A328258 -id:A328258 - cp's OEIS frontend

A348586 Numbers k such that abs(A328258(k)) = abs(A328258(k+1)).

1, 11, 40, 179, 695, 928, 991, 1079, 2772, 2799, 2839, 6687, 7632, 7739, 7960, 8568, 9347, 10703, 11008, 11472, 12847, 12935, 13580, 14064, 16000, 16260, 17135, 20944, 26432, 27999, 35399, 37236, 42251, 42756, 44199, 55308, 56419, 68976, 70127, 74671, 77748, 83099

Offset: 1

Amiram Eldar, Oct 24 2021

nonn

Equivalently, numbers k such that A328258(k) = -A328258(k+1).

			1 is a term since abs(A328258(1)) = abs(A328258(2)) = 1.
11 is a term since abs(A328258(11)) = abs(A328258(12)) = 12.

Amiram Eldar, Table of n, a(n) for n = 1..1000

The unitary version of A348585.
Cf. A328258.
Similar sequences: A002961, A064125, A206368, A333261.

f[p_, e_] := 1 - (-1)^p*(p^e); s[1] = 1; s[n_] := Abs[Times @@ (f @@@ FactorInteger[n])]; Select[Range[10^5], s[#] == s[# + 1] &]

f(n) = sumdiv(n, d, if (gcd(d, n/d) == 1, (-1)^(d + 1) * d)); \\ A328258
isok(k) = f(k) + f(k+1) == 0; \\ Michel Marcus, Oct 24 2021

A348584 Numbers k such that k | A328258(k).

Original entry on oeis.org

1, 12, 56, 180, 992, 16256, 127400, 441000, 2646000, 67100672, 325458000, 2758909440, 17179738112, 274877382656

Offset: 1

Amiram Eldar, Oct 24 2021

The corresponding ratios A113184(k)/k are 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -2, -1, -1, ...
If p is a Mersenne exponent (A000043), then 2^p*(2^p-1) (twice an even perfect number) is a term with ratio A328258(k)/k = -1.
If there exists an odd term k, then it is a unitary multiply-perfect number (A327158), since A328258(k) = A034448(k) for an odd k.

			12 is a term since A328258(12) = -12 is divisible by 12.

The unitary version of A348583.
A139256 is a subsequence.
Cf. A000043, A034448, A327158, A328258.

f[p_, e_] := 1 - (-1)^p*(p^e); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[3*10^6], Divisible[s[#], #] &]

A360156 a(n) is the sum of the even unitary divisors of 2*n.

Original entry on oeis.org

2, 4, 8, 8, 12, 16, 16, 16, 20, 24, 24, 32, 28, 32, 48, 32, 36, 40, 40, 48, 64, 48, 48, 64, 52, 56, 56, 64, 60, 96, 64, 64, 96, 72, 96, 80, 76, 80, 112, 96, 84, 128, 88, 96, 120, 96, 96, 128, 100, 104, 144, 112, 108, 112, 144, 128, 160, 120, 120, 192, 124, 128

Offset: 1

Amiram Eldar, Jan 28 2023

a(n) is the unitary analog of A146076(2*n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Octavio A. Agustín-Aquino, Wang-Sun Formula in GL(Z/2kZ), INTEGERS, Vol. 23 (2023), #A37; arXiv preprint, arXiv:2207.14495 [math.NT], 2022.

Cf. A002117, A034448, A100008, A146076, A171977, A192066, A328258.

usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; a[n_] := Module[{e = IntegerExponent[n, 2]}, 2^(e + 1) * usigma[n/2^e]]; Array[a, 100]

usigma(n) = {my(f = factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + 1)} ;
a(n) = {my(e = valuation(n, 2)); (1 << (e+1)) * usigma(n >> e); }

a(n) = Sum_{even d|(2*n), gcd(d, 2*n/d)=1} d.
a(n) = A034448(2*n) - A192066(2*n).
a(n) = A192066(2*n) - A328258(2*n).
a(n) = A171977(n) * A192066(n).
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / (7*zeta(3)).
Dirichlet g.f. of b(n): (zeta(s)*zeta(s-1)/zeta(2*s-1))*(2^(s+1)-2)/(2^(2*s)-2), where b(n) is the sum of the even unitary divisors of n: b(n) = a(n/2) if n is even and 0 otherwise.

cp's OEIS Frontend

A348586 Numbers k such that abs(A328258(k)) = abs(A328258(k+1)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A348584 Numbers k such that k | A328258(k).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A360156 a(n) is the sum of the even unitary divisors of 2*n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula