A337050 -id:A337050 - cp's OEIS frontend

A134193 a(1) = 1; for n>1, a(n) = the smallest positive integer not occurring among the exponents in the prime-factorization of n.

1, 2, 2, 1, 2, 2, 2, 1, 1, 2, 2, 3, 2, 2, 2, 1, 2, 3, 2, 3, 2, 2, 2, 2, 1, 2, 1, 3, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 1, 3, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 1, 2, 2, 2, 3, 2, 2, 2, 1, 2, 2, 3, 3, 2, 2, 2, 2, 1, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 3, 1, 2, 2, 2, 2, 2, 2

Offset: 1

Leroy Quet, Jan 13 2008

nonn

From Amiram Eldar, Jun 30 2025: (Start)
The first position of k = 1, 2, 3, ... is A006939(k-1).
Let d(k) be the asymptotic density of the occurrences of k = 1, 2, ... in this sequence.
d(1) = 0 = the density of the powerful numbers (A001694).
d(2) = Product_{primes p} (1 - 1/p^2 + 1/p^3) = 0.748535... (A330596) = the density of A337050.
d(3) = Product_{primes p} (1 - 1/p^3 + 1/p^4) - Product_{primes p} (1 - 1/p^2 + 1/p^4) = 0.23548870893364493209...
d(4) = Product_{primes p} (1 - 1/p^4 + 1/p^5) - Product_{primes p} (1 - 1/p^3 + 1/p^5) - Product_{primes p} (1 - 1/p^2 + 1/p^3 - 1/p^4 + 1/p^5) + Product_{primes p} (1 - 1/p^2 + 1/p^5) = 0.01580134256336122613... .
d(5) = 0.000174471282..., d(6) = 0.000000217516..., etc.
In general, d(k) = Sum_{s subset of {2, 3, ..., k-1}} (-1)^card(s) * Product_{p prime} (1 -Sum_{i=1..card(s)} 1/p^s(i) + 1/p^(s(i)+1) - 1/p^k + 1/p^(k+1)).
The asymptotic mean of this sequence is Sum_{k>=1} k*d(k) = 2.26761567808299143335... . (End)

			The prime factorization of 24 is 2^3 * 3^1. The exponents are 3 and 1. Therefore a(24) = 2 is the smallest positive integer not occurring among (3,1).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A001694, A006939, A136567, A181819, A257993, A330596, A337050.

Join[{1}, Table[Complement[Range[n], Table[FactorInteger[n][[i, 2]], {i, 1, Length[FactorInteger[n]]}]][[1]], {n, 2, 120}]] (* Stefan Steinerberger, Jan 21 2008 *)

a(n) = if (n==1, 1, my(f=factor(n)); ve = vecsort(f[,2],,8); k = 1; while(vecsearch(ve, k), k++); k;); \\ Michel Marcus, Jul 28 2017

(define (A134193 n) (A257993 (A181819 n))) ;; Antti Karttunen, Jul 28 2017

a(n) = A257993(A181819(n)). - Antti Karttunen, Jul 28 2017

More terms from Stefan Steinerberger, Jan 21 2008

A329376 Multiplicative with a(p^e) = p when e = 2, otherwise a(p^e) = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 5, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 7, 5, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 5, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 7, 3, 10, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Nov 16 2019

Product of those distinct prime factors that occur exactly twice in the prime factorization of n, that is, whose exponent is 2.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000196, A062503, A337050.

