A061142 -id:A061142 - cp's OEIS frontend

A162510 Dirichlet inverse of A076479.

1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 2, 1, 1, 1, 8, 1, 2, 1, 2, 1, 1, 1, 4, 2, 1, 4, 2, 1, 1, 1, 16, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 8, 2, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 2, 1, 1, 2, 32, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 2, 2, 1, 1, 1, 8, 8, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 16, 1, 2, 2, 4, 1, 1, 1, 4, 1

Offset: 1

Gerard P. Michon, Jul 05 2009

Apart from signs, this sequence is identical to A162512.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Generalizations of Mertens's Formula and k-Free and s-Full Numbers with Prime Divisors in Arithmetic Progression, ResearchGate, 2024.
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms).
Gérard P. Michon, Multiplicative functions.

Cf. A076479, A162511, A162512.
Cf. A000079, A001222, A005117, A034444, A046660, A061142.
Cf. A167864, A347195.

A162510 := proc(n)
    local a,f;
    a := 1;
    for f in ifactors(n)[2] do
        a := a*2^(op(2,f)-1) ;
    end do:
    return a;
end proc: # R. J. Mathar, May 20 2017

a[n_] := 2^(PrimeOmega[n] - PrimeNu[n]); Array[a, 100] (* Jean-François Alcover, Apr 24 2017, after R. J. Mathar *)

a(n)=my(f=factor(n)[,2]); 2^(vecsum(f)-#f) \\ Charles R Greathouse IV, Nov 02 2016

from sympy import factorint
from operator import mul
def a(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [2**(f[i] - 1) for i in f]) # Indranil Ghosh, May 20 2017

Multiplicative with a(p^e) = 2^(e-1) for any prime p and any positive exponent e.
a(n) = n/2 when n is a power of 2 (A000079).
a(n) = 1 when n is a squarefree number (A005117).
a(n) = 2^A046660(n) = A061142(n)/A034444(n). - R. J. Mathar, Nov 02 2016
a(n) = Sum_{d|n} mu(d) * 2^A001222(n/d). - Daniel Suteu, May 21 2020
a(1) = 1; a(n) = -Sum_{d|n, d < n} (-1)^omega(n/d) * a(d). - Ilya Gutkovskiy, Mar 10 2021
Dirichlet g.f.: (1/zeta(s)) * Product_{p prime} (1/(1 - 2/p^s)). - Amiram Eldar, Sep 16 2023
Sum_{k=1..n} 1/a(k) = c * n + o(n), where c = Product_{p prime} (1 - 1/(p*(2*p-1))) = 0.74030830284678515949... (Jakimczuk, 2024, Theorem 2.4, p. 16). - Amiram Eldar, Mar 08 2024
From Vaclav Kotesovec, Mar 08 2024: (Start)
Dirichlet g.f.: zeta(s) * (1 + 1/(2^s*(2^s - 2))) * f(s), where f(s) = Product_{p prime, p>2} (1 + 1/(p^s*(p^s - 2))).
Sum_{k=1..n} a(k) ~ (f(1)*n / (4*log(2))) * (log(n) - 1 + gamma + 5*log(2)/2 + f'(1)/f(1)), where
f(1) = Product_{p prime, p>2} (1 + 1/(p*(p-2))) = A167864 = 1.51478012813749125771853381230067247330485921179389884042843306025133959...,
f'(1) = f(1) * Sum_{p prime, p>2} (-2*log(p)/((p-1)*(p-2))) = -2*f(1)*A347195 = -2.6035805486753944250682818932032862770113061830543948257159113584026980...
and gamma is the Euler-Mascheroni constant A001620. (End)

A165826 Totally multiplicative sequence with a(p) = 5.

Original entry on oeis.org

1, 5, 5, 25, 5, 25, 5, 125, 25, 25, 5, 125, 5, 25, 25, 625, 5, 125, 5, 125, 25, 25, 5, 625, 25, 25, 125, 125, 5, 125, 5, 3125, 25, 25, 25, 625, 5, 25, 25, 625, 5, 125, 5, 125, 125, 25, 5, 3125, 25, 125

Offset: 1

Jaroslav Krizek, Sep 28 2009

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

Cf. A000351, A001222, A061142, A165824, A165825.

