A358346 -id:A358346 - cp's OEIS frontend

A358347 a(n) is the sum of the unitary divisors of n that are squares.

1, 1, 1, 5, 1, 1, 1, 1, 10, 1, 1, 5, 1, 1, 1, 17, 1, 10, 1, 5, 1, 1, 1, 1, 26, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 50, 1, 1, 1, 1, 1, 1, 1, 5, 10, 1, 1, 17, 50, 26, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 10, 65, 1, 1, 1, 5, 1, 1, 1, 10, 1, 1, 26, 5, 1, 1, 1, 17, 82, 1

Offset: 1

Amiram Eldar, Nov 11 2022

The number of unitary divisors of n that are squares is A056624(n).

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

Cf. A056624, A077610, A078434, A247041, A268335, A350388, A351568.

Similar sequences: A033634, A034448, A035316, A358346.

f[p_, e_] := If[OddQ[e], 1, p^e + 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, 1, f[i,1]^f[i,2] + 1));}

a(n) >= 1 with equality if and only if n is an exponentially odd number (A268335).
Multiplicative with a(p^e) = p^e + 1 if e is even, and 1 otherwise.
a(n) = A034448(n)/A358346(n).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = zeta(3/2)/(3*zeta(5/2)) = 0.6491241554... .
Dirichlet g.f.: zeta(s)*zeta(2*s-2)/zeta(3*s-2). - Amiram Eldar, Jan 29 2023
a(n) = A034448(A350388(n)). - Amiram Eldar, Sep 09 2023

A385043 The sum of the unitary divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 1, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 4, 26, 42, 1, 40, 30, 72, 32, 1, 48, 54, 48, 50, 38, 60, 56, 6, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 3, 72, 8, 80, 90, 60, 120, 62, 96, 80, 1, 84, 144, 68, 90, 96

Offset: 1

Amiram Eldar, Jun 16 2025

The number of these divisors is A385042(n), and the largest of them is A367168(n).

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

The unitary analog of A353900.
Cf. A005117, A034448, A138302, A209229, A385042.
The sum of unitary divisors of n that are: A092261 (squarefree), A192066 (odd), A358346 (exponentially odd), A358347 (square), A360720 (powerful), A371242 (cubefree), A380396 (cube), A383763 (exponentially squarefree), this sequence (exponentially 2^n), A385045 (5-rough), A385046 (3-smooth), A385047 (power of 2), A385048 (cubefull), A385049 (biquadratefree).

f[p_, e_] := If[e == 2^IntegerExponent[e, 2], p^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~, if(f[i, 2] == 1<

Multiplicative with a(p^e) = p^(A209229(e)) + 1.
a(n) <= A034448(n), with equality if and only if n is in A138302.
a(n) <= A353900(n), with equality if and only if n is squarefree (A005117).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1/(p*(p+1)) + Sum_{k>=2} (1/p^(2^k)-1/p^(2^k-1))) = 1.21427559551509410114... .

A385045 The sum of the unitary divisors of n that are 5-rough numbers (A007310).

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 8, 1, 1, 6, 12, 1, 14, 8, 6, 1, 18, 1, 20, 6, 8, 12, 24, 1, 26, 14, 1, 8, 30, 6, 32, 1, 12, 18, 48, 1, 38, 20, 14, 6, 42, 8, 44, 12, 6, 24, 48, 1, 50, 26, 18, 14, 54, 1, 72, 8, 20, 30, 60, 6, 62, 32, 8, 1, 84, 12, 68, 18, 24, 48, 72, 1, 74, 38

Offset: 1

Amiram Eldar, Jun 16 2025

First differs from A186099 at n = 25; a(25) = 26, while A186099(25) = 31.

The number of these divisors is A385044(n), and the largest of them is A065330(n).

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

The unitary analog of A186099.
Cf. A002117, A007310, A034448, A065330, A385044.
The sum of unitary divisors of n that are: A092261 (squarefree), A192066 (odd), A358346 (exponentially odd), A358347 (square), A360720 (powerful), A371242 (cubefree), A380396 (cube), A383763 (exponentially squarefree), A385043 (exponentially 2^n), this sequence (5-rough), A385046 (3-smooth), A385047 (power of 2), A385048 (cubefull), A385049 (biquadratefree).

f[p_, e_] := If[p <= 3, 1, p^e + 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,1] < 5, 1, f[i, 1]^f[i, 2] + 1));}

Multiplicative with a(p^e) = 1 if p <= 3, and p^e + 1 if p >= 5.
a(n) = A034448(n)/A385046(n).
a(n) <= A034448(n), with equality if and only if n is 5-rough.
a(n) <= A186099(n).
Dirichlet g.f.: (zeta(s)*zeta(s-1)/zeta(2*s-1)) * ((1-1/2^(s-1))/(1-1/2^(2*s-1))) * ((1-1/3^(s-1))/(1-1/3^(2*s-1))).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 3*Pi^2/(91*zeta(3)) = 0.270679... .

