A048785 -id:A048785 - cp's OEIS frontend

A360910 Multiplicative with a(p^e) = 3*e - 1.

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

Offset: 1

Vaclav Kotesovec, Feb 25 2023

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A048785, A226602, A360909, A360911, A360908.

a[n_] := Times @@ ((3*Last[#] - 1) & /@ FactorInteger[n]); a[1] = 1; Array[a, 100] (* Amiram Eldar, Feb 25 2023 *)

for(n=1, 100, print1(direuler(p=2, n, (1+2*X^2)/(1-X)^2)[n], ", "))

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.

A360911 Multiplicative with a(p^e) = 3*e - 2.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Feb 25 2023

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A048785, A226602, A360909, A360910.

a[n_] := Times @@ ((3*Last[#] - 2) & /@ FactorInteger[n]); a[1] = 1; Array[a, 100] (* Amiram Eldar, Feb 25 2023 *)

for(n=1, 100, print1(direuler(p=2, n, (1-X+3*X^2)/(1-X)^2)[n], ", "))

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...

A059911 a(n) = |{m : multiplicative order of n mod m = 6}|.

Original entry on oeis.org

0, 3, 10, 16, 37, 10, 42, 24, 58, 53, 164, 26, 68, 38, 32, 68, 169, 22, 222, 38, 42, 50, 328, 40, 180, 219, 108, 26, 334, 82, 460, 82, 92, 72, 220, 108, 449, 86, 128, 80, 192, 22, 336, 110, 222, 218, 540, 84, 778, 129, 150, 80, 270, 54, 328, 356, 132, 68, 348, 22

Offset: 1

Vladeta Jovovic, Feb 08 2001

The multiplicative order of a mod m, gcd(a,m) = 1, is the smallest natural number d for which a^d = 1 (mod m).

			a(2) = |{9,21,63}| = 3, a(3) = |{7,14,28,52,56,91,104,182,364,728}| = 10, a(4) = |{13,35,39,45,65,91,105,117,195,273,315,455,585,819,1365,4095}| = 16,...

Cf. A059907-A059910, A059499, A059885-A059892, A002326, A053446-A053452, A002329, A055205, A048691, A048785.

a[n_] := Total[{1, -1, -1, 1} * DivisorSigma[0, n^{6, 3, 2, 1} - 1]]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jan 25 2025*)

a(n) = if(n == 1, 0, numdiv(n^6-1) - numdiv(n^3-1) - numdiv(n^2-1) + numdiv(n-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = tau(n^6-1)-tau(n^3-1)-tau(n^2-1)+tau(n-1), where tau(n) = number of divisors of n A000005. Generally, if b(n, r) = |{m : multiplicative order of n mod m = r}| then b(n, r) = Sum_{d|r} mu(d)*tau(n^(r/d)-1), where mu(n) = Moebius function A008683.

A092520 Number of square divisors of n-th cube: a(n) = A046951(n^3).

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 5, 4, 4, 2, 8, 2, 4, 4, 7, 2, 8, 2, 8, 4, 4, 2, 10, 4, 4, 5, 8, 2, 8, 2, 8, 4, 4, 4, 16, 2, 4, 4, 10, 2, 8, 2, 8, 8, 4, 2, 14, 4, 8, 4, 8, 2, 10, 4, 10, 4, 4, 2, 16, 2, 4, 8, 10, 4, 8, 2, 8, 4, 8, 2, 20, 2, 4, 8, 8, 4, 8, 2, 14, 7, 4, 2, 16, 4, 4, 4, 10, 2, 16, 4, 8, 4, 4, 4, 16

Offset: 1

Reinhard Zumkeller, Apr 06 2004

Apparently the inverse Mobius transform of A056624 (and therefore multiplicative). - R. J. Mathar, Feb 07 2011

			For n=12, the divisors of 12^3 = 1728 are 1 = 1^2, 2, 3, 4 = 2^2, 6, 8, 9 = 3^2, 12, 16 = 4^2, 18, 24, 27, 32, 36 = 6^2, 48, 54, 64 = 8^2, 72, 96, 108, 144 = 12^2, 192, 216, 288, 432, 576 = 24^2, 864 and 1728: eight of them are squares, therefore a(12) = 8.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A000290, A000578, A005117, A046951, A048785, A056624.