5^PrimeOmega[Range[100]] (* G. C. Greubel, Apr 09 2016 *)

for(n=1, 100, print1(direuler(p=2, n, 1/(1-5*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 28 2023

a(n) = A000351(A001222(n)) = 5^bigomega(n) = 5^A001222(n).

Dirichlet g.f.: Product_{p prime} 1 / (1 - 5 * p^(-s)). - Ilya Gutkovskiy, Oct 30 2019

A167864 Decimal expansion of Selberg-Delange constant Product_{prime p > 2} (1 + 1/(p(p-2))).

Original entry on oeis.org

1, 5, 1, 4, 7, 8, 0, 1, 2, 8, 1, 3, 7, 4, 9, 1, 2, 5, 7, 7, 9, 0, 9, 1, 9, 2, 5, 5, 6, 4, 9, 4, 7, 4, 8, 9, 2, 4, 1, 5, 2, 7, 0, 1, 5, 8, 2, 8, 6, 2, 1, 4, 3, 9, 5, 3, 5, 7, 4, 8, 4, 2, 7, 1, 4, 8, 4, 9, 3, 2, 2, 0, 9, 8, 1, 5, 6, 1, 1, 5, 8, 1, 0, 8, 7, 7, 5, 8, 5, 3, 8, 2, 7, 6, 9, 8, 0, 7, 6, 7, 7, 6, 5, 6, 2

Offset: 1

Jonathan Sondow, Nov 13 2009, Nov 17 2009

Coefficient in formulas for the distribution of integers with a fixed number of prime factors.
Reciprocal of the twin prime constant A005597. See A005597 for links and additional references and comments.
Numerators of partial products are A062271. Denominators are A062270.
An analog for primes of Wallis' product pi/2 = Product_{n >=1} (2n)^2/(2n-1)(2n+1), because A167864 = Product_{prime p>2} (p-1)^2/(p-2)p.
Grosswald (see links) proves that Sum_{k<=x} 2^Omega(k) ~ (1/(8*log(2))) * c * x * (log(x))^2 + O(x * log(x)) where c is this constant. - Amiram Eldar, Jun 06 2020
The asymptotic density of numbers m with A046660(m) = Omega(m) - omega(m) = k is asymptotically ~ c/2^(k+2) as k -> oo, where c is this constant (Rényi, 1955). - Amiram Eldar, Aug 08 2020
Named after the Norwegian mathematician Atle Selberg (1917-2007) and the French mathematician Hubert Delange (1914-2003). - Amiram Eldar, Jun 20 2021

			Product_{prime p > 2} (1 + 1/(p(p-2))) = 1.5147801281374912577909192556...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, pp. 84-93.
Atle Selberg, Note on a paper by L. G. Sathe, J. Indian Math. Soc., Vol. 18, No. 1 (1954), pp. 83-87.
Gérald Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995, p. 206.

Michel Balazard, Hubert Delange and Jean-Louis Nicolas, Sur le nombre de facteurs premiers des entiers, C. R. Acad. Sci., Paris, Ser. I, Vol. 306 (1988), pp. 511-514. [From Jonathan Sondow, Nov 17 2009]
Hubert Delange, Sur des formules de Atle Selberg, Acta Arith., Vol. 19 (1971), pp. 105-146.
Steven R. Finch, Mathematical Constants, Errata and Addenda, Sec. 2.1.
Emil Grosswald, The average order of an arithmetic function, Duke Mathematical Journal, Vol. 23, No. 1 (1956), pp. 41-44.
Jean-Louis Nicolas, Sur la distribution des nombres entiers ayant une quantite fixée de facteurs premiers, Acta Arith., Vol. 44 (1984), pp. 191-200.
Alfred Rényi, On the density of certain sequences of integers, Publications de l'Institut Mathématique, Vol. 8 (1955), pp. 157-162.