A385046 The sum of the unitary divisors of n that are 3-smooth numbers (A003586).

Original entry on oeis.org

1, 3, 4, 5, 1, 12, 1, 9, 10, 3, 1, 20, 1, 3, 4, 17, 1, 30, 1, 5, 4, 3, 1, 36, 1, 3, 28, 5, 1, 12, 1, 33, 4, 3, 1, 50, 1, 3, 4, 9, 1, 12, 1, 5, 10, 3, 1, 68, 1, 3, 4, 5, 1, 84, 1, 9, 4, 3, 1, 20, 1, 3, 10, 65, 1, 12, 1, 5, 4, 3, 1, 90, 1, 3, 4, 5, 1, 12, 1, 17

Offset: 1

Amiram Eldar, Jun 16 2025

The number of these divisors is A382488(n), and the largest of them is A065331(n).

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

The unitary analog of A072079.
Cf. A001620, A003586, A034448, A065331, A082633, A382488.
The sum of unitary divisors of n that are: A092261 (squarefree), A192066 (odd), A358346 (exponentially odd), A358347 (square), A360720 (powerful), A371242 (cubefree), A380396 (cube), A383763 (exponentially squarefree), A385043 (exponentially 2^n), A385045 (5-rough), this sequence (3-smooth), A385047 (power of 2), A385048 (cubefull), A385049 (biquadratefree).

f[n_, p_] := If[Divisible[n, p], p^IntegerExponent[n, p] + 1, 1]; a[n_] := f[n, 2]*f[n, 3]; Array[a, 100]

a(n) = if(n%2, 1, 2^valuation(n, 2)+1) * if(!(n%3), 3^valuation(n, 3)+1, 1);

Multiplicative with a(p^e) = p^e + 1 if p <= 3, and 1 if p >= 5.
a(n) = A034448(n)/A385045(n).
a(n) <= A034448(n), with equality if and only if n 3-smooth.
a(n) <= A072079(n).
Dirichlet g.f.: zeta(s) * ((1-1/2^(2*s-1))/(1-1/2^(s-1))) * ((1-1/3^(2*s-1))/(1-1/3^(s-1))).
Sum_{k=1..n} a(k) ~ (n/(6*log(2)*log(3))) * (log(n)^2 + c1*log(n) + c2), where c1 = 2*gamma - 2 + 7*log(2) + 5*log(3) - 2*log(6) = 5.916004..., c2 = 2 - 5*log(2) - 11*log(2)^2/6 - 3*log(3) - 5*log(3)^2/6 + 15*log(2)*log(3)/2 + (5*log(2) + 3*log(3) - 2)*gamma - 2*gamma_1 = 1.957142..., gamma is Euler's constant (A001620), and gamma_1 is the 1st Stieltjes constant (A082633).

A385047 The sum of the unitary divisors of n that are powers of 2.

Original entry on oeis.org

1, 3, 1, 5, 1, 3, 1, 9, 1, 3, 1, 5, 1, 3, 1, 17, 1, 3, 1, 5, 1, 3, 1, 9, 1, 3, 1, 5, 1, 3, 1, 33, 1, 3, 1, 5, 1, 3, 1, 9, 1, 3, 1, 5, 1, 3, 1, 17, 1, 3, 1, 5, 1, 3, 1, 9, 1, 3, 1, 5, 1, 3, 1, 65, 1, 3, 1, 5, 1, 3, 1, 9, 1, 3, 1, 5, 1, 3, 1, 17, 1, 3, 1, 5, 1, 3

Offset: 1

Amiram Eldar, Jun 16 2025

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

The unitary analog of A038712.
Cf. A001620, A006519, A034448.
The sum of unitary divisors of n that are: A092261 (squarefree), A192066 (odd), A358346 (exponentially odd), A358347 (square), A360720 (powerful), A371242 (cubefree), A380396 (cube), A383763 (exponentially squarefree), A385043 (exponentially 2^n), A385045 (5-rough), A385046 (3-smooth), this sequence (power of 2), A385048 (cubefull), A385049 (biquadratefree).

a[n_] := If[OddQ[n], 1, 2^IntegerExponent[n, 2] + 1]; Array[a, 100]

PARI
```
a(n) = if(n%2, 1, 2^valuation(n, 2)+1);
```

Multiplicative with a(2^e) = 2^e + 1, and a(p^e) = 1 for an odd prime p.
a(n) = A034448(n) / A192066(n).
a(n) = A059841(n) + A006519(n), i.e., a(n) = A006519(n) + 1 if n is even, and 1 is n is odd.
Dirichlet g.f.: zeta(s) * ((1-1/2^(2*s-1))/(1-1/2^(s-1))).
Sum_{k=1..n} a(k) ~ (n/(2*log(2))) * (log(n) + gamma - 1 + 5*log(2)/2), where gamma is Euler's constant (A001620).

