A061142 -id:A061142 - cp's OEIS frontend

A167332 Totally multiplicative sequence with a(p) = 2(4p-1) = 8*p-2 for prime p.

1, 14, 22, 196, 38, 308, 54, 2744, 484, 532, 86, 4312, 102, 756, 836, 38416, 134, 6776, 150, 7448, 1188, 1204, 182, 60368, 1444, 1428, 10648, 10584, 230, 11704, 246, 537824, 1892, 1876, 2052, 94864, 294, 2100, 2244, 104272, 326, 16632, 342, 16856

Offset: 1

Jaroslav Krizek, Nov 01 2009

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001222, A061142, A166653.

f:=n -> mul((8*t[1]-2)^t[2],t=ifactors(n)[2]):
map(f, [$1..100]); # Robert Israel, Jun 06 2016

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((4*fi[[All, 1]] - 1)^fi[[All, 2]])); Table[a[n]*2^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 06 2016 *)
f[p_, e_] := (8*p-2)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 18 2023 *)

a(n) = {my(f=factor(n)); for (k=1, #f~, f[k,1] = 8*f[k,1]-2;); factorback(f);} \\ Michel Marcus, Jun 06 2016

Multiplicative with a(p^e) = (2*(4*p-1))^e. If n = Product p(k)^e(k) then a(n) = Product (2*(4*p(k)-1))^e(k).

a(n) = A061142(n) * A166653(n) = 2^bigomega(n) * A166653(n) = 2^A001222(n) * A166653(n).

A167333 Totally multiplicative sequence with a(p) = 2*(5p-1) = 10p-2 for prime p.

Original entry on oeis.org

1, 18, 28, 324, 48, 504, 68, 5832, 784, 864, 108, 9072, 128, 1224, 1344, 104976, 168, 14112, 188, 15552, 1904, 1944, 228, 163296, 2304, 2304, 21952, 22032, 288, 24192, 308, 1889568, 3024, 3024, 3264, 254016, 368, 3384, 3584, 279936, 408, 34272, 428

Offset: 1

Jaroslav Krizek, Nov 01 2009

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001222, A061142, A166654.

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((5*fi[[All, 1]] - 1)^fi[[All, 2]])); Table[a[n]*2^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 06 2016 *)
f[p_, e_] := (10*p-2)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 18 2023 *)

a(n) = {my(f=factor(n)); for (k=1, #f~, f[k,1] = 10*f[k,1]-2;); factorback(f);} \\ Michel Marcus, Jun 06 2016

Multiplicative with a(p^e) = (2*(5p-1))^e. If n = Product p(k)^e(k) then a(n) = Product (2*(5*p(k)-1))^e(k).

a(n) = A061142(n) * A166654(n) = 2^bigomega(n) * A166654(n) = 2^A001222(n) * A166654(n).

A167334 Totally multiplicative sequence with a(p) = 2*(2p+1) = 4p+2 for prime p.

Original entry on oeis.org

1, 10, 14, 100, 22, 140, 30, 1000, 196, 220, 46, 1400, 54, 300, 308, 10000, 70, 1960, 78, 2200, 420, 460, 94, 14000, 484, 540, 2744, 3000, 118, 3080, 126, 100000, 644, 700, 660, 19600, 150, 780, 756, 22000, 166, 4200, 174, 4600, 4312, 940, 190, 140000, 900

Offset: 1

Jaroslav Krizek, Nov 01 2009

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001222, A061142, A166660.

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((2*fi[[All, 1]] + 1)^fi[[All, 2]])); Table[a[n]*2^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 06 2016 *)
f[p_, e_] := (4*p+2)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 19 2023 *)

Multiplicative with a(p^e) = (2*(2p+1))^e. If n = Product p(k)^e(k) then a(n) = Product (2*(2*p(k)+1))^e(k).

a(n) = A061142(n) * A166660(n) = 2^bigomega(n) * A166660(n) = 2^A001222(n) * A166660(n).

A167335 Totally multiplicative sequence with a(p) = 2*(3p+1) = 6p+2 for prime p.

