A068068 -id:A068068 - cp's OEIS frontend

A385042 The number of unitary divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

1, 2, 2, 2, 2, 4, 2, 1, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 2, 2, 4, 1, 4, 2, 8, 2, 1, 4, 4, 4, 4, 2, 4, 4, 2, 2, 8, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 2, 4, 2, 4, 4, 2, 8, 2, 4, 4, 1, 4, 8, 2, 4, 4, 8, 2, 2, 2, 4, 4, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4

Offset: 1

Amiram Eldar, Jun 16 2025

First differs from A367515 at n = 128.

The sum of these divisors is A385043(n), and the largest of them is A367168(n).

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

The unitary analog of A353898.
Cf. A005117, A034444, A138302, A209229, A367168, A367515, A385043.
The number of unitary divisors of n that are: A000034 (power of 2), A055076 (exponentially odd), A056624 (square), A056671 (squarefree), A068068 (odd), A323308 (powerful), A365498 (cubefree), A365499 (biquadratefree), A368248 (cubefull), A380395 (cube), A382488 (3-smooth), this sequence (exponentially 2^n), A385044 (5-rough).

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

PARI
```
a(n) = vecprod(apply(x -> (x == 1<
				
```

Multiplicative with a(p^e) = A209229(e) + 1.
a(n) <= A034444(n), with equality if and only if n is in A138302.
a(n) <= A353898(n), with equality if and only if n is squarefree (A005117).

A385044 The number of unitary divisors of n that are 5-rough numbers (A007310).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 16 2025

The sum of these divisors is A385045(n), and the largest of them is A065330(n).

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

The unitary analog of A035218.
Cf. A007310, A034444, A065330, A385045.
Cf. A001620, A013661, A306016.
The number of unitary divisors of n that are: A000034 (power of 2), A055076 (exponentially odd), A056624 (square), A056671 (squarefree), A068068 (odd), A323308 (powerful), A365498 (cubefree), A365499 (biquadratefree), A368248 (cubefull), A380395 (cube), A382488 (3-smooth), A385042 (exponentially 2^n), this sequence (5-rough).

f[p_, e_] := If[p <= 3, 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x <= 3, 1, 2), factor(n)[, 1]));

Multiplicative with a(p^e) = 1 if p <= 3, and 2 if p >= 5.
a(n) = A034444(n)/A382488(n).
a(n) <= A034444(n), with equality if and only if n is 5-rough.
a(n) <= A035218(n).
Dirichlet g.f.: (zeta(s)^2/zeta(2*s)) * (1/((1+1/2^s)*(1+1/3^s))).
Sum_{k=1..n} a(k) ~ (n / (2 * zeta(2))) *(log(n) + 2*gamma - 1 + log(2)/3 + log(3)/4 - 2*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620).

A068067 Number of integers m, 0 < m <= n, such that n divides m(m+1)/2.

Original entry on oeis.org

1, 0, 2, 0, 2, 1, 2, 0, 2, 1, 2, 1, 2, 1, 4, 0, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 0, 4, 1, 4, 1, 2, 1, 4, 1, 2, 3, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 4, 1, 2, 3, 2, 1, 4, 0, 4, 3, 2, 1, 4, 3, 2, 1, 2, 1, 4, 1, 4, 3, 2, 1, 2, 1, 2, 3, 4, 1, 4, 1, 2, 3, 4, 1, 4, 1, 4, 1, 2, 1, 4, 1, 2, 3, 2, 1, 8

Offset: 1

Robert G. Wilson v, Feb 18 2002

nonn

Least n with a(n) = 2^k is prime(k+1)#/2 = A002110(A000040(k+1))/2. Least n with a(n) = 2^k-1 != 1 is p(k+1)#.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000040, A001221, A002110, A068068.

a[n_] := Length[Select[Range[n], Mod[ #(#+1)/2, n]==0&]]

a(n) = {my(c = 0); for(k = 1, n, c += !((k*(k+1)/2) % n)); c;} \\ Amiram Eldar, Sep 15 2024

a(n) = 0 iff n = 2^k with k >= 1.
If n is even, a(n) = 2^(omega(n)-1) - 1; if n is odd, a(n) = 2^omega(n). Here omega(n) = A001221(n) is the number of distinct prime divisors of n.
A068068(n) - a(n) = 0 if n is odd, 1 if n is even.

