cp's OEIS Frontend

a(n) = lcm(20, n)/20. - Zerinvary Lajos, Jun 12 2009

a(n) = n/gcd(n, 20). - Andrew Howroyd, Jul 25 2018

From Luce ETIENNE, Nov 04 2018: (Start)

a(n) = 9*a(n-20) - 36*a(n-40) + 84*a(n-60) - 126*a(n-80) + 126*a(n-100) - 84*a(n-120) + 36*a(n-140) - 9*a(n-160) + a(n-180).

a(n) = (5*(119*m^9 - 4923*m^8 + 86250*m^7 - 832230*m^6 + 4807887*m^5 - 16882299*m^4 + 34770400*m^3 - 37855620m^2 + 16581744*m + 54432)*floor(n/10) + 72*m*(3*m^8 - 120*m^7 + 2030*m^6 - 18900*m^5 + 105329*m^4 - 356580*m^3 + 706220*m^2 - 733200*m + 300258) + ((19*m^9 - 855*m^8 + 15810*m^7 - 154350*m^6 + 849387*m^5 - 2597175*m^4 + 4037840*m^3 - 2600100*m^2 + 540144*m - 90720)*floor(n/10) - 72*m*(m^7 - 35*m^6 + 490*m^5 - 3500*m^4 + 13489*m^3 - 27335*m^2 + 26340*m - 9450))*(-1)^floor(n/10))/362880 where m = (n mod 10). (End)

From Peter Bala, Feb 24 2019: (Start)

a(n) = n/gcd(n,20) is a quasi-polynomial in n since gcd(n,20) is a purely periodic sequence of period 20.

O.g.f.: F(x) - F(x^2) - F(x^4) - 4*F(x^5) + 4*F(x^10) + 4*F(x^20), where F(x) = x/(1 - x)^2.

O.g.f. for reciprocals: Sum_{n >= 1} x^n/a(n) = Sum_{d divides 20} (phi(d)/d) * log(1/(1 - x^d)) = log(1/(1 - x)) + (1/2)*log(1/(1 - x^2)) + (2/4)*log(1/(1 - x^4)) + (4/5)*log(1/(1 - x^5)) + (4/10)*log(1/(1 - x^10)) + (8/20)*log(1/(1 - x^20)), where phi(n) denotes the Euler totient function A000010. (End)

From Amiram Eldar, Nov 25 2022: (Start)

Multiplicative with a(2^e) = 2^max(0, e-2), a(5^e) = 5^max(0,e-1), and a(p^e) = p^e otherwise.

Dirichlet g.f.: zeta(s-1)*(1 - 1/2^s - 1/4^s - 4/5^s + 4/10^s + 4/20^s).

Sum_{k=1..n} a(k) ~ (231/800) * n^2. (End)

a(n) = 2*a(n-8) - a(n-16).

G.f.: x* (x^2-x+1) * (x^12 +2*x^11 +4*x^10 +3*x^9 +4*x^8 +4*x^7 +7*x^6 +4*x^5 +4*x^4 +3*x^3 +4*x^2 +2*x +1) / ( (x-1)^2 *(x+1)^2 *(x^2+1)^2 *(x^4+1)^2 ). - R. J. Mathar, Dec 02 2010

From R. J. Mathar, Apr 18 2011: (Start)

a(n) = A109049(n)/8.

Dirichlet g.f. zeta(s-1)*(1-1/2^s-1/2^(2s)-1/2^(3s)).

Multiplicative with a(2^e) = 2^max(0,e-3). a(p^e) = p^e if p>2. (End)

a(n) = n/gcd(n,8), n >= 0. See the harmonic mean comment above. - Wolfdieter Lang, Jul 04 2013

a(n) = n if n is odd; for n == 0 (mod 8) it is n/8, for n == 2 or 6 (mod 8) it is n/2 and for n == 4 (mod 8) it is n/4. - Wolfdieter Lang, Dec 04 2013

From Peter Bala, Feb 20 2019: (Start)

O.g.f.: Sum_{n >= 0} a(n)*x^n = F(x) - F(x^2) - F(x^4) - F(x^8), where F(x) = x/(1 - x)^2.

More generally, for m >= 1, Sum_{n >= 0} (a(n)^m)*x^n = F(m,x) + (1 - 2^m)*( F(m,x^2) + F(m,x^4) + F(m,x^8) ), where F(m,x) = A(m,x)/(1 - x)^(m+1) with A(m,x) the m-th Eulerian polynomial: A(1,x) = x, A(2,x) = x*(1 + x), A(3,x) = x*(1 + 4*x + x^2) - see A008292.

