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Multiplicative with a(p^e) = 3*e - 1.
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6
1, 2, 2, 5, 2, 4, 2, 8, 5, 4, 2, 10, 2, 4, 4, 11, 2, 10, 2, 10, 4, 4, 2, 16, 5, 4, 8, 10, 2, 8, 2, 14, 4, 4, 4, 25, 2, 4, 4, 16, 2, 8, 2, 10, 10, 4, 2, 22, 5, 10, 4, 10, 2, 16, 4, 16, 4, 4, 2, 20, 2, 4, 10, 17, 4, 8, 2, 10, 4, 8, 2, 40, 2, 4, 10, 10, 4, 8, 2
FORMULA
Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 2/p^(2*s)).
Let f(s) = Product_{primes p} (1 + 2/p^(2*s)), then Sum_{k=1..n} a(k) ~ n*(f(1)*(log(n) + 2*gamma - 1) + f'(1)), where f(1) = Product_{primes p} (1 + 2/p^2) = 2.1908700855532557963501937947188223715671192999357721091330157224657649571..., f'(1) = f(1) * Sum_{primes p} (-4*log(p)/(p^2 + 2)) = -3.559220569509264750413960031425742000438433285978558703470289340806139902... and gamma is the Euler-Mascheroni constant A001620.
MATHEMATICA
a[n_] := Times @@ ((3*Last[#] - 1) & /@ FactorInteger[n]); a[1] = 1; Array[a, 100] (* Amiram Eldar, Feb 25 2023 *)
PROG
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1+2*X^2)/(1-X)^2)[n], ", "))
Multiplicative with a(p^e) = 3*e - 2.
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6
1, 1, 1, 4, 1, 1, 1, 7, 4, 1, 1, 4, 1, 1, 1, 10, 1, 4, 1, 4, 1, 1, 1, 7, 4, 1, 7, 4, 1, 1, 1, 13, 1, 1, 1, 16, 1, 1, 1, 7, 1, 1, 1, 4, 4, 1, 1, 10, 4, 4, 1, 4, 1, 7, 1, 7, 1, 1, 1, 4, 1, 1, 4, 16, 1, 1, 1, 4, 1, 1, 1, 28, 1, 1, 4, 4, 1, 1, 1, 10, 10, 1, 1, 4
FORMULA
Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 - 1/p^s + 3/p^(2*s)).
Dirichlet g.f.: zeta(s) * Product_{primes p} (1 + 3/(p^s*(p^s-1))).
Sum_{k=1..n} a(k) ~ c*n, where c = Product_{primes p} (1 + 3/(p*(p-1))) = 5.092999766083306437144607885642959667401184716827970969797879646796872425...
MATHEMATICA
a[n_] := Times @@ ((3*Last[#] - 2) & /@ FactorInteger[n]); a[1] = 1; Array[a, 100] (* Amiram Eldar, Feb 25 2023 *)
PROG
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1-X+3*X^2)/(1-X)^2)[n], ", "))
Multiplicative with a(p^e) = 5*e, p prime and e > 0.
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2
1, 5, 5, 10, 5, 25, 5, 15, 10, 25, 5, 50, 5, 25, 25, 20, 5, 50, 5, 50, 25, 25, 5, 75, 10, 25, 15, 50, 5, 125, 5, 25, 25, 25, 25, 100, 5, 25, 25, 75, 5, 125, 5, 50, 50, 25, 5, 100, 10, 50, 25, 50, 5, 75, 25, 75, 25, 25, 5, 250, 5, 25, 50, 30, 25, 125, 5, 50, 25, 125, 5, 150
FORMULA
Dirichlet g.f.: Product_{primes p} (1 + 5*p^s/(p^s - 1)^2).
MATHEMATICA
g[p_, e_] := 5*e; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1+3*X+X^2)/(1-X)^2)[n], ", "))
CROSSREFS
Cf. A005361 (multiplicative with a(p^e) = e), A000005 (e+1), A343443 (e+2), A360997 (e+3), A322327 (2*e), A048691 (2*e+1), A360908 (2*e-1), A226602 (3*e), A048785 (3*e+1), A360910 (3*e-1), A360909 (3*e+2), A360911 (3*e-2), A322328 (4*e).
Multiplicative with a(p^e) = e + 3.
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5
1, 4, 4, 5, 4, 16, 4, 6, 5, 16, 4, 20, 4, 16, 16, 7, 4, 20, 4, 20, 16, 16, 4, 24, 5, 16, 6, 20, 4, 64, 4, 8, 16, 16, 16, 25, 4, 16, 16, 24, 4, 64, 4, 20, 20, 16, 4, 28, 5, 20, 16, 20, 4, 24, 16, 24, 16, 16, 4, 80, 4, 16, 20, 9, 16, 64, 4, 20, 16, 64, 4, 30, 4, 16
FORMULA
Dirichlet g.f.: Product_{primes p} (1 + (4*p^s - 3)/(p^s - 1)^2).
Dirichlet g.f.: zeta(s)^4 * Product_{primes p} (1 - 5/p^(2*s) + 6/p^(3*s) - 2/p^(4*s)).
MATHEMATICA
g[p_, e_] := e+3; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1+2*X-2*X^2)/(1-X)^2)[n], ", "))
CROSSREFS
Cf. A005361 (multiplicative with a(p^e) = e), A000005 (e+1), A343443 (e+2), this sequence (e+3), A322327 (2*e), A048691 (2*e+1), A360908 (2*e-1), A226602 (3*e), A048785 (3*e+1), A360910 (3*e-1), A360909 (3*e+2), A360911 (3*e-2), A322328 (4*e), A360996 (5*e).
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