Edited by David W. Wilson and Dean Hickerson, Jun 08 2002

A100007 Number of unitary divisors of 2n-1 (d such that d divides 2n-1, GCD(d,(2n-1)/d)=1). Bisection of A034444.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 2, 2, 4, 4, 2, 4, 2, 2, 4, 2, 2, 4, 2, 4, 4, 2, 2, 4, 4, 2, 4, 2, 2, 4, 4, 2, 2, 2, 4, 4, 2, 4, 4, 4, 2, 4, 2, 2, 8, 2, 2, 4, 2, 4, 4, 4, 2, 4, 2, 2, 4, 2, 4, 4, 2, 2, 4, 4, 4, 4, 2, 2, 4, 4, 2, 4, 4, 2, 8, 2, 2, 4, 2, 4, 4, 2, 2, 4, 4, 4, 4, 2, 2, 8, 2, 2, 4, 4, 4, 4, 4

Offset: 1

N. J. A. Sloane, Nov 20 2004

			a(13)=2 because among the three divisors of 25 only 1 and 25 are unitary.

Cf. A001620, A034444, A068068, A100008.

with(numtheory): for n from 1 to 120 do printf(`%d,`,2^nops(ifactors(2*n-1)[2])) od: # Emeric Deutsch, Dec 24 2004

a[n_] := 2^PrimeNu[2*n-1]; Array[a, 100] (* Amiram Eldar, Jan 28 2023 *)

a(n) = 2^omega(2*n-1); \\ Amiram Eldar, Jan 28 2023

From Ilya Gutkovskiy, Apr 28 2017: (Start)
a(n) = [x^(2*n-1)] Sum_{k>=1} mu(k)^2*x^k/(1 - x^k).
a(n) = 2^omega(2*n-1). (End)
From Amiram Eldar, Jan 28 2023: (Start)
a(n) = A034444(2*n-1) = A068068(2*n-1).
Sum_{k=1..n} a(k) ~ 4*n*((log(n) + 2*gamma - 1 + 7*log(2)/3 - 2*zeta'(2)/zeta(2)) / Pi^2, where gamma is Euler's constant (A001620). (End)

More terms from Emeric Deutsch, Dec 24 2004

A120891 Number of primitive Pythagorean triangles with odd leg 2n-1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 1, 1, 4, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 4, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 4, 1, 1, 2, 2

Offset: 1

Lekraj Beedassy, Jul 12 2006

nonn

Bisection of A024361.

Bisection of even-numbered terms of A024361 results in alternating zero terms; removing zeros gives A068068. - Ray Chandler, Feb 04 2020

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A024361, A068068, A092523, A120890.

a(n)=2^(k-1), where k=A092523(n) for n > 1.

a(1)=0 inserted by Ray Chandler, Feb 04 2020

A156688 The total number of distinct Pythagorean triples with an area numerically equal to n times their perimeters.

Original entry on oeis.org

2, 3, 6, 4, 6, 9, 6, 5, 10, 9, 6, 12, 6, 9, 18, 6, 6, 15, 6, 12, 18, 9, 6, 15, 10, 9, 14, 12, 6, 27, 6, 7, 18, 9, 18, 20, 6, 9, 18, 15, 6, 27, 6, 12, 30, 9, 6, 18, 10, 15, 18, 12, 6, 21, 18, 15, 18, 9, 6, 36, 6, 9, 30, 8, 18, 27, 6, 12, 18, 27, 6, 25, 6, 9, 30, 12, 18, 27, 6, 18, 18, 9, 6, 36, 18, 9, 18, 15, 6, 45, 18, 12, 18, 9, 18

Offset: 1

Ant King, Feb 18 2009

The members of this sequence are also 1/2 the number of divisors of 8n^2. The corresponding results for primitive triangles only are in A068068.

Also, the total number of distinct "areas with equal border", that is: Let x, y be positive integers so that the area xy equals the border around it with thickness n. As a formula it is: 2xy = (x+2n)(y+2n). To compare with the original, the areas at thickness 5 are 11x210, 12x110, 14x60, 15x50, 18x35, 20x30. - Juhani Heino, Jul 22 2012

			There are 6 Pythagorean triples whose area is 5 times their perimeters - (21,220,221), (22,120,122), (24,70,74), (25,60,65),(28,45,53) and (30,40,50) - hence a(5)=6.