Original entry on oeis.org

1, 14, 20, 196, 32, 280, 44, 2744, 400, 448, 68, 3920, 80, 616, 640, 38416, 104, 5600, 116, 6272, 880, 952, 140, 54880, 1024, 1120, 8000, 8624, 176, 8960, 188, 537824, 1360, 1456, 1408, 78400, 224, 1624, 1600, 87808, 248, 12320, 260, 13328, 12800

Offset: 1

Jaroslav Krizek, Nov 01 2009

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001222, A061142, A166661.

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((3*fi[[All, 1]] + 1)^fi[[All, 2]])); Table[a[n]*2^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 06 2016 *)
f[p_, e_] := (6*p+2)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 19 2023 *)

Multiplicative with a(p^e) = (2*(3p+1))^e. If n = Product p(k)^e(k) then a(n) = Product (2*(3*p(k)+1))^e(k).

a(n) = A061142(n) * A166661(n) = 2^bigomega(n) * A166661(n) = 2^A001222(n) * A166661(n).

A167336 Totally multiplicative sequence with a(p) = 2*(4p+1) = 8p+2 for prime p.

Original entry on oeis.org

1, 18, 26, 324, 42, 468, 58, 5832, 676, 756, 90, 8424, 106, 1044, 1092, 104976, 138, 12168, 154, 13608, 1508, 1620, 186, 151632, 1764, 1908, 17576, 18792, 234, 19656, 250, 1889568, 2340, 2484, 2436, 219024, 298, 2772, 2756, 244944, 330, 27144, 346

Offset: 1

Jaroslav Krizek, Nov 01 2009

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001222, A061142, A166662.

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((4*fi[[All, 1]] + 1)^fi[[All, 2]])); Table[a[n]*2^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 06 2016 *)
f[p_, e_] := (8*p+2)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 19 2023 *)

Multiplicative with a(p^e) = (2*(4p+1))^e. If n = Product p(k)^e(k) then a(n) = Product (2*(4*p(k)+1))^e(k).

a(n) = A061142(n) * A166662(n) = 2^bigomega(n) * A166662(n) = 2^A001222(n) * A166662(n).

A167337 Totally multiplicative sequence with a(p) = 2*(5p+1) = 10p+2 for prime p.

Original entry on oeis.org

1, 22, 32, 484, 52, 704, 72, 10648, 1024, 1144, 112, 15488, 132, 1584, 1664, 234256, 172, 22528, 192, 25168, 2304, 2464, 232, 340736, 2704, 2904, 32768, 34848, 292, 36608, 312, 5153632, 3584, 3784, 3744, 495616, 372, 4224, 4224, 553696, 412, 50688

Offset: 1

Jaroslav Krizek, Nov 01 2009

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001222, A061142, A166663.

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((5*fi[[All, 1]] + 1)^fi[[All, 2]])); Table[a[n]*2^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 06 2016 *)
f[p_, e_] := (10*p+2)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 19 2023 *)

Multiplicative with a(p^e) = (2*(5p+1))^e. If n = Product p(k)^e(k) then a(n) = Product (2*(5*p(k)+1))^e(k).

a(n) = A061142(n) * A166663(n) = 2^bigomega(n) * A166663(n) = 2^A001222(n) * A166663(n).

A328854 Dirichlet g.f.: Product_{p prime} 1 / (1 - 2 * p^(-s))^2.

Original entry on oeis.org

1, 4, 4, 12, 4, 16, 4, 32, 12, 16, 4, 48, 4, 16, 16, 80, 4, 48, 4, 48, 16, 16, 4, 128, 12, 16, 32, 48, 4, 64, 4, 192, 16, 16, 16, 144, 4, 16, 16, 128, 4, 64, 4, 48, 48, 16, 4, 320, 12, 48, 16, 48, 4, 128, 16, 128, 16, 16, 4, 192, 4, 16, 48, 448, 16, 64, 4, 48, 16, 64, 4, 384, 4, 16, 48

Offset: 1

Ilya Gutkovskiy, Oct 28 2019

Dirichlet convolution of A061142 with itself.