A385048 The sum of the unitary divisors of n that are cubefull numbers (A036966).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 28, 1, 1, 1, 1, 33, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 28, 1, 9, 1, 1, 1, 1, 1, 1, 1, 65, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 17, 82, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 16 2025

The number of these divisors is A368248(n), and the largest of them is A360540(n).

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

The unitary analog of A385005.
Cf. A034448, A036966, A046100, A360540, A368248.
The sum of unitary divisors of n that are: A092261 (squarefree), A192066 (odd), A358346 (exponentially odd), A358347 (square), A360720 (powerful), A371242 (cubefree), A380396 (cube), A383763 (exponentially squarefree), A385043 (exponentially 2^n), A385045 (5-rough), A385046 (3-smooth), A385047 (power of 2), this sequence (cubefull), A385049 (biquadratefree).

f[p_, e_] := If[e <= 2, 1, p^e + 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] < 3, 1, f[i, 1]^f[i, 2] + 1));}

Multiplicative with a(p^e) = 1 if e <= 2, and a(p^e) = p^e + 1 if e >= 3.
a(n) = A034448(n) / A371242(n).
a(n) <= A034448(n), with equality if and only if n is cubefull (A036966).
a(n) <= A385005(n), with equality if and only if n is biquadratefree (A046100).
Dirichlet g.f.: zeta(s)*zeta(s-1)*Product_{p prime} (1 - 1/p^(s-1) + 1/p^(3*s-3) - 1/p^(4*s-3)).

A385049 The sum of the unitary divisors of n that are biquadratefree numbers (A046100).

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 9, 10, 18, 12, 20, 14, 24, 24, 1, 18, 30, 20, 30, 32, 36, 24, 36, 26, 42, 28, 40, 30, 72, 32, 1, 48, 54, 48, 50, 38, 60, 56, 54, 42, 96, 44, 60, 60, 72, 48, 4, 50, 78, 72, 70, 54, 84, 72, 72, 80, 90, 60, 120, 62, 96, 80, 1, 84, 144, 68, 90

Offset: 1

Amiram Eldar, Jun 16 2025

First differs from A383763 at n = 32.

The number of these divisors is A365499(n), and the largest of them is A385007(n).

D. Suryanarayana, The number and sum of k-free integers <= x which are prime to n, Indian J. Math., Vol. 11 (1969), pp. 131-139.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Francesco Pappalardi, A survey on k-freeness, Number Theory, Ramanujan Math. Soc. Lect. Notes Ser., Vol. 1 (2003), pp. 71-88.

The unitary analog of A385006.
Cf. A036967, A046100, A365499, A385007.
The sum of unitary divisors of n that are: A092261 (squarefree), A192066 (odd), A358346 (exponentially odd), A358347 (square), A360720 (powerful), A371242 (cubefree), A380396 (cube), A383763 (exponentially squarefree), A385043 (exponentially 2^n), A385045 (5-rough), A385046 (3-smooth), A385047 (power of 2), A385048 (cubefull), this sequence (biquadratefree).

f[p_, e_] := If[e < 4, p^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~, if(f[i, 2] < 4, f[i, 1]^f[i, 2] + 1, 1)); }

Multiplicative with a(p^e) = p^e + 1 for e <= 3, and a(p^e) = 1 for e >= 4.
a(n) = 1 if and only if n is 4-full (A036967).
a(n) <= A034448(n), with equality if and only if n is biquadratefree.
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^(2*s-2) - 1/p^(2*s-1) + 1/p^(3*s-3) - 1/p^(3*s-2) - 1/p^(4*s-3)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p^2 + p) - 1/p^4) = 1.27769267395905900191... .

A374537 a(n) is the sum of the squares of the divisors of n that are exponentially odd numbers.

Original entry on oeis.org

1, 5, 10, 5, 26, 50, 50, 69, 10, 130, 122, 50, 170, 250, 260, 69, 290, 50, 362, 130, 500, 610, 530, 690, 26, 850, 739, 250, 842, 1300, 962, 1093, 1220, 1450, 1300, 50, 1370, 1810, 1700, 1794, 1682, 2500, 1850, 610, 260, 2650, 2210, 690, 50, 130, 2900, 850, 2810

Offset: 1

Amiram Eldar, Jul 11 2024

The number of divisors of n that are exponentially odd is A322483(n) and their sum is A033634(n).