Chi, Henjin and Killgrove, Raymond; Problem 1447, Crux Math 15(5), May 1989.
Chi, Henjin and Killgrove, Raymond; Solution to Problem 1447, Crux Math 16(7), September 1990.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Ron Knott, Right-angled Triangles and Pythagoras' Theorem

Cf. A000005, A068068.

Mathematica
```
1/2 DivisorSigma[0,8#^2] &/@Range[75]
```

A156688(n) = (numdiv(8*n*n)/2); \\ Antti Karttunen, Sep 27 2018

a(n) = A000005(8n^2)/2 = A078644(2n).

More terms from Antti Karttunen, Sep 27 2018

A384557 The number of exponential unitary (or e-unitary) divisors of n that are exponentially odd numbers (A268335).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 03 2025

First differs from A359411 at n = 2097152 = 2^21: a(2097152) = 4, while A359411(2097152) = 2.
First differs from A368979 at n = 512 = 2^9: a(512) = 2, while A368979(512) = 3.
First differs from A367516 at n = 128 = 2^7: a(128) = 2, while A367516(128) = 1.
First differs from A382291 at n = 128 = 2^7: a(128) = 2, while A382291(128) = 4.
First differs from A368168 at n = 64 = 2^6: a(64) = 2, while A368168(64) = 1.
The sum of these divisors is A384559(n), and the largest of them is A331737(n).
The number of exponential unitary (or e-unitary) divisors of n is A278908(n) and the number of divisors of n that are exponentially odd numbers is A322483(n).
All the terms are powers of 2. The first term that is greater than 2 is a(32768) = 4.

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

Cf. A068068, A138302, A268335, A278908, A322483, A331737, A359411, A367516, A368168, A368979, A382291, A384559.

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

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

Multiplicative with a(p^e) = A068068(e).
a(n) >= 1, with equality if and only if n is in A138302.
a(n) <= A278908(n), with equality if and only if n is an exponentially odd number.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=2} (d(k) - d(k-1))/p^k) = 1.13551542615965557947..., where d(k) is the number of odd unitary divisors of k (A068068).

A346773 a(n) = Sum_{d|n} möbius(d)^n.

Original entry on oeis.org

1, 2, 0, 2, 0, 4, 0, 2, 0, 4, 0, 4, 0, 4, 0, 2, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 8, 0, 2, 0, 4, 0, 4, 0, 4, 0, 4, 0, 8, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 8, 0, 4, 0, 2, 0, 8, 0, 4, 0, 8, 0, 4, 0, 4, 0, 4, 0, 8, 0, 4, 0, 4, 0, 8, 0, 4, 0, 4, 0, 8, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 8, 0

Offset: 1

Seiichi Manyama, Aug 03 2021

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A008683, A023887, A034444, A068068, A080323, A100008, A264782.

Table[Sum[MoebiusMu[d]^n,{d,Divisors[n]}],{n,103}] (* Stefano Spezia, Aug 03 2021 *)

PARI
```
a(n) = sumdiv(n, d, moebius(d)^n);
```
PARI
```
a(n) = if(n%2, 0^(n-1), 2^omega(2*n));
```

N=99; x='x+O('x^N); Vec(sum(k=1, N, (moebius(k)*x)^k/(1-(moebius(k)*x)^k)))

G.f.: Sum_{k>=1} (mu(k)*x)^k/(1 - (mu(k)*x)^k).

a(2*n-1) = 0^(n-1) and a(2*n) = A034444(2*n) = A100008(n) for n > 0.

cp's OEIS Frontend

A385042 The number of unitary divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

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A385044 The number of unitary divisors of n that are 5-rough numbers (A007310).

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A068067 Number of integers m, 0 < m <= n, such that n divides m(m+1)/2.

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A100007 Number of unitary divisors of 2n-1 (d such that d divides 2n-1, GCD(d,(2n-1)/d)=1). Bisection of A034444.

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A120891 Number of primitive Pythagorean triangles with odd leg 2n-1.

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A156688 The total number of distinct Pythagorean triples with an area numerically equal to n times their perimeters.

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A384557 The number of exponential unitary (or e-unitary) divisors of n that are exponentially odd numbers (A268335).

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