Row 3 of array A106177, and the square roots of its row 9.

f[p_, e_] := If[e == 2, p, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 11 2023 *)

A329376(n) = { my(f = factor(n)); prod(i=1,#f~,f[i, 1]^(2 == f[i, 2])); };

for(n=1, 100, print1(direuler(p=2, n, (1 + p*X^2 + X + X^2/(-1 + 1/X)))[n], ", ")) \\ Vaclav Kotesovec, May 31 2024

Multiplicative with a(p^e) = p when e = 2, otherwise a(p^e) = 1.
a(n) <= A000196(n).
From Amiram Eldar, Feb 11 2023: (Start)
a(n) <= sqrt(n), with equality if and only if n is in A062503.
a(n) = 1 if and only if n is in A337050. (End)
From Vaclav Kotesovec, May 31 2024: (Start)
Dirichlet g.f.: zeta(2*s-1) * zeta(s) * Product_{p prime} (1 - 1/p^(5*s-1) + 1/p^(5*s-2) + 1/p^(4*s-1) - 1/p^(4*s-2) - 1/p^(3*s-1) + 1/p^(3*s) - 1/p^(2*s)).
Let f(s) = Product_{p prime} (1 - 1/p^(5*s-1) + 1/p^(5*s-2) + 1/p^(4*s-1) - 1/p^(4*s-2) - 1/p^(3*s-1) + 1/p^(3*s) - 1/p^(2*s)).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 3*gamma - 1 + f'(1)/f(1)) / 2, where
f(1) = Product_{p prime} (1 - 3/p^2 + 3/p^3 - 1/p^4) = 0.33718787379158997196169281615215824494915412775816393888028828465611936...,
f'(1) = f(1) * Sum_{p prime} (9*p^2 - 12*p + 5) * log(p) / (p^4 - 3*p^2 + 3*p - 1) = f(1) * 3.78385641685861932254178374972226733621783278751462026270346293...
and gamma is the Euler-Mascheroni constant A001620. (End)

A366763 The number of divisors of n that have no exponent 2 in their prime factorization.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 21 2023

The number of terms of A337050 that divide n.

The sum of these divisors is A366764(n), and the largest of them is A366765(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000005, A004709, A005117, A034444, A337050, A366764, A366765.

Cf. A330596.

f[p_, e_] := Max[e, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> max(x, 2), factor(n)[, 2]));

Multiplicative with a(p^e) = max(e, 2);
a(n) <= A000005(n), with equality if and only if n is squarefree (A005117).
a(n) >= A034444(n), with equality if and only if n is cubefree (A004709).
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s)).
From Vaclav Kotesovec, Apr 20 2025: (Start)
Let f(s) = Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s)).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 1/p^2 + 1/p^3) = A330596 = 0.74853525968236356464421504863791060164164034300532440451585279392592558689...,
f'(1) = f(1) * Sum_{p prime} (2*p-3)*log(p)/(p^3-p+1) = f(1) * 0.560697508735949606541137451100554649565120075155278833396722097786365686597...
and gamma is the Euler-Mascheroni constant A001620. (End)

A366764 The sum of divisors of n that have no exponent 2 in their prime factorization.

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 11, 4, 18, 12, 12, 14, 24, 24, 27, 18, 12, 20, 18, 32, 36, 24, 44, 6, 42, 31, 24, 30, 72, 32, 59, 48, 54, 48, 12, 38, 60, 56, 66, 42, 96, 44, 36, 24, 72, 48, 108, 8, 18, 72, 42, 54, 93, 72, 88, 80, 90, 60, 72, 62, 96, 32, 123, 84, 144, 68

Offset: 1

Amiram Eldar, Oct 21 2023

The sum of terms of A337050 that divide n.

The number of these divisors is A366763(n), and the largest of them is A366765(n).

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

Cf. A000203, A004709, A005117, A034448, A065487, A337050, A366763, A366765.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - p^2; f[p_, 1] := p + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, f[i, 1] + 1, (f[i, 1]^(f[i,2] + 1) - 1)/(f[i, 1] - 1) - f[i, 1]^2));}

Multiplicative with a(p) = p + 1, and a(p^e) = (p^(e+1) - 1)/(p - 1) - p^2 for e >= 2.
a(n) <= A000203(n), with equality if and only if n is squarefree (A005117).
a(n) >= A034448(n), with equality if and only if n is cubefree (A004709).
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^(2*s-2) + 1/p^(3*s-3)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p^3-p)) = 1.231291... (A065487).

