A076400 -id:A076400 - cp's OEIS frontend

A366438 The number of divisors of the exponentially odd numbers (A268335).

1, 2, 2, 2, 4, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 8, 4, 4, 2, 8, 2, 6, 4, 4, 4, 2, 4, 4, 8, 2, 8, 2, 4, 2, 4, 2, 8, 4, 8, 4, 4, 2, 2, 4, 4, 8, 2, 4, 8, 2, 2, 4, 4, 8, 2, 4, 2, 4, 4, 4, 8, 2, 4, 4, 4, 4, 12, 2, 2, 8, 2, 8, 8, 4, 2, 2, 8, 4, 2, 8, 4, 4, 4, 16, 4

Offset: 1

Amiram Eldar, Oct 10 2023

1 is the only odd term in this sequence.

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

Cf. A000005, A268335, A366439.

Similar sequences: A048691, A072048, A076400, A358040, A363194, A363195.

f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, OddQ], Times @@ (e + 1), Nothing]]; f[1] = 1; Array[f, 150]

lista(max) = for(k = 1, max, my(e = factor(k)[, 2], isexpodd = 1); for(i = 1, #e, if(!(e[i] % 2), isexpodd = 0; break)); if(isexpodd, print1(vecprod(apply(x -> x+1, e)), ", ")));

from math import prod
from itertools import count, islice
from sympy import factorint
def A366438_gen(): # generator of terms
    for n in count(1):
        f = factorint(n).values()
        if all(e&1 for e in f):
            yield prod(e+1 for e in f)
A366438_list = list(islice(A366438_gen(),30)) # Chai Wah Wu, Oct 10 2023

a(n) = A000005(A268335(n)).

A363194 Number of divisors of the n-th powerful number A001694(n).

Original entry on oeis.org

1, 3, 4, 3, 5, 3, 4, 6, 9, 3, 7, 12, 5, 9, 12, 3, 4, 8, 15, 3, 9, 12, 16, 9, 6, 9, 18, 3, 15, 4, 3, 12, 15, 20, 9, 9, 12, 10, 3, 21, 5, 20, 12, 9, 7, 15, 18, 3, 24, 27, 3, 12, 18, 16, 11, 9, 12, 24, 9, 9, 25, 12, 4, 12, 3, 12, 9, 9, 18, 21, 3, 28, 27, 36, 3, 15

Offset: 1

Amiram Eldar, May 21 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (8).

Cf. A000005, A001694, A180114, A306458, A343443.

Similar sequences: A072048, A076400, A363195.

DivisorSigma[0, Select[Range[3000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]]

apply(numdiv, select(ispowerful, [1..3000]))

from itertools import count, islice
from math import prod
from sympy import factorint
def A363194_gen(): # generator of terms
    for n in count(1):
        f = factorint(n).values()
        if all(e>1 for e in f):
            yield prod(e+1 for e in f)
A363194_list = list(islice(A363194_gen(),20)) # Chai Wah Wu, May 21 2023

a(n) = A000005(A001694(n)).
Sum_{A001694(k) < x} a(k) = c_1 * sqrt(x) * log(x)^2 + c_2 * sqrt(x) * log(x) + c_3 * sqrt(x) + O(x^(5/12 + eps)), where c_1, c_2 and c_3 are constants. c_1 = Product_{p prime} (1 + 4/p^(3/2) - 1/p^2 - 6/p^(5/2) + 2/p^(7/2))/8 = 0.516273682988566836609... . [corrected Sep 21 2024]
a(n) = A343443(A306458(n)). - Amiram Eldar, Sep 01 2023

A076399 Number of prime factors of n-th perfect power (with repetition).