Sum_{n >= 1} (1/n)*a(n)*x^n = G(x) - (1/2)*G(x^2) - (1/4)*G(x^4) - (1/8)*G(x^8), where G(x) = x/(1 - x).

Sum_{n >= 1} (1/n^2)*a(n)*x^n = L(x) - (1/2^2)*L(x^2) - (1/4)^2*L(x^4) - (1/8)^2*L(x^8), where L(x) = Log(1/(1 - x)).

Sum_{n >= 1} (1/a(n))*x^n = L(x) + (1/2)*L(x^2) + (1/2)*L(x^4) + (1/2)*L(x^8). (End)

Sum_{k=1..n} a(k) ~ (43/128) * n^2. - Amiram Eldar, Nov 25 2022

a(n) = 2*a(n-16) - a(n-32) for n > 31. - Paul Curtz, Apr 12 2011

Octosections: a(8*n) = A026741(n). a(2+8*n) = 1+4*n. a(4+8*n) = 1+2*n. a(6+8*n) = 3+4*n. Bisection: a(1+2*n) = 1+2*n. - Paul Curtz, Apr 12 2011

Dirichlet g.f.: zeta(s-1)*(1-1/2^s-1/4^s-1/8^s-1/16^s). - R. J. Mathar, Apr 18 2011

a(n) = numerator of n/(2^(2*n+1)). - Ralf Steiner, Feb 09 2017

The previous comment is incorrect, a(n) first differs from the numerator of n/(2^(2*n+1)) at n = 32. - Peter Bala, Feb 27 2019

From Peter Bala, Feb 21 2019: (Start)

a(n) = n/gcd(n,16), where gcd(n,16) = [1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 16, ...] is a periodic sequence of period 16: a(n) is thus quasi_polynomial in n.

O.g.f.: Sum_{n >= 0} a(n)*x^n = F(x) - F(x^2) - F(x^4) - F(x^8) - F(x^16), where F(x) = x/(1 - x)^2.

More generally, Sum_{n >= 0} (a(n)^m)*x^n = F(m,x) + (1 - 2^m)*( F(m,x^2) + F(m,x^4) + F(m,x^8) + F(m,x^16) ), where F(m,x) = A(m,x)/(1 - x)^(m+1) with A(m,x) the m-th Eulerian polynomial: A(1,x) = x, A(2,x) = x*(1 + x), A(3,x) = x*(1 + 4*x + x^2) - see A008292.

Repeatedly applying the Euler operator x*d/dx or its inverse operator to the o.g.f. for the sequence produces generating functions for the sequences ( (n^m)*a(n) )n>=1 for m in Z. Some examples are given below. (End)

From Amiram Eldar, Nov 25 2022: (Start)

Multiplicative with a(2^e) = 2^max(0,e-4), and a(p^e) = p^e if p>2.

Sum_{k=1..n} a(k) ~ (171/512) * n^2. (End)

a(n) = 2*a(n-18) - a(n-36). - Paul Curtz, Feb 27 2011

Nonasection: a(9*n) = A026741(n). - Paul Curtz, Mar 21 2011

Dirichlet g.f.: zeta(s-1)*(1 - 2/3^s - 2/9^s - 1/2^s + 2/6^s + 2/18^s). - R. J. Mathar, Apr 18 2011

a(n) = n/gcd(n,18), n >= 0. See the harmonic mean comment above, and the Zerinvary Lajos program below. - Wolfdieter Lang, Jul 04 2013

a(n+3) = A227042(n+3,3), n >= 0. - Wolfdieter Lang, Jul 04 2013

From Amiram Eldar, Nov 25 2022: (Start)

Multiplicative with a(2^e) = 2^max(0, e-1), a(3^e) = 3^max(0,e-2), and a(p^e) = p^e otherwise.

Sum_{k=1..n} a(k) ~ (61/216) * n^2. (End)

a(n) = 3*n/gcd(n,3) = 3*n/A109007(n).

From Bruno Berselli, Mar 11 2011: (Start)

G.f.: 3*x*(1+2*x+x^2+2*x^3+x^4)/(1-x^3)^2.

a(n) = 3*A051176(n);

a(n) = n*(7-2*A099837(n))/3 for n>0. (End)

From Wesley Ivan Hurt, Jul 24 2016: (Start)

a(n) = 2*a(n-3) - a(n-6) for n>5.

a(n) = 9*n/(5 + 4*cos(2*n*Pi/3)).