A366765 The largest divisor of n that have no exponent 2 in their prime factorization.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 8, 3, 10, 11, 6, 13, 14, 15, 16, 17, 6, 19, 10, 21, 22, 23, 24, 5, 26, 27, 14, 29, 30, 31, 32, 33, 34, 35, 6, 37, 38, 39, 40, 41, 42, 43, 22, 15, 46, 47, 48, 7, 10, 51, 26, 53, 54, 55, 56, 57, 58, 59, 30, 61, 62, 21, 64, 65, 66, 67, 34, 69

Offset: 1

Amiram Eldar, Oct 21 2023

The largest term of A337050 that divides n.

The number of these divisors is A366763(n), and their sum is A366764(n).

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

Cf. A337050, A366763, A366764.

Similar sequences: A055231, A057521, A008833, A350390.

f[p_, e_] := p^If[e < 3, 1, e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1] ^ if(f[i, 2] < 3, 1, f[i, 2]));}

Multiplicative with a(p^e) = p if e <= 2 and p^e otherwise.
a(n) <= n, with equality if and only if n is in A337050.
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - 1/p^(2*s-2) + 1/p^(2*s-1) + 1/p^(3*s-3) - 1/p^(3*s-2)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 2/p^3 - 1/p^4) = 0.83234421330425224469... .

A368884 The largest unitary divisor of n that is the square of a squarefree number (A062503).

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 1, 9, 1, 1, 4, 1, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 1, 25, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 36, 1, 1, 1, 1, 1, 1, 1, 4, 9, 1, 1, 1, 49, 25, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 9, 1, 1, 1, 1, 4, 1, 1, 1, 9, 1, 1, 25, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1

Offset: 1

Amiram Eldar, Jan 09 2024

The number of these divisors is A368885(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A005117, A062503, A337050, A368885, A368886.

f[p_, e_] := If[e == 2, p^2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 2, f[i,1]^f[i,2], 1));}

from math import prod
from sympy import factorint
def A368884(n): return prod(p**e for p, e in factorint(n).items() if e==2) # Chai Wah Wu, Jan 09 2024

Multiplicative with a(p^e) = p^2 if e = 2, and 1 otherwise.
a(n) = n / A368886(n).
a(n) >= 1, with equality if and only if n is in A337050.
a(n) <= n, with equality if and only if n is in A062503.
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(2*s-2) - 1/p^(2*s) - 1/p^(3*s-2) + 1/p^(3*s)).
From Vaclav Kotesovec, Jan 09 2024: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (p^(2*s) - p^2) * (1 + (p^s - 1) * (p^2 + p^s + p^(2*s))) / p^(5*s).
Sum_{k=1..n} a(k) ~ c * zeta(3/2) * n^(3/2)/3, where c = Product_{p prime} (1 - 1/p^(11/2) + 1/p^(9/2) + 1/p^4 + 1/p^(7/2) - 1/p^3 - 1/p^(5/2) - 1/p^2) = 0.45021226373776124069206513259105992151602618717147857709105849... (End)

A368885 The number of unitary divisors of n that are squares of a squarefree number (A062503).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Jan 09 2024

First differs from A294932 at n = 32.

The largest of these divisors is A368884(n).

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

Cf. A034444, A062503, A294932, A337050, A368884.

f[p_, e_] := If[e == 2, 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x->if(x==2, 2, 1), factor(n)[, 2]));

from sympy import factorint
def A368885(n): return 1<Chai Wah Wu, Jan 09 2024

Multiplicative with a(p^e) = 2 if e = 2, and 1 otherwise.
a(n) >= 1, with equality if and only if n is in A337050.
a(n) <= A034444(n), with equality if and only if n is in A062503.
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(2*s) - 1/p^(3*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/p^2 - 1/p^3) = 1.30596827416754083231... .

