A365480 -id:A365480 - cp's OEIS frontend

A365479 The sum of unitary divisors of the smallest square divisible by n.

1, 5, 10, 5, 26, 50, 50, 17, 10, 130, 122, 50, 170, 250, 260, 17, 290, 50, 362, 130, 500, 610, 530, 170, 26, 850, 82, 250, 842, 1300, 962, 65, 1220, 1450, 1300, 50, 1370, 1810, 1700, 442, 1682, 2500, 1850, 610, 260, 2650, 2210, 170, 50, 130, 2900, 850, 2810, 410

Offset: 1

Amiram Eldar, Sep 05 2023

The number of unitary divisors of the smallest square divisible by n is the same as the number of unitary divisors of n, A034444(n).

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

Cf. A002117, A034444, A034448, A053143, A365346, A365480, A365481.

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

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

from math import prod
from sympy import factorint
def A365479(n): return prod(p**(e+(e&1))+1 for p,e in factorint(n).items()) # Chai Wah Wu, Sep 05 2023

a(n) = A034448(A053143(n)).
Multiplicative with a(p^e) = p^(e + (e mod 2)) + 1.
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(2*s-2) - 1/p^(3*s-2)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (Pi^2/45) * zeta(3) * Product_{p prime} (1 - 1/p^4 + 1/p^5 - 1/p^6) = 0.248414056414... .

A365481 The sum of unitary divisors of the smallest number whose square is divisible by n.

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 5, 4, 18, 12, 12, 14, 24, 24, 5, 18, 12, 20, 18, 32, 36, 24, 20, 6, 42, 10, 24, 30, 72, 32, 9, 48, 54, 48, 12, 38, 60, 56, 30, 42, 96, 44, 36, 24, 72, 48, 20, 8, 18, 72, 42, 54, 30, 72, 40, 80, 90, 60, 72, 62, 96, 32, 9, 84, 144, 68, 54, 96

Offset: 1

Amiram Eldar, Sep 05 2023

The number of unitary divisors of the smallest number whose square is divisible by n is the same as the number of unitary divisors of n, A034444(n).

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

Cf. A002117, A013661, A019554, A034444, A034448, A365347, A365479, A365480.

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

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

from math import prod
from sympy import factorint
def A365481(n): return prod(p**((e>>1)+(e&1))+1 for p,e in factorint(n).items()) # Chai Wah Wu, Sep 05 2023

a(n) = A034448(A019554(n)).
Multiplicative with a(p^e) = p^(ceiling(e/2)) + 1.
Dirichlet g.f.: zeta(s) * zeta(2*s-1) * Product_{p prime} (1 + 1/p^(s-1) - 1/p^(2*s-1) - 1/p^(3*s-1)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * zeta(2) * zeta(3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4 - 1/p^5 + 1/p^6) = 0.515959523197... .

A369759 The sum of unitary divisors of the smallest cubefull exponentially odd number that is divisible by n.

Original entry on oeis.org

1, 9, 28, 9, 126, 252, 344, 9, 28, 1134, 1332, 252, 2198, 3096, 3528, 33, 4914, 252, 6860, 1134, 9632, 11988, 12168, 252, 126, 19782, 28, 3096, 24390, 31752, 29792, 33, 37296, 44226, 43344, 252, 50654, 61740, 61544, 1134, 68922, 86688, 79508, 11988, 3528, 109512

Offset: 1

Amiram Eldar, Jan 31 2024

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

Cf. A034448, A335988, A356192, A365480, A369757, A369758.

Cf. A013661, A013662, A013664.

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

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

a(n) = A034448(A356192(n)).
Multiplicative with a(p) = p^3 + 1, a(p^e) = p^e + 1 for an odd e >= 3, and a(p^e) = p^(e+1) + 1 for an even e.
a(n) >= A034448(n), with equality if and only if n is cubefull exponentially odd number (A335988).
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-3) - 1/p^(2*s-2) - 1/p^(3*s-5) + 1/p^(4*s-5) - 1/p^(4*s-3)).
Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = (zeta(4)*zeta(6)/zeta(2)) * Product_{p prime} (1 - 1/p^6 + 1/p^11 - 1/p^12) = 0.65813930591740259189... .

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A365479 The sum of unitary divisors of the smallest square divisible by n.

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A365481 The sum of unitary divisors of the smallest number whose square is divisible by n.

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A369759 The sum of unitary divisors of the smallest cubefull exponentially odd number that is divisible by n.

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Mathematica

PARI

Formula