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

Cf. A001157, A005117, A033634, A322483, A351265, A358346, A374458, A374538.

Cf. A002117, A013661, A065464, A183699.

f[p_, e_] := 1 + p^2 * (p^(4*Floor[(e-1)/2]+4) - 1) / (p^4 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p = f[, 1], e = f[, 2]); prod(i = 1, #p, 1 + p[i]^2 * (p[i]^(4*((e[i]-1)\2)+4) - 1) / (p[i]^4 - 1));}

a(n) = A001157(n) if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = 1 + p^2 * (p^(4*floor((e-1)/2)+4) - 1) / (p^4 - 1).
Dirichlet g.f.: zeta(s) * zeta(2*s-4) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(2*s-4)).
Sum_{k=1..n} a(k) = c * n^3 / 3, where c = zeta(2) * zeta(3) * Product_{p prime} (1 - 2/p^2 + 1/p^3) = A183699 * A065464 = 0.84677961058798544766... .

A374538 a(n) is the sum of the squares of the unitary divisors of n that are exponentially odd numbers (A268335).

Original entry on oeis.org

1, 5, 10, 1, 26, 50, 50, 65, 1, 130, 122, 10, 170, 250, 260, 1, 290, 5, 362, 26, 500, 610, 530, 650, 1, 850, 730, 50, 842, 1300, 962, 1025, 1220, 1450, 1300, 1, 1370, 1810, 1700, 1690, 1682, 2500, 1850, 122, 26, 2650, 2210, 10, 1, 5, 2900, 170, 2810, 3650, 3172

Offset: 1

Amiram Eldar, Jul 11 2024

The number of unitary divisors of n that are exponentially odd is A055076(n) and their sum is A358346(n).

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

Cf. A000290, A005117, A034676, A055076, A268335, A350389, A358346, A374537.

Cf. A002117, A013661, A183699.

f[p_, e_] := 1 + If[OddQ[e], p^(2*e), 0]; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + if(f[i, 2]%2,  f[i, 1]^(2*f[i, 2]), 0));}

a(n) = A034676(A350389(n)).
a(n) >= 1 with equality if and only if n is a square (A000290).
a(n) <= A374537(n) with equality if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = p^(2*e) + 1 if e is odd, and 1 otherwise.
Dirichlet g.f.: zeta(s) * zeta(2*s-4) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(2*s-4) - 1/p^(2*s-2)).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(2) * zeta(3) * Product_{p prime} (1 - 2/p^2 + 1/p^3 - 1/p^4 + 1/p^5) = 0.79482441214759383925... .

A382660 The unitary totient function applied to the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 1, 2, 4, 2, 6, 7, 4, 10, 12, 6, 8, 16, 18, 12, 10, 22, 14, 12, 26, 28, 8, 30, 31, 20, 16, 24, 36, 18, 24, 28, 40, 12, 42, 22, 46, 32, 52, 26, 40, 42, 36, 28, 58, 60, 30, 48, 20, 66, 44, 24, 70, 72, 36, 60, 24, 78, 40, 82, 64, 42, 56, 70, 88, 72, 60, 46, 72

Offset: 1

Amiram Eldar, Apr 02 2025

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

Cf. A013662, A047994, A065463, A268335.

Similar sequences: A358346, A363825, A366438, A366439, A366534, A366535, A367417, A368711, A368979, A374456, A382661.

f[p_, e_] := p^e-1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; expOddQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], OddQ]; uphi /@ Select[Range[100], expOddQ]

uphi(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^f[i, 2]-1);}
isexpodd(n) = {my(f = factor(n)); for(i=1, #f~, if(!(f[i, 2] % 2), return (0))); 1;}
list(lim) = apply(uphi, select(isexpodd, vector(lim, i, i)));

a(n) = A047994(A268335(n)).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(4)/(2*d^2)) * Product_{p prime} (1 - 2/p^2 + 2/p^3 - 2/p^4 + 1/p^5) = 0.504949539649594981601..., and d = A065463 is the asymptotic density of the exponentially odd numbers.

cp's OEIS Frontend

A358347 a(n) is the sum of the unitary divisors of n that are squares.

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A385043 The sum of the unitary divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

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A385045 The sum of the unitary divisors of n that are 5-rough numbers (A007310).

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A385046 The sum of the unitary divisors of n that are 3-smooth numbers (A003586).

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A385047 The sum of the unitary divisors of n that are powers of 2.

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A385048 The sum of the unitary divisors of n that are cubefull numbers (A036966).

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A385049 The sum of the unitary divisors of n that are biquadratefree numbers (A046100).

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A374537 a(n) is the sum of the squares of the divisors of n that are exponentially odd numbers.

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