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

Cf. A000005, A001222, A061142, A123667, A322328.

Table[2^PrimeOmega[n] DivisorSigma[0, n], {n, 1, 75}]
f[p_, e_] := (e + 1)*2^e; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)

a(n) = numdiv(n)*2^bigomega(n); \\ Michel Marcus, Dec 02 2020

for(n=1, 100, print1(direuler(p=2, n, 1/(1 - 2*X)^2)[n], ", ")) \\ Vaclav Kotesovec, Aug 22 2021

If n = Product (p_j^k_j) then a(n) = Product (2^k_j * (k_j + 1)).

a(n) = 2^bigomega(n) * tau(n), where bigomega = A001222 and tau = A000005.

A353172 a(n) is the least k > 1 such that Omega(n) = Omega(n mod k), where Omega = A001222.

Original entry on oeis.org

2, 3, 4, 5, 3, 7, 4, 9, 5, 6, 3, 13, 5, 5, 9, 17, 3, 10, 4, 12, 11, 6, 3, 25, 7, 10, 15, 10, 3, 11, 4, 33, 9, 5, 13, 20, 5, 8, 5, 24, 3, 15, 4, 9, 25, 6, 3, 49, 5, 14, 9, 11, 3, 19, 7, 20, 12, 6, 3, 22, 7, 7, 11, 65, 11, 18, 4, 10, 5, 25, 3, 40, 5, 5, 19, 16

Offset: 1

Thomas Scheuerle, Apr 28 2022

It appears that a(m) = m*k/p if m = p*2^n ... . Are these formulas related to some well-known sequence of rational numbers?

			a(10) = 6 because 10 = 5*2 and 10 mod 6 = 4 = 2*2.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000040, A001222, A003627, A029744, A061142, A077065, A095925.

a(n) = my(k=2); while(bigomega(n) != bigomega(max(n%k,1)), k++); k

from itertools import count
from sympy.ntheory.factor_ import primeomega
def A353172(n):
    a = primeomega(n)
    for k in count(2):
        if (m := n % k) > 0 and primeomega(m) == a:
            return k # Chai Wah Wu, Jun 20 2022

a(A029744(n)) = A029744(n) + 1.
a(A003627(n)) = 3.
a(A000040(n)) = A095925(n).
a(A077065(n)) = 6. For n > 2.
If a(n) = 10, then n mod 10 is in most cases 8 and seldom 6.
a(m) = m*3/5 if m = 5*2^n or m = 15. This formula is valid for all positive n because (5*2^n) mod (5*2^n)*(3/5) = 2^(n+1). If the sequence of solutions does not create powers of two in the modulo operation, it will be of finite length. See next two formulas:
a(m) = m*3/11 if m = 11, 22, 33 or 66.
a(m) = m*4/43 if m = 43*2^n for n < 4. This series of solutions terminates because of the next formula which replaces the powers of two:
a(m) = m*41/(43*2^4) if m = 43*2^4*2^n. This formula is valid for all positive n.
a(m) = m*5/9 if m = 9*2^n or m = 27 or 45. This formula is valid for all positive n.
For each k = a(p) if k < p and gcd(k, p) = 1 such a formula, of the form a(m) = m*k/p, if m = p*2^n ..., can be developed.

A375286 a(n) = f(1) + f(2) + ... + f(n), where f(n) = (-2)^Omega(n) = A165872(n).

Original entry on oeis.org

1, -1, -3, 1, -1, 3, 1, -7, -3, 1, -1, -9, -11, -7, -3, 13, 11, 3, 1, -7, -3, 1, -1, 15, 19, 23, 15, 7, 5, -3, -5, -37, -33, -29, -25, -9, -11, -7, -3, 13, 11, 3, 1, -7, -15, -11, -13, -45, -41, -49, -45, -53, -55, -39, -35, -19, -15, -11, -13, 3, 1, 5, -3, 61

Offset: 1

Charles R Greathouse IV, Aug 09 2024

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Daniel R. Johnston, Nicol Leong, and Sebastian Tudzi, New bounds and progress towards a conjecture on the summatory function of (-2)^Omega(n), arXiv:2408.04143 [math.NT], 2024.
Michael J. Mossinghoff and Timothy S. Trudgian, Oscillations in weighted arithmetic sums, arXiv:2007.14537 [math.NT], 2020.
Zhi-Wei Sun, On a pair of zeta functions, arXiv:1204.6689 [math.NT], 2012.