A374291 Squares of powerful numbers.

Original entry on oeis.org

1, 16, 64, 81, 256, 625, 729, 1024, 1296, 2401, 4096, 5184, 6561, 10000, 11664, 14641, 15625, 16384, 20736, 28561, 38416, 40000, 46656, 50625, 59049, 65536, 82944, 83521, 104976, 117649, 130321, 153664, 160000, 186624, 194481, 234256, 250000, 262144, 279841, 331776

Offset: 1

Amiram Eldar, Jul 02 2024

First differs from A340588 at n = 12.
4-full (or 3-full) squares.
Numbers whose exponents in their prime factorization are all even numbers >= 4.
This sequence is closed under multiplication.
The sequence {A000290(n)*A078615(A000290(n)), n>=1} is a permutation of this sequence, and the sequence {a(n)/A078615(a(n)), n>=1} is a permutation of {A000290(n), n>=1}.
The sequence {A335988(n)*A007947(A335988(n)), n>=1} is a permutation of this sequence, and the sequence {a(n)/A007947(a(n)), n>=1} is a permutation of A335988.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to powerful numbers.

Intersection of A000290 and A036967 (or A036966).
Intersection of A000290 and A337050.
Subsequence of A322449.
Cf. A000290, A001694, A007947, A078615, A335988, A340588.

powQ[n_] := n==1 || AllTrue[FactorInteger[n][[;; , 2]], # > 1 &]; Select[Range[600], powQ]^2

is(k) = issquare(k) && ispowerful(sqrtint(k));

from math import isqrt
from sympy import mobius, integer_nthroot
def A374291(n):
    def squarefreepi(n):
        return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l = n+x, 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2, 3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        c -= squarefreepi(integer_nthroot(x, 3)[0])-l
        return c
    return bisection(f,n,n)**2 # Chai Wah Wu, Sep 10 2024

a(n) = A000290(A001694(n)) = A001694(n)^2.
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^2*(p^2-1))) = zeta(4)*zeta(6)/zeta(12) = 15015/(1382*Pi^2) = 1.10082313486953808844... .
Sum_{n>=1} 1/a(n)^s =  Product_{p prime} (1 + 1/(p^(2*s)*(p^(2*s)-1))) = zeta(4*s)*zeta(6*s)/zeta(12*s), for s > 1/4.

A375847 The maximum exponent in the prime factorization of the largest unitary cubefree divisor of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 0, 2, 1, 1, 2, 1, 1, 1, 0, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 0, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 0, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Aug 31 2024

Cf. A004709, A036966, A038109, A051903, A330596, A337050, A360539.

Cf. A007424 (analogous with the largest cubefree divisor, for n >= 2).

a[n_] := Max[Join[{0}, Select[FactorInteger[n][[;; , 2]], # <= 2 &]]]; a[1] = 0; Array[a, 100]

a(n) = {my(e = select(x -> x <= 2, factor(n)[,2])); if(#e == 0, 0, vecmax(e));}

a(n) = A051903(A360539(n)).
a(n) = 0 if and only if n is cubefull (A036966).
a(n) = 1 if and only if n is in A337050 \ A036966.
a(n) = 2 if and only if n is in A038109.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2 - A330596 = 1.25146474031763643535... .

cp's OEIS Frontend

A134193 a(1) = 1; for n>1, a(n) = the smallest positive integer not occurring among the exponents in the prime-factorization of n.

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A329376 Multiplicative with a(p^e) = p when e = 2, otherwise a(p^e) = 1.

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A366763 The number of divisors of n that have no exponent 2 in their prime factorization.

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A366764 The sum of divisors of n that have no exponent 2 in their prime factorization.

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A366765 The largest divisor of n that have no exponent 2 in their prime factorization.

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A368884 The largest unitary divisor of n that is the square of a squarefree number (A062503).

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A368885 The number of unitary divisors of n that are squares of a squarefree number (A062503).

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