Cf. A005597.

Cf. A001222 (Omega), A046660, A061142 (2^Omega), A069205 (partial sums of 2^Omega).

s[n_] := (1/n)* N[Sum[MoebiusMu[d]*2^(n/d), {d, Divisors[n]}], 160]; C2 = (175/256)*Product[(Zeta[ n]*(1 - 2^(-n))*(1 - 3^(-n))*(1 - 5^(-n))*(1 - 7^(-n)))^(-s[ n]), {n, 2, 160}]; RealDigits[1/C2][[1]][[1 ;; 105]] (* Jean-François Alcover, Oct 30 2012, after Pari program in A005597 *)
$MaxExtraPrecision = 300; digits = 105; terms = 600; P[n_] := PrimeZetaP[n] - 1/2^n; LR = Join[{0, 0}, LinearRecurrence[{3, -2}, {2, 6}, terms + 10]]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*P[n-1]/(n-1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits+10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 19 2016 *)

prodeulerrat((1 + 1/(p*(p-2))),,3) \\ Hugo Pfoertner, Aug 08 2020

Equals 1/A005597.

Equals Product_{prime p>2} (p-1)^2/(p-2)p = (2^2/1*3)(4^2/3*5)(6^2/5*7)(10^2/9*11) ....

A124508 a(n) = 2^BigO(n) * 3^omega(n), where BigO = A001222 and omega = A001221, the numbers of prime factors of n with and without repetitions.

Original entry on oeis.org

1, 6, 6, 12, 6, 36, 6, 24, 12, 36, 6, 72, 6, 36, 36, 48, 6, 72, 6, 72, 36, 36, 6, 144, 12, 36, 24, 72, 6, 216, 6, 96, 36, 36, 36, 144, 6, 36, 36, 144, 6, 216, 6, 72, 72, 36, 6, 288, 12, 72, 36, 72, 6, 144, 36, 144, 36, 36, 6, 432, 6, 36, 72, 192, 36, 216, 6, 72, 36, 216, 6, 288, 6

Offset: 1

Reinhard Zumkeller, Nov 04 2006

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Factorization.
Eric Weisstein's World of Mathematics, Smooth number.

Cf. A000079, A000244, A000400, A001221, A001222, A003586, A005117, A007283, A061142, A074816, A124509, A124510, A124511, A124512.

Table[2^PrimeOmega[n] 3^PrimeNu[n],{n,80}] (* Harvey P. Dale, Mar 26 2013 *)

a(n) = my(f = factor(n)); 2^bigomega(f) * 3^omega(f); \\ Amiram Eldar, Jul 11 2023

Multiplicative with p^e -> 3*2^e, p prime and e>0.
a(n) = A061142(n)*A074816(n) = A000079(A001222(n))*A000244(A001221(n)).
A124509 gives the range: A124509(n) = a(A124510(n)) and a(m) <> a(A124510(n)) for m < A124510(n).
For primes p, q with p <> q: a(p) = 6; a(p*q) = 36; a(p^k) = 3*2^k, k>0.
For squarefree numbers m: a(m) = 6^omega(m).
A001222(a(n)) = A001222(n)+1; A001221(a(n)) = 2 for n > 1.
A124511(n) = a(a(n)); A124512(n) = a(a(a(n))).

A226177 a(n) = mu(n)*d(n), where mu(n) = A008683 and d(n) = A000005.