Partial sums of A165872.

Cf. A001222, A002321, A212496, A061142, A069205.

a:= proc(n) option remember; `if`(n<1, 0,
      a(n-1)+(-2)^numtheory[bigomega](n))
    end:
seq(a(n), n=1..64);  # Alois P. Heinz, Apr 25 2025

PARI
```
s=0; vector(60,n,s+=(-2)^bigomega(n))
```

Johnston, Leong, & Tudzi prove that |a(n)| < 2260n. Sun conjectures that |a(n)| < n for n >= 3078. Mossinghoff & Trudgian verify this to 2.5 * 10^14.

Because of powers of two, |a(n)| >= n/2 infinitely often.

A354273 Square array read by ascending antidiagonals: A(n,k) = k^Omega(n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 4, 3, 4, 1, 1, 2, 9, 4, 5, 1, 1, 4, 3, 16, 5, 6, 1, 1, 2, 9, 4, 25, 6, 7, 1, 1, 8, 3, 16, 5, 36, 7, 8, 1, 1, 4, 27, 4, 25, 6, 49, 8, 9, 1, 1, 4, 9, 64, 5, 36, 7, 64, 9, 10, 1, 1, 2, 9, 16, 125, 6, 49, 8, 81, 10, 11, 1, 1, 8, 3, 16, 25, 216, 7, 64, 9, 100, 11, 12, 1

Offset: 1

Stefano Spezia, May 22 2022

			Array begins:
    1, 1,  1,  1,   1,   1,   1,   1, ...
    1, 2,  3,  4,   5,   6,   7,   8, ...
    1, 2,  3,  4,   5,   6,   7,   8, ...
    1, 4,  9, 16,  25,  36,  49,  64, ...
    1, 2,  3,  4,   5,   6,   7,   8, ...
    1, 4,  9, 16,  25,  36,  49,  64, ...
    1, 2,  3,  4,   5,   6,   7,   8, ...
    1, 8, 27, 64, 125, 216, 343, 512, ...
    ...

K. L. Verma, On an arithmetical functions involving general exponential, Palestine Journal of Mathematics Vol. 11(2)(2022), 496-504.

Cf. A001222, A051129.

Cf. A000012 (n = 1 or k = 1), A061142 (k = 2), A165824 - A165871 (k = 3..50), A176029 (diagonal).

A[n_,k_]:=k^PrimeOmega[n]; Flatten[Table[A[n-k+1,k],{n,13},{k,n}]]

A(n, k) = A051129(A001222(n), k).

The columns are totally multiplicative: A(i*j, k) = A(i, k)*A(j, k).

cp's OEIS Frontend

A167332 Totally multiplicative sequence with a(p) = 2*(4*p-1) = 8*p-2 for prime p.

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A167333 Totally multiplicative sequence with a(p) = 2*(5p-1) = 10p-2 for prime p.

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A167334 Totally multiplicative sequence with a(p) = 2*(2p+1) = 4p+2 for prime p.

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A167335 Totally multiplicative sequence with a(p) = 2*(3p+1) = 6p+2 for prime p.

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A167336 Totally multiplicative sequence with a(p) = 2*(4p+1) = 8p+2 for prime p.

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A167337 Totally multiplicative sequence with a(p) = 2*(5p+1) = 10p+2 for prime p.

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A328854 Dirichlet g.f.: Product_{p prime} 1 / (1 - 2 * p^(-s))^2.

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A353172 a(n) is the least k > 1 such that Omega(n) = Omega(n mod k), where Omega = A001222.

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A167332 Totally multiplicative sequence with a(p) = 2(4p-1) = 8*p-2 for prime p.