A363195 -id:A363195 - 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

A364154 Lexicographically earliest sequence of distinct positive integers such that a(n) is least novel multiple m of the product of all primes less than the greatest prime factor of a(n-1) which do not divide a(n-1); a(1) = 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 30, 8, 9, 10, 12, 11, 210, 13, 2310, 14, 15, 16, 17, 30030, 18, 19, 510510, 20, 21, 40, 24, 22, 105, 26, 1155, 28, 45, 32, 23, 9699690, 25, 36, 27, 34, 15015, 38, 255255, 42, 35, 48, 29, 223092870, 31, 6469693230, 33, 70, 39, 770, 51, 10010

Offset: 1

David James Sycamore, Jul 11 2023

nonn

It follows from the definition that the sequence is infinite, and that the records (outside of the first 7 terms) are all primorial numbers, meaning that it grows very quickly.
When there are no primes less than the greatest prime factor of a(n-1) which do not divide a(n-1) then m is the least novel multiple of 1, the empty product, and therefore a(n) = u, the least unused number in the sequence so far. The only way a prime can enter the sequence is as u. When a(n-1) = prime(k), a(n) is A002110(k-1), and any primorial term is followed by u. Thus: prime —> primorial —> u.
Sequence is a permutation of the positive integers since by the definition no number appears more than once and m = 1 eventually introduces any number not already placed by the first part of the definition (m > 1).

			a(1) = 1 and there are no primes < 1 which divide 1 therefore m = 1 so a(2) = 2, the least unused number. Likewise a(3) = 3.
a(4) = 2*2 = 4 since 2 is the only prime < 3 which does not divide 3 and 2 has already occurred.
Since a(7) = 7, a(8) = 2*3*5 = 30.

Michael De Vlieger, Table of n, a(n) for n = 1..4992
Michael De Vlieger, Log log scatterplot of log_10(a(n)), n = 1..2^16.
Michael De Vlieger, Plot p^e | a(n) at (x,y) = (n, pi(p)) for n = 1..2^11, with a color function representing e = 1 as black, e = 2 as red, ..., the maximum value of e in the dataset as magenta. Under the plot, we indicate the empty product in black, primes in red, composite prime powers in gold, squarefree composites in green, and all other numbers in blue.

Cf. A002110, A083720, A351495, A359804, A363195.

nn = 120; c[] := False; m[] := 1; a[1] = j = 1; c[1] = True;
Do[k = Times @@ Complement[Prime@ Range[PrimePi@ Last[#] - 1], #] &[
   FactorInteger[j][[All, 1]] ];
 While[c[k m[k]], m[k]++]; k *= m[k];
 Set[{a[n], c[k], j}, {k, True, k}], {n, 2, nn}];
Array[a, nn]

lista(nn) = my(c, m, v=List([1, 2])); for(k=3, nn, c=m=1; forprime(p=2, vecmax(factor(v[k-1])[, 1]), if(v[k-1]%p, m*=p)); while(setsearch(Set(v), c*m), c++); listput(v, c*m)); Vec(v) \\ Jinyuan Wang, Jul 11 2023

More terms from Jinyuan Wang, Jul 11 2023

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).

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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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A364154 Lexicographically earliest sequence of distinct positive integers such that a(n) is least novel multiple m of the product of all primes less than the greatest prime factor of a(n-1) which do not divide a(n-1); a(1) = 1.

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

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