Original entry on oeis.org

0, 2, 3, 2, 4, 2, 3, 5, 4, 2, 6, 4, 4, 2, 3, 7, 6, 2, 4, 6, 4, 5, 8, 2, 6, 3, 2, 6, 4, 4, 9, 2, 8, 4, 4, 6, 6, 2, 6, 2, 6, 10, 4, 4, 4, 8, 3, 2, 4, 4, 8, 2, 9, 6, 2, 6, 6, 11, 4, 7, 3, 2, 10, 4, 6, 4, 6, 6, 2, 8, 4, 5, 8, 4, 4, 6, 2, 8, 2, 4, 6, 12, 4, 6, 2, 6, 4, 6, 3, 2, 10, 2, 4, 6, 6, 9, 4, 6, 2, 10, 8

Offset: 1

Reinhard Zumkeller, Oct 09 2002

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, The Number of Prime Factors on Average in Certain Integer Sequences, Journal of Integer Sequences, Vol. 25 (2022), Article 22.2.3.
Eric Weisstein's World of Mathematics, Perfect Powers.
Eric Weisstein's World of Mathematics, Prime Factor.

Cf. A001222, A025478, A025479, A076398, A076400, A083342.

a076399 n = a001222 (a025478 n) * a025479 n
-- Reinhard Zumkeller, Mar 28 2014

PrimeOmega[Select[Range[10^4], # == 1 || GCD @@ FactorInteger[#][[;; , 2]] > 1 &]] (* Amiram Eldar, Feb 18 2023 *)

is(n) = n==1 || ispower(n);
apply(bigomega, select(is, [1..5000])) \\ Amiram Eldar, Feb 18 2023

from sympy import mobius, integer_nthroot, primeomega
def A076399(n):
    def f(x): return int(n-2+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) 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 int(primeomega(kmax)) # Chai Wah Wu, Aug 14 2024

a(n) = A001222(A001597(n)).
a(n) = A001222(A025478(n))*A025479(n).
Sum_{A001597(k) <= x} a(k) = 2*sqrt(x)*log(log(x)) + 2*(B_2 - log(2))*sqrt(x) + O(sqrt(x)/log(x)), where B_2 = A083342 (Jakimczuk and Lalín, 2022). - Amiram Eldar, Feb 18 2023, corrected Sep 21 2024

A352475 Numbers m such that gcd(d(m),6) = 1.

Original entry on oeis.org

1, 16, 64, 81, 625, 729, 1024, 1296, 2401, 4096, 5184, 10000, 11664, 14641, 15625, 28561, 38416, 40000, 46656, 50625, 59049, 65536, 82944, 83521, 117649, 130321, 153664, 194481, 234256, 250000, 262144, 279841, 331776, 455625, 456976, 531441, 640000, 707281, 746496

Offset: 1

Michael De Vlieger, Mar 26 2022

All terms are square since numbers coprime to 6 are odd.
The square roots of terms are in A001694.
Intersection of A000290 and A336590, i.e., numbers whose prime factorization has only exponents that are congruent to {0, 4} mod 6 (A047233). - Amiram Eldar, Mar 31 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000
Titus Hilberdink, How often is d(n) a power of a given integer?, Journal of Number Theory, Vol. 236 (2022), pp. 261-279.

Cf. A000005, A000290, A001694, A007310, A047233, A076400, A100044, A336590, A350014.

Select[Range[864]^2, GCD[DivisorSigma[0, #], 6] == 1 &] (* or, more efficiently, *)
With[{nn = 864}, Select[Union[Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}]]^2, Mod[DivisorSigma[0, #], 3] > 0 &]]

isok(m) = gcd(numdiv(m), 6) == 1; \\ Michel Marcus, Mar 29 2022

m = 100000; seq = direuler(p=2, m, (1 - X^8)/(1 - X^4)/(1 - X^6)); for(n=1, m, if(seq[n] != 0, print1(n, ", "))) \\ Vaclav Kotesovec, May 19 2022

a(n) = A350014(n)^2.
Sum_{n>=1} 1/a(n) = Pi^2/9 (A100044). - Amiram Eldar, Mar 31 2022
The number of terms <= x is (zeta(3/2)/zeta(2))*x^(1/4) + (zeta(2/3)/zeta(4/3))*x^(1/6) + O(x^(1/8 + eps)), for all eps > 0 (Hilberdink, 2022). - Amiram Eldar, May 18 2022

A363195 Number of divisors of the n-th cubefull number A036966(n).