Original entry on oeis.org

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

Offset: 1

Wesley Ivan Hurt, May 29 2013

The prime numbers are the only solutions to mu(n)*d(n) = -2.
Multiplicative with a(p) = -2, a(p^e) = 0, e > 1.
The Moebius inverse is A076479, and the Dirichlet inverse A061142. - R. J. Mathar, Jun 03 2013
Möbius transform of (-1)^omega(n). - Wesley Ivan Hurt, Jun 22 2024

			a(5) = mu(5)*d(5) = (-1)(2) = -2.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A000040, A001358, A008683, A074823 (absolute values), A001221.

with(numtheory); a:=n->mobius(n)*tau(n); seq(a(k),k=1..100);

Table[MoebiusMu[n] DivisorSigma[0, n], {n, 105}] (* Michael De Vlieger, Jul 23 2017 *)

A226177(n) = moebius(n)*numdiv(n); \\ Antti Karttunen, Jul 23 2017

for(n=1, 100, print1(direuler(p=2, n, (1 - 2*X))[n], ", ")); \\ Vaclav Kotesovec, Aug 21 2021

(define (A226177 n) (if (= 1 n) n (* (if (= 1 (A067029 n)) -2 0) (A226177 (A028234 n))))) ;; Antti Karttunen, Jul 23 2017

a(n) = mu(n)*d(n) = A008683(n)*A000005(n).
Sum_{n>0} a(n)/n^s = Product_{p prime} (1 - 2p^(-s)). - Ralf Stephan, Jul 07 2013
a(n) = mu(n) * 2^omega(n) = |mu(n)| * (-2)^omega(n), where omega = A001221. - Álvar Ibeas, Dec 30 2018
a(n) = Sum_{d|n} (-1)^omega(d) * mu(n/d). - Wesley Ivan Hurt, Jun 22 2024

More terms from Antti Karttunen, Jul 23 2017

Name changed by David A. Corneth, Jul 23 2017

A318307 Multiplicative with a(p^e) = 2^A002487(e).

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 8, 2, 8, 4, 4, 4, 4, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 8, 4, 8, 4, 4, 2, 8, 2, 4, 4, 4, 4, 8, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4, 8, 2, 8, 4, 4, 4, 4, 4, 16, 2, 4, 4, 4, 2, 8, 2, 8, 8

Offset: 1

Antti Karttunen, Aug 29 2018

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A318306, A318316, A318470, A318667.

Differs from A037445 for the first time at n=32, where a(32) = 8, while A037445(32) = 4.

f[m_] := Module[{a = 1, b = 0, n = m}, While[n > 0, If[OddQ[n], b += a, a += b]; n = Floor[n/2]]; b]; Array[Times @@ Map[2^f@ # &, FactorInteger[#][[All, -1]] ] - Boole[# == 1] &, 105] (* after Jean-François Alcover at A002487 *)

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A318307(n) = factorback(apply(e -> 2^A002487(e),factor(n)[,2]));

from functools import reduce
from sympy import factorint
def A318307(n): return 1<Chai Wah Wu, May 18 2023

a(n) = 2^A318306(n).
a(n) = A061142(A318470(n)).
a(n^2) = a(n).
a(A003557(n^2)) = A318316(n).
Dirichlet convolution square of A318667(n)/A317934(n).

A322838 Number of positive integers less than n with more prime factors than n, counted with multiplicity.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 2, 0, 1, 1, 5, 0, 6, 2, 2, 0, 9, 1, 10, 1, 5, 5, 13, 0, 6, 6, 2, 2, 18, 2, 19, 0, 10, 10, 10, 1, 24, 11, 11, 1, 27, 5, 28, 5, 5, 15, 31, 0, 16, 6, 17, 6, 36, 2, 19, 2, 20, 20, 41, 2, 42, 21, 9, 0, 23, 10, 47, 10, 25, 10, 50, 1, 51, 27, 11, 11

Offset: 1

Gus Wiseman, Dec 28 2018

nonn

			Column n lists the a(n) positive integers less than n with more prime factors than n:
  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20
  ---------------------------------------------------------------------
              4     6     8  8   10      12  12  12      16  16  18  16
                    4            9       10  8   8       15      16
                                 8       9               14      15
                                 6       8               12      14
                                 4       6               10      12
                                         4               9       10
                                                         8       9
                                                         6       8
                                                         4       6
                                                                 4

Positions of zeros appear to be A029744.

Cf. A000010, A000961, A001221, A001222, A045920, A058933, A061142, A067003, A302242, A322839, A322841.