A092520 := proc(n)
    A046951(n^3) ;
end proc:
seq(A092520(n),n=1..50) ; # R. J. Mathar, Jul 09 2016

a[n_] := Count[Divisors[n^3], d_ /; IntegerQ[Sqrt[d]]]; Array[a, 100] (* Jean-François Alcover, Feb 13 2018 *)
f[p_, e_] := Floor[(3*e+2)/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2020 *)

A046951(n) = factorback(apply(e->e\2+1, factor(n)[, 2])) \\ This function from Charles R Greathouse IV, Sep 17 2015
A092520(n) = A046951(n^3); \\ Antti Karttunen, May 25 2017

a(n) = sumdiv(n^3, d, issquare(d)); \\ Michel Marcus, Apr 08 2018

a(n) = vecprod(apply(x->(3*x+2)\2, factor(n)[, 2])); \\ Amiram Eldar, Aug 18 2024

a(n) = A000005(n) iff n is squarefree.
From Werner Schulte, Feb 19 2018: (Start)
Multiplicative with a(p^e) = floor((3*e+2)/2) = A001651(e+1), p prime and e >= 0.
Dirichlet g.f.: Sum_{n>0} a(n)/n^s = (zeta(s))^2 * zeta(2*s) / zeta(3*s). (End)
Sum_{k=1..n} a(k) ~ Pi^2 * n/(6*zeta(3)) * (log(n) - 1 + 2*gamma + 12*zeta'(2)/Pi^2 - 3*zeta'(3)/zeta(3)) + zeta(1/2)^2 * sqrt(n) / zeta(3/2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 08 2019

A360909 Multiplicative with a(p^e) = 3*e + 2.

Original entry on oeis.org

1, 5, 5, 8, 5, 25, 5, 11, 8, 25, 5, 40, 5, 25, 25, 14, 5, 40, 5, 40, 25, 25, 5, 55, 8, 25, 11, 40, 5, 125, 5, 17, 25, 25, 25, 64, 5, 25, 25, 55, 5, 125, 5, 40, 40, 25, 5, 70, 8, 40, 25, 40, 5, 55, 25, 55, 25, 25, 5, 200, 5, 25, 40, 20, 25, 125, 5, 40, 25, 125

Offset: 1

Vaclav Kotesovec, Feb 25 2023

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

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), this sequence (3*e+2), A360911 (3*e-2), A322328 (4*e), A360996 (5*e).

a[n_] := Times @@ ((3*Last[#] + 2) & /@ FactorInteger[n]); a[1] = 1; Array[a, 100] (* Amiram Eldar, Feb 25 2023 *)

for(n=1, 100, print1(direuler(p=2, n, (1+3*X-X^2)/(1-X)^2)[n], ", "))

Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 3/p^s - 1/p^(2*s)).
Dirichlet g.f.: zeta(s)^5 * Product_{primes p} (1 - 7/p^(2*s) + 11/p^(3*s) - 6/p^(4*s) + 1/p^(5*s)), (with a product that converges for s=1).
Sum_{k=1..n} a(k) ~ c * n * log(n)^4 / 24, where c = Product_{primes p} (1 - 7/p^2 + 11/p^3 - 6/p^4 + 1/p^5) = 0.091414252314317101861531055690354339957600046..., more precise (but very complicated) asymptotics can be obtained (in Mathematica notation) as Residue[Zeta[s]^5 * f[s] * n^s / s, {s, 1}], where f[s] = Product_{primes p} (1 - 7/p^(2*s) + 11/p^(3*s) - 6/p^(4*s) + 1/p^(5*s)).

