A368250 -id:A368250 - cp's OEIS frontend

A327527 Number of uniform divisors of n.

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 5, 2, 4, 4, 5, 2, 5, 2, 5, 4, 4, 2, 6, 3, 4, 4, 5, 2, 8, 2, 6, 4, 4, 4, 7, 2, 4, 4, 6, 2, 8, 2, 5, 5, 4, 2, 7, 3, 5, 4, 5, 2, 6, 4, 6, 4, 4, 2, 9, 2, 4, 5, 7, 4, 8, 2, 5, 4, 8, 2, 8, 2, 4, 5, 5, 4, 8, 2, 7, 5, 4, 2, 9, 4, 4, 4, 6, 2, 9, 4, 5, 4, 4, 4, 8, 2, 5, 5, 7, 2, 8, 2, 6, 8

Offset: 1

Gus Wiseman, Sep 17 2019

nonn

A number is uniform if its prime multiplicities are all equal, meaning it is a power of a squarefree number. Uniform numbers are listed in A072774. The maximum uniform divisor of n is A327526(n).

			The uniform divisors of 40 are {1, 2, 4, 5, 8, 10}, so a(40) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Gus Wiseman, Sequences counting and encoding certain classes of multisets.

See link for additional cross-references.
Cf. A000005, A000961, A005117, A006530, A007947, A071625, A112798.
Cf. A072774, A327526.
Cf. A001620, A013661, A306016, A368250, A368251.

Table[Length[Select[Divisors[n],SameQ@@Last/@FactorInteger[#]&]],{n,100}]
a[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, 1 + Total[2^Accumulate[Count[e, #] & /@ Range[Max[e], 1, -1]] - 1]]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Dec 19 2023 *)

isA072774(n) = { ispower(n, , &n); issquarefree(n); }; \\ From A072774
A327527(n) = sumdiv(n,d,isA072774(d)); \\ Antti Karttunen, Nov 13 2021

From Amiram Eldar, Dec 19 2023: (Start)
a(n) = A034444(n) + A368251(n).
Sum_{k=1..n} a(k) ~ (n/zeta(2)) * (log(n) + 2*gamma - 1 - 2*zeta'(2)/zeta(2) + c * zeta(2)), where gamma is Euler's constant (A001620) and c = A368250. (End)

Data section extended up to 105 terms by Antti Karttunen, Nov 13 2021

A072777 Powers of squarefree numbers that are not squarefree.

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 169, 196, 216, 225, 243, 256, 289, 343, 361, 441, 484, 512, 529, 625, 676, 729, 841, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1681, 1764, 1849

Offset: 1

Reinhard Zumkeller, Jul 10 2002

For all n exists k: a(n) = A072774(k) and A072776(k) > 1.

Numbers k such that every prime in the prime factorization of k is raised to the same power > 1; k is a term iff k/A007947(k)^m = 1 for some m > 1. - David James Sycamore, Jun 12 2024

			The number 144 = 12^2 is not a member because 12 is not squarefree.
64 = 2^6 and 49 = 7^2 are members because, though not squarefree, they are powers of the squarefree numbers 2 and 7, respectively. Note that 64 is included even though it is also a square of a nonsquarefree number. - _Stanislav Sykora_, Jul 11 2014

Stanislav Sykora and Reinhard Zumkeller, Table of n, a(n) for n = 1..20000 (first 10000 terms from Reinhard Zumkeller)

Cf. A013929, A368250.
Cf. A005117, subsequence of A001597 and A072774.
Cf. A007947.

import Data.Map (singleton, findMin, deleteMin, insert)
a072777 n = a072777_list !! (n-1)
a072777_list = f 9 (drop 2 a005117_list) (singleton 4 (2, 2)) where
   f vv vs'@(v:ws@(w:_)) m
    | xx < vv = xx : f vv vs' (insert (bx*xx) (bx, ex+1) $ deleteMin m)
    | xx > vv = vv : f (w*w) ws (insert (v^3) (v, 3) m)
    where (xx, (bx, ex)) = findMin m
-- Reinhard Zumkeller, Apr 06 2014

Select[Range[2000], Length[u = Union[FactorInteger[#][[All, 2]]]] == 1 && u[[1]] > 1 &] (* Jean-François Alcover, Mar 27 2013 *)

BelongsToA(n) = {my(f, k, e); if(n == 1, return(0));
f = factor(n); e = f[1, 2]; if(e == 1, return(0));
for(k = 2, #f[, 2], if(f[k, 2] != e, return(0))); return(1);}
Ntest(nmax, test) = {my(k = 1, n = 0, v); v = vector(nmax); while(1, n++; if(test(n), v[k] = n; k++; if(k > nmax, break)); ); return(v); }
a = Ntest(20000, BelongsToA) \\ Note: not very efficient. - Stanislav Sykora, Jul 11 2014

is(n)=ispower(n,,&n) && issquarefree(n) \\ Charles R Greathouse IV, Oct 16 2015

from math import isqrt
from sympy import mobius, integer_nthroot
def A072777(n):
    def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))-1
    def f(x): return n-1+x-sum(g(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length()))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 19 2024

Sum_{n>=1} 1/a(n) = Sum_{n>=2} mu(n)^2/(n*(n-1)) = Sum_{n>=2} (zeta(n)/zeta(2*n) - 1) = 0.8486338679... (A368250). - Amiram Eldar, Jul 22 2020

A368249 a(n) = A002378(A005117(n)-1).

Original entry on oeis.org

0, 2, 6, 20, 30, 42, 90, 110, 156, 182, 210, 272, 342, 420, 462, 506, 650, 812, 870, 930, 1056, 1122, 1190, 1332, 1406, 1482, 1640, 1722, 1806, 2070, 2162, 2550, 2756, 2970, 3192, 3306, 3422, 3660, 3782, 4160, 4290, 4422, 4692, 4830, 4970, 5256, 5402, 5852, 6006

Offset: 1

Amiram Eldar, Dec 19 2023

The squarefree oblong numbers (A229882) are all terms of this sequence, and their relative asymptotic density in it is A065474/A059956 = 0.530711... (A065469).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002378, A005117, A229882.

Cf. A059956, A065469, A065474, A368250.

Table[n*(n - 1), {n, Select[Range[100], SquareFreeQ]}]

lista(kmax) = forsquarefree(k=1, kmax, print1(k[1]*(k[1]-1), ", "));

from math import isqrt
from sympy import mobius
def A368249(n):
    def f(x): return int(n-sum(mobius(k)*(x//k**2) for k in range(2, isqrt(x)+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m*(m-1) # Chai Wah Wu, Dec 23 2024

Sum_{n>=2} 1/a(n) = Sum_{k>=2} (zeta(k)/zeta(2*k) - 1) = 0.848633... (A368250).

A368251 The number of nonsquarefree divisors of n that are powers of squarefree numbers (A072777).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 3, 1, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 1, 0, 0, 1, 5, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 1, 1, 0, 0, 0, 3, 3, 0, 0, 1, 0, 0, 0

Offset: 1

Amiram Eldar, Dec 19 2023

nonn

First differs from A046660 and A066301 at n = 36, and from A183094 at n = 72.

Let b(n, k) be the sequence that counts the divisors of n that are k-th powers of squarefree numbers. Then, b(n, 1) = A034444(n), b(n, 2) = A323308(n), b(n, 3) = A368248(n). b(n, k) is multiplicative with b(p^e, k) = 2 if e >= k, and 1 otherwise. The asymptotic mean of b(n, k) for k >= 2 is lim_{m->oo} (1/m) * Sum_{n=1..m} b(n, k) = zeta(k)/zeta(2*k). Since a(n) = Sum_{k>=2} (b(n, k) - 1), the formula for the asymptotic mean of this sequence follows (see the Formula section).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A048105, A072777, A327527, A368250.
Cf. A034444, A323308, A368248.
Cf. A046660, A066301, A183094.