Table[Length[Select[Range[n],PrimeOmega[#]>PrimeOmega[n]&]],{n,100}]

A351347 Dirichlet g.f.: Product_{p prime} 1 / (1 - p^(-s) - 2p^(-2s)).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 5, 3, 1, 1, 3, 1, 1, 1, 11, 1, 3, 1, 3, 1, 1, 1, 5, 3, 1, 5, 3, 1, 1, 1, 21, 1, 1, 1, 9, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 11, 3, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 3, 1, 1, 3, 43, 1, 1, 1, 3, 1, 1, 1, 15, 1, 1, 3, 3, 1, 1, 1, 11, 11, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 21, 1, 3, 3, 9

Offset: 1

Ilya Gutkovskiy, Feb 08 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (2*10^8 terms)

Cf. A001045, A061142, A351219, A351346, A351348.

f[p_, e_] := (2^(e + 1) + (-1)^e)/3; a[n_] := Times @@ f @@@ FactorInteger[n]; Table[a[n], {n, 1, 100}]

for(n=1, 100, print1(direuler(p=2, n, 1/(1 - X - 2*X^2))[n], ", ")) \\ Vaclav Kotesovec, Feb 10 2022

Multiplicative with a(p^e) = Jacobsthal(e+1).
From Vaclav Kotesovec, Feb 11 2022: (Start)
Let f(s) = Product_{prime p>2} (1 - 3/p^(2*s) + 2/p^(3*s))/(1 - 4/p^(2*s)), then
Sum_{k=1..n} a(k) ~ n*((2 * Pi^2 * log(n) + Pi^2 * (5*log(2) + 2*gamma - 2) + 24*zeta'(2))*f(1) + 2*Pi^2 * f'(1)) / (48*log(2)), where
f(1) = Product_{prime p > 2} (1 + 1/(p*(p-2))) = A167864 = 1.5147801281374912577909192556494748924152701582862143953574842714849322098...,
f'(1) = -f(1) * Sum_{primes p > 2} 2*log(p) / (2 - 3*p + p^2) = -2*f(1)*A347195 = -2.603580548675394425068281893203286277011306183054394825715911358402698051... and gamma is the Euler-Mascheroni constant A001620. (End)

A123667 a(n) = n * 2^bigomega(n).

Original entry on oeis.org

1, 4, 6, 16, 10, 24, 14, 64, 36, 40, 22, 96, 26, 56, 60, 256, 34, 144, 38, 160, 84, 88, 46, 384, 100, 104, 216, 224, 58, 240, 62, 1024, 132, 136, 140, 576, 74, 152, 156, 640, 82, 336, 86, 352, 360, 184, 94, 1536, 196, 400, 204, 416, 106, 864, 220, 896, 228, 232, 118

Offset: 1

Franklin T. Adams-Watters, Oct 04 2006

Rearrangement of A123666.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A061142, A001222 (bigomega), A123666.

A123667:=n->n*2^numtheory[bigomega](n): seq(A123667(n), n=1..100); # Wesley Ivan Hurt, Apr 09 2017

Table[n*2^PrimeOmega[n],{n,60}] (* Harvey P. Dale, Jun 11 2014 *)

a(n) = n*2^bigomega(n); \\ Michel Marcus, Apr 10 2017

Totally multiplicative with a(p) = 2p, p prime.
a(n) = n * A061142(n).
Dirichlet g.f.: Product_{p prime} 1 / (1 - 2 * p^(1 - s)). - Ilya Gutkovskiy, Oct 29 2019

A342767 Array T(n, k), n, k > 0, read by antidiagonals; a variant of lunar multiplication (A087062) based on prime factorizations of numbers (see Comments section for precise definition).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 4, 3, 4, 1, 1, 2, 4, 4, 2, 1, 1, 4, 3, 8, 3, 4, 1, 1, 2, 6, 4, 4, 6, 2, 1, 1, 8, 3, 8, 5, 8, 3, 8, 1, 1, 4, 8, 4, 6, 6, 4, 8, 4, 1, 1, 4, 9, 16, 5, 12, 5, 16, 9, 4, 1, 1, 2, 6, 8, 8, 6, 6, 8, 8, 6, 2, 1, 1, 8, 3, 8, 9, 16, 7, 16, 9, 8, 3, 8, 1