A360997 Multiplicative with a(p^e) = e + 3.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Feb 28 2023

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

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).

Cf. A000005, A007425, A007426, A074816.

g[p_, e_] := e+3; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

for(n=1, 100, print1(direuler(p=2, n, (1+2*X-2*X^2)/(1-X)^2)[n], ", "))

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)).
From Amiram Eldar, Sep 01 2023: (Start)
a(n) = A000005(A361264(n)).
a(n) = A074816(n)*A007426(n)/A007425(n). (End)

A059908 a(n) = |{m : multiplicative order of n mod m = 3}|.

Original entry on oeis.org

0, 1, 2, 4, 3, 2, 8, 2, 12, 5, 12, 2, 12, 2, 4, 20, 5, 6, 10, 2, 6, 14, 12, 2, 40, 9, 4, 6, 18, 10, 16, 6, 6, 8, 12, 12, 39, 2, 12, 8, 8, 6, 16, 6, 18, 26, 12, 6, 50, 3, 18, 8, 18, 2, 32, 12, 8, 20, 4, 6, 60, 2, 12, 26, 21, 4, 64, 10, 6, 8, 8, 6, 20, 14, 4, 12, 6, 4, 64, 2, 70, 7, 12, 6, 24

Offset: 1

Vladeta Jovovic, Feb 08 2001

The multiplicative order of a mod m, gcd(a,m) = 1, is the smallest natural number d for which a^d = 1 (mod m).

			a(2) = |{7}| = 1, a(3) = |{13,26}| = 2, a(4) = |{7,9,21,63}| = 4, a(5) = |{31,62,124}| = 3, a(6) = |{43,215}| = 2, a(7) = |{9,18,19,38,57,114,171,342}| = 8,...

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A059907, A059909-A059911, A059499, A059885-A059892, A002326, A053446-A053452, A002329, A055205, A048691, A048785.

Row n=3 of A212957. - Alois P. Heinz, Oct 24 2012

Table[DivisorSigma[0,n^3-1]-DivisorSigma[0,n-1],{n,90}] (* Harvey P. Dale, Feb 03 2015 *)

a(n) = if(n == 1, 0, numdiv(n^3-1) - numdiv(n-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = tau(n^3-1)-tau(n-1), where tau(n) = number of divisors of n A000005. Generally, if b(n, r) = |{m : multiplicative order of n mod m = r}| then b(n, r) = Sum_{d|r} mu(d)*tau(n^(r/d)-1), where mu(n) = Moebius function A008683.

A059909 a(n) = |{m : multiplicative order of n mod m = 4}|.

Original entry on oeis.org

0, 2, 6, 4, 12, 4, 26, 18, 14, 6, 24, 12, 64, 8, 16, 8, 66, 20, 36, 8, 64, 24, 76, 6, 28, 12, 64, 24, 48, 12, 100, 40, 50, 48, 36, 8, 96, 40, 28, 8, 104, 12, 208, 36, 24, 36, 200, 18, 48, 36, 36, 24, 128, 8, 152, 16, 172, 24, 48, 12, 48, 36, 56, 72, 40, 8, 128, 56, 48, 40

Offset: 1

Vladeta Jovovic, Feb 08 2001

The multiplicative order of a mod m, gcd(a,m) = 1, is the smallest natural number d for which a^d = 1 (mod m).

			a(2) = |{5, 15}| = 2, a(3) = |{5, 10, 16, 20, 40, 80}| = 6, a(4) = |{17, 51, 85, 255}| = 4, a(5) = |{13, 16, 26, 39, 48, 52, 78, 104, 156, 208, 312, 624}| = 12, ...

Cf. A059907, A059908, A059910-A059911, A059499, A059885-A059892, A002326, A053446-A053452, A002329, A055205, A048691, A048785.