Original entry on oeis.org

1, 4, 5, 4, 6, 7, 5, 4, 8, 16, 6, 9, 4, 20, 10, 5, 20, 7, 24, 16, 11, 25, 4, 28, 24, 20, 12, 8, 4, 5, 30, 16, 6, 16, 32, 30, 24, 13, 4, 20, 35, 20, 28, 9, 4, 36, 36, 28, 14, 16, 25, 20, 40, 16, 24, 35, 4, 40, 5, 42, 7, 32, 15, 6, 20, 32, 16, 20, 10, 30, 45, 20

Offset: 1

Amiram Eldar, May 21 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (8).

Cf. A000005, A036966, A362986.

Similar sequences: A072048, A076400, A363194.

DivisorSigma[0, Select[Range[25000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 2 &]]

lista(kmax) = for(k = 1, kmax, if(k==1 || vecmin(factor(k)[, 2]) > 2, print1(numdiv(k), ", ")));

from itertools import count, islice
from math import prod
from sympy import factorint
def A363195_gen(): # generator of terms
    for n in count(1):
        f = factorint(n).values()
        if all(e>2 for e in f):
            yield prod(e+1 for e in f)
A363195_list = list(islice(A363195_gen(),20)) # Chai Wah Wu, May 21 2023

a(n) = A000005(A036966(n)).

Sum_{A036966(k) < x} a(k) = c_1 * x^(1/3) * log(x)^3 + c_2 * x^(1/3) * log(x)^2 + c_3 * x^(1/3) * log(x) + c_4 * x^(1/3) + O(x^(7/24 + eps)), where c_1, c_2, c_3 and c_4 are constants. c_1 = Product_{p prime} ((1-1/p)^4 * (1 + 1/((p^(1/3) - 1)^2*p^(1/3)) + 3/(p-p^(2/3))))/162 = 0.1346652397135839416... . [corrected Sep 21 2024]

A076398 Number of distinct prime factors of n-th perfect power.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 2, 3, 1, 2, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 1, 3

Offset: 1

Reinhard Zumkeller, Oct 09 2002

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Perfect Powers.
Eric Weisstein's World of Mathematics, Distinct Prime Factors.

Cf. A001221, A001597, A025478, A076399, A076400.

a076398 = a001221 . a025478  -- Reinhard Zumkeller, Mar 28 2014

ppQ[1] = True; ppQ[n_] := GCD @@ FactorInteger[n][[All, 2]] > 1; PrimeNu /@ Select[Range[10^4], ppQ] (* Jean-François Alcover, Jul 15 2017 *)

lista(nn) = for(n=1, nn, if ((n==1) || ispower(n), print1(omega(n), ", "))); \\ Michel Marcus, Jul 15 2017

from sympy import mobius, integer_nthroot, primenu
def A076398(n):
    def f(x): return int(n-2+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) 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 int(primenu(kmax)) # Chai Wah Wu, Aug 14 2024

a(n) = A001221(A001597(n)).

a(n) = A001221(A025478(n)).

A076401 Sum of divisors of n-th perfect power.

Original entry on oeis.org

1, 7, 15, 13, 31, 31, 40, 63, 91, 57, 127, 121, 217, 133, 156, 255, 403, 183, 399, 600, 403, 364, 511, 307, 847, 400, 381, 961, 741, 931, 1023, 553, 1651, 781, 1281, 1093, 1767, 871, 2821, 993, 2340, 2047, 1729, 2149, 1767, 3751, 1464, 1407, 2667, 2379

Offset: 1

Reinhard Zumkeller, Oct 09 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Perfect Powers.
Eric Weisstein's World of Mathematics, Divisor Function.

Cf. A000203, A001597, A076400.