Offset: 1

Rémy Sigrist, Apr 02 2021

To compute T(n, k):
- write the prime factors of n and of k in ascending order with multiplicities on two lines, right aligned,
- to "multiply" two prime numbers: take the smallest,
- to "add" two prime numbers: take the largest,
- for example, for T(12, 14):
     12   ->   2 2 3
     14   -> x   2 7
             -------
               2 2 3
           + 2 2 2
           ---------
             2 2 2 3   ->  24 = T(12, 14)
This sequence is closely related to lunar multiplication (A087062):
- let n and k be two p-smooth numbers,
- let f be the function that associates to a p-smooth number, say m, the unique number whose (p+1)-base digits are prime, nondecreasing and whose product is m,
- let g be the inverse of f,
- then for any p-smooth numbers n and k, T(n, k) = g(f(n) "*" f(k)) where "*" denotes lunar product in base p+1,
- as T(n, p) = n for any prime number >= A006530(n), we don't have prime numbers here,
- however, if we consider only p-smooth numbers (for some prime number p), then p is the "unit" and the semiprimes p*q (with q <= p) are "prime".

			Array T(n, k) begins:
  n\k|  1  2   3   4   5   6   7   8   9  10  11  12  13  14
  ---+------------------------------------------------------
    1|  1  1   1   1   1   1   1   1   1   1   1   1   1   1
    2|  1  2   2   4   2   4   2   8   4   4   2   8   2   4  ->  A061142
    3|  1  2   3   4   3   6   3   8   9   6   3  12   3   6  ->  A079065
    4|  1  4   4   8   4   8   4  16   8   8   4  16   4   8
    5|  1  2   3   4   5   6   5   8   9  10   5  12   5  10
    6|  1  4   6   8   6  12   6  16  18  12   6  24   6  12
    7|  1  2   3   4   5   6   7   8   9  10   7  12   7  14
    8|  1  8   8  16   8  16   8  32  16  16   8  32   8  16
    9|  1  4   9   8   9  18   9  16  27  18   9  36   9  18
   10|  1  4   6   8  10  12  10  16  18  20  10  24  10  20
   11|  1  2   3   4   5   6   7   8   9  10  11  12  11  14
   12|  1  8  12  16  12  24  12  32  36  24  12  48  12  24
   13|  1  2   3   4   5   6   7   8   9  10  11  12  13  14
   14|  1  4   6   8  10  12  14  16  18  20  14  24  14  28

Rémy Sigrist, Table of n, a(n) for n = 1..10011
Rémy Sigrist, PARI program for A342767
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A006530, A061142, A079065, A087062, A342765, A342768.

PARI
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T(n, k) = T(k, n).
T(n, n) = A342768(n).
T(n, 1) = 1.
T(n, 2) = A061142(n).
T(n, 3) = A079065(n).
T(n, p) = n for any prime number p >= A006530(n).

cp's OEIS Frontend

A162510 Dirichlet inverse of A076479.

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A165826 Totally multiplicative sequence with a(p) = 5.

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A167864 Decimal expansion of Selberg-Delange constant Product_{prime p > 2} (1 + 1/(p(p-2))).

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A124508 a(n) = 2^BigO(n) * 3^omega(n), where BigO = A001222 and omega = A001221, the numbers of prime factors of n with and without repetitions.

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A226177 a(n) = mu(n)*d(n), where mu(n) = A008683 and d(n) = A000005.

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A318307 Multiplicative with a(p^e) = 2^A002487(e).

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A322838 Number of positive integers less than n with more prime factors than n, counted with multiplicity.

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A351347 Dirichlet g.f.: Product_{p prime} 1 / (1 - p^(-s) - 2p^(-2s)).