Table[DivisorSigma[0,n^4-1]-DivisorSigma[0,n^2-1],{n,70}] (* Harvey P. Dale, Nov 30 2011 *)

a(n) = if(n == 1, 0, numdiv(n^4-1) - numdiv(n^2-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = tau(n^4-1)-tau(n^2-1), where tau(n) = number of divisors of n A000005. More generally, if b(n, r) = |{m : multiplicative order of n mod m = r}| then b(n, r) = Sum_{d|r} mu(d)*tau(n^(r/d)-1), where mu(n) = Moebius function A008683.

A059910 a(n) = |{m : multiplicative order of n mod m = 5}|.

Original entry on oeis.org

0, 1, 4, 6, 9, 4, 4, 6, 20, 9, 8, 2, 6, 6, 12, 44, 5, 6, 18, 14, 12, 4, 4, 2, 56, 13, 20, 4, 6, 2, 40, 6, 18, 12, 12, 44, 63, 6, 28, 4, 16, 14, 8, 2, 18, 12, 28, 14, 70, 3, 42, 12, 42, 6, 24, 8, 56, 44, 60, 6, 60, 2, 4, 90, 21, 20, 24, 2, 18, 60, 88, 6, 12, 2, 28, 26, 6, 28, 8, 14, 170

Offset: 1

Vladeta Jovovic, Feb 08 2001

The multiplicative order of a mod m, gcd(a,m) = 1, is the smallest natural number d for which a^d = 1 (mod m).

Cf. A059907-A059909, A059911, A059499, A059885-A059892, A002326, A053446-A053452, A002329, A055205, A048691, A048785.

a[n_] := Subtract @@ DivisorSigma[0, {n^5-1, n-1}]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jan 25 2025 *)

a(n) = if(n == 1, 0, numdiv(n^5-1) - numdiv(n-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = tau(n^5-1)-tau(n-1), where tau(n) = number of divisors of n A000005. Generally, if b(n, r) = |{m : multiplicative order of n mod m = r}| then b(n, r) = Sum_{d|r} mu(d)*tau(n^(r/d)-1), where mu(n) = Moebius function A008683.

A144943 a(n) = number of divisors of n^3 (excluding 1 and n^3).

Original entry on oeis.org

0, 2, 2, 5, 2, 14, 2, 8, 5, 14, 2, 26, 2, 14, 14, 11, 2, 26, 2, 26, 14, 14, 2, 38, 5, 14, 8, 26, 2, 62, 2, 14, 14, 14, 14, 47, 2, 14, 14, 38, 2, 62, 2, 26, 26, 14, 2, 50, 5, 26, 14, 26, 2, 38, 14, 38, 14, 14, 2, 110, 2, 14, 26, 17, 14, 62, 2, 26, 14, 62, 2, 68, 2, 14, 26, 26, 14, 62, 2, 50

Offset: 1

Mark Taggart (mt2612f(AT)aol.com), Sep 26 2008

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1001

Cf. A000005, A048785.

Join[{0},(DivisorSigma[0,#]-2)&/@(Range[2,90]^3)]  (* Harvey P. Dale, Apr 02 2011 *)

A144943(n) = if(1==n,0,numdiv(n^3)-2); \\ Antti Karttunen, May 19 2017

For n > 1, a(n) = A048785(n) - 2. - Antti Karttunen, May 19 2017

cp's OEIS Frontend

A360910 Multiplicative with a(p^e) = 3*e - 1.

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A360911 Multiplicative with a(p^e) = 3*e - 2.

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A059911 a(n) = |{m : multiplicative order of n mod m = 6}|.

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A092520 Number of square divisors of n-th cube: a(n) = A046951(n^3).

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A360909 Multiplicative with a(p^e) = 3*e + 2.

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A360997 Multiplicative with a(p^e) = e + 3.

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A059908 a(n) = |{m : multiplicative order of n mod m = 3}|.

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A059909 a(n) = |{m : multiplicative order of n mod m = 4}|.