a[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, 1 + Total[2^Accumulate[Count[e, #] & /@ Range[Max[e], 1, -1]] - 1] - 2^Length[e]]; a[1] = 0; Array[a, 100]

a(n) = {my(f = factor(n), e, m, h, c); if(n == 1, 0, e = f[,2]; m = vecmax(e); h = vector(m); for(i = 1,m, c = 0; for(j = 1, #e, if(e[j] == (m+1-i), c++)); h[i] = c); for(i = 2, m, h[i] += h[i-1]); for(i = 1, m, h[i] = 2^h[i]-1); 1 + vecsum(h) - 1<<#e);}

a(n) = A327527(n) - A034444(n).
a(n) = 0 if and only if n is squarefree (A005117).
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=2} (zeta(k)/zeta(2*k) - 1) = 0.848633... (A368250).

A370494 Oblong numbers of the form (k-1)*k where k is the product of an odd number of distinct primes.

Original entry on oeis.org

2, 6, 20, 42, 110, 156, 272, 342, 506, 812, 870, 930, 1332, 1640, 1722, 1806, 2162, 2756, 3422, 3660, 4290, 4422, 4830, 4970, 5256, 6006, 6162, 6806, 7832, 9312, 10100, 10302, 10506, 10920, 11342, 11772, 11990, 12656, 12882, 16002, 16770, 17030, 18632, 18906, 19182

Offset: 1

Amiram Eldar, Feb 20 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Complement of A370495 within A368249.

Cf. A002378, A030059, A033150, A368250.

Table[n*(n - 1), {n, Select[Range[150], MoebiusMu[#] == -1 &]}]

lista(kmax) = forsquarefree(k=1, kmax, if(moebius(k) == -1, print1(k[1]*(k[1]-1), ", ")));

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A370494(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+x-primepi(x)-sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,i)) for i in range(3,x.bit_length(),2)))
    return (k:=bisection(f,n,n))*(k-1) # Chai Wah Wu, Jan 28 2025

a(n) = A002378(A030059(n)-1).

Sum_{n>=1} 1/a(n) = (A368250 + A033150 - 1)/2 = 0.776922504035... .

A370495 Oblong numbers of the form (k-1)*k where k is the product of an even number of distinct primes.

Original entry on oeis.org

0, 30, 90, 182, 210, 420, 462, 650, 1056, 1122, 1190, 1406, 1482, 2070, 2550, 2970, 3192, 3306, 3782, 4160, 4692, 5402, 5852, 6642, 7140, 7310, 7482, 8190, 8556, 8742, 8930, 11130, 12210, 13110, 13806, 14042, 14762, 15006, 16512, 17556, 17822, 19740, 20022, 20306

Offset: 1

Amiram Eldar, Feb 20 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Complement of A370494 within A368249.

Cf. A002378, A030059, A033150, A368250.

Table[n*(n - 1), {n, Select[Range[150], MoebiusMu[#] == 1 &]}]

lista(kmax) = forsquarefree(k=1, kmax, if(moebius(k) == 1, print1(k[1]*(k[1]-1), ", ")));

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A370495(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n-1+x-sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,i)) for i in range(2,x.bit_length(),2)))
    return (k:=bisection(f,n,n))*(k-1) # Chai Wah Wu, Jan 28 2025

a(n) = A002378(A030229(n)-1).

Sum_{n>=2} 1/a(n) = (A368250 - A033150 + 1)/2 = 0.071711363929... .

cp's OEIS Frontend

A327527 Number of uniform divisors of n.

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A072777 Powers of squarefree numbers that are not squarefree.

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A368249 a(n) = A002378(A005117(n)-1).

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A368251 The number of nonsquarefree divisors of n that are powers of squarefree numbers (A072777).

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A370494 Oblong numbers of the form (k-1)*k where k is the product of an odd number of distinct primes.

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A370495 Oblong numbers of the form (k-1)*k where k is the product of an even number of distinct primes.

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