in combinations(range(1, N + 1), size):
if valid_subset(subset):
max_size = max(max_size, size)
return max_size
for N in range(1, 11):
print(largest_subset_size(N))
from math import prod from functools import lru_cache from itertools import combinations from sympy.ntheory.primetest import is_square @lru_cache(maxsize=None) def A372306(n): if n==1: return 1 i = A372306(n-1)+1 if sum(1 for p in combinations(range(1,n),2) if is_square(n*prod(p))) > 0: a = [set(p) for p in combinations(range(1,n+1),3) if is_square(prod(p))] for q in combinations(range(1,n),i-1): t = set(q)|{n} if not any(s<=t for s in a): return i else: return i-1 else: return i # Chai Wah Wu, May 30 2024
Table[MoebiusMu[n]^2 * 3^PrimeNu[n], {n, 1, 100}]
for(n=1, 100, print1(direuler(p=2, n, (1 + 3*X))[n], ", "))
for(n=1, 100, print1(direuler(p=2, n, (1 - 6*X^2 + 8*X^3 - 3*X^4)/(1 - X)^3)[n], ", "))
with(numtheory); M:=10000; c:=0; for n from 1 to M do if mobius(n) = 1 then c:=c+1; fi; lprint(n,c); od; # N. J. A. Sloane, Sep 14 2008
a[n_] := If[MoebiusMu[n] == 1, 1, 0]; Accumulate@ Array[a, 100] (* Amiram Eldar, Oct 01 2023 *)
for(n=1,150,print1(sum(i=1,n,if(moebius(i)-1,0,1)),","))
ListTools:-PartialSums([seq(-min(numtheory:-mobius(n),0),n=1..100)]); # Robert Israel, Jan 08 2018
a[n_]:=Sum[Boole[MoebiusMu[k]==-1],{k,n}]; Array[a,78] (* Stefano Spezia, Jan 30 2023 *)
for(n=1,150,print1(sum(i=1,n,if(moebius(i)+1,0,1)),","))
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