N:= 10^4: # to use all perfect powers <= N
L:= sort(convert({seq(seq(i^k, i=1..floor(N^(1/k))), k=2..ilog2(N))},list)):
map(numtheory:-sigma,L); # Robert Israel, Oct 02 2014

lista(nn) = {for (n=1, nn, if ((n==1) || ispower(n), print1(sigma(n), ", ")););} \\ Michel Marcus, Oct 02 2014

a(n) = A000203(A001597(n)).

A366441 The number of divisors of the 5-rough numbers (A007310).

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 4, 2, 2, 2, 2, 3, 2, 4, 2, 2, 4, 2, 2, 2, 4, 2, 2, 4, 2, 4, 4, 2, 2, 2, 2, 2, 2, 4, 4, 3, 4, 2, 2, 4, 2, 2, 4, 4, 2, 2, 4, 2, 4, 2, 2, 3, 2, 6, 2, 2, 4, 4, 2, 2, 2, 2, 4, 4, 4, 2, 4, 4, 4, 2, 2, 2, 2, 4, 2, 2, 6, 4, 2, 4, 2, 4

Offset: 1

Amiram Eldar, Oct 10 2023

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

Cf. A000005, A001620, A007310, A366442.

Similar sequences: A048691, A072048, A076400, A099774, A358040, A363194, A363195.

a[n_] := DivisorSigma[0, 2*Floor[3*n/2] - 1]; Array[a, 100]

PARI
```
a(n) = numdiv((3*n)\2 << 1 - 1)
```

from sympy import divisor_count
def A366441(n): return divisor_count((n+(n>>1)<<1)-1) # Chai Wah Wu, Oct 10 2023

a(n) = A000005(A007310(n)).

Sum_{k=1..n} a(k) ~ (log(n) + 2*gamma - 1 + 2*log(6)) * n / 3, where gamma is Euler's constant (A001620).

A354799 Integers m in A001694 such that 3 | d(m^2), where d(n) = A000005(n).

Original entry on oeis.org

16, 81, 128, 144, 324, 400, 432, 625, 648, 784, 1024, 1152, 1296, 1936, 2000, 2025, 2187, 2401, 2500, 2592, 2704, 3200, 3456, 3600, 3888, 3969, 4624, 5000, 5184, 5488, 5625, 5776, 6272, 7056, 8100, 8192, 8464, 8748, 9216, 9604, 9801, 10000, 10125, 10368, 10800

Offset: 1

Michael De Vlieger, Jun 21 2022

			A001694(5) = 16 is a term since d(16^2) = d(256) = 9, and 9 is a multiple of 3.
A001694(13) = 81 is a term since d(81^2) = d(6561) = 9, and 9 is a multiple of 3.
A001694(3) = 8 is not a term since d(8^2) = d(64) = 7, which is not divisible by 3.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A000005, A001694, A008585, A076400, A350014.

With[{nn = 10800}, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], Mod[DivisorSigma[0, #^2], 3] == 0 &]]

isok(m) = ispowerful(m) && !(numdiv(m^2) % 3); \\ Michel Marcus, Jun 27 2022

from sympy import divisor_count as d, factorint as f
def ok(k): return k > 1 and min(f(k).values()) > 1 and d(k*k)%3 == 0
print([k for k in range(11000) if ok(k)]) # Michael S. Branicky, Jun 28 2022

Equals { A001694 \ A350014 }.
Equals { m in A001694 : d(m^2) mod 3 = 0 }.
Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - 5*zeta(3)/(2*zeta(2)) = 0.1166890133... . - Amiram Eldar, Jun 28 2022

cp's OEIS Frontend

A366438 The number of divisors of the exponentially odd numbers (A268335).

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A363194 Number of divisors of the n-th powerful number A001694(n).

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A076399 Number of prime factors of n-th perfect power (with repetition).

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A352475 Numbers m such that gcd(d(m),6) = 1.

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A363195 Number of divisors of the n-th cubefull number A036966(n).

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A076398 Number of distinct prime factors of n-th perfect power.

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A076401 Sum of divisors of n-th perfect power.

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