If n mod 3 = 0 then 3*floor(n/3), else 3*n. (End)

a(n) = n*(1 + 2*((n^2) mod 3)). - Timothy Hopper, Feb 23 2017

From Michael Somos, Mar 04 2017: (Start)

G.f.: 3 * x / (1 - x)^2 - 6 * x^3 / (1 - x^3)^2. -

a(n) = a(-n) for all n in Z. (End)

Sum_{k=1..n} a(k) ~ (7/6) * n^2. - Amiram Eldar, Nov 26 2022

Sum_{n>=1} (-1)^(n+1)/a(n) = 5*log(2)/9. - Amiram Eldar, Sep 08 2023

From R. J. Mathar, Apr 18 2011: (Start)

a(n) = A109051(n)/10.

Dirichlet g.f.: zeta(s-1)*(1 - 4/5^s - 1/2^s + 4/10^s).

Multiplicative with a(2^e) = 2^max(0,e-1), a(5^e) = 5^max(0,e-1), a(p^e) = p^e if p = 3 or p >= 7. (End)

From Peter Bala, Feb 17 2019: (Start)

a(n) = numerator(n/((n + 2)*(n + 5))).

a(n) = n/b(n), where b(n) = [1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, ...] is a purely periodic sequence of period 10. Thus a(n) is a quasi-polynomial in n.

If gcd(n,m) = 1 then a( a(n)*a(m) ) = a(a(n)) * a(a(m)), a( a(a(n))*a(a(m)) ) = a(a(a(n))) * a(a(a(m))) and so on.

O.g.f.: Sum_{d divides 10} A023900(d)*x^d/(1 - x^d)^2 = x/(1 - x)^2 - x^2/(1 - x^2)^2 - 4*x^5/(1 - x^5)^2 + 4*x^10/(1 - x^10)^2.

(End)

Sum_{k=1..n} a(k) ~ (63/200) * n^2. - Amiram Eldar, Nov 25 2022

Dirichlet g.f.: zeta(s-1)*(1 - 6/7^s - 1/2^s + 6/14^s). - R. J. Mathar, Apr 18 2011

a(n) = 2*a(n-14) - a(n-28). - G. C. Greubel, Feb 19 2019

From Peter Bala, Feb 22 2019: (Start)

a(n) = n/gcd(n,14).

a(n) = n/b(n), where b(n) = [1, 2, 1, 2, 1, 2, 7, 2, 1, 2, 1, 2, 1, 14, ...] is a purely periodic sequence of period 14. Thus a(n) is a quasi-polynomial in n.

If gcd(n,m) = 1 then a( a(n)*a(m) ) = a(a(n)) * a(a(m)), a( a(a(n))*a(a(m)) ) = a(a(a(n))) * a(a(a(m))) and so on.

O.g.f.: Sum_{d divides 14} A023900(d)*x^d/(1 - x^d)^2 = x/(1 - x)^2 - x^2/(1 - x^2)^2 - 6*x^7/(1 - x^7)^2 + 6*x^14/(1 - x^14)^2.

O.g.f. for reciprocals: Sum_{n >= 1} (1/a(n))*x^n = L(x) + 1/2*L(x^2) + 6/7*L(x^7) + 6/14*L(x^14), where L(x) = log (1/(1 - x)). (End)

From Amiram Eldar, Nov 25 2022: (Start)

Multiplicative with a(2^e) = 2^max(0,e-1), a(7^e) = 7^max(0,e-1), and a(p^e) = p^e otherwise.

Sum_{k=1..n} a(k) ~ (129/392) * n^2. (End)

T(n,k) = (n-k)/gcd(n,k), 1 <= k <= n.

T(n,k) = A025581(n,k)/A050873(n,k);

T(n,1) = A001477(n-1);

T(n,2) = A026741(n-2) for n > 1;

T(n,3) = A051176(n-3) for n > 2;

T(n,4) = A060819(n-4) for n > 4;

T(n,n-3) = A144437(n) for n > 3;

T(n,n-2) = A000034(n) for n > 2;

T(n,n-1) = A000012(n);

T(n,n) = A000004(n).

Dirichlet g.f.: zeta(s-1)*(1 - 18/19^s). - R. J. Mathar, Apr 18 2011

a(n) = 2*a(n-19) - a(n-38). - G. C. Greubel, Feb 19 2019

From Amiram Eldar, Nov 25 2022: (Start)

Multiplicative with a(19^e) = 19^(e-1), and a(p^e) = p^e if p != 19.

Sum_{k=1..n} a(k) ~ (343/722) * n^2. (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = 37*log(2)/19. - Amiram Eldar, Sep 08 2023

a(n) = tau(n)+3*tau(n/3)+A035191(n) if n is congruent to 0 mod 3 else tau(n)+A035191(n), where A035191(n) is the number of divisors of n that are not congruent to 0 mod 3.

a(n)/2 is multiplicative with a(3^e)=2e+1 and a(p^e)=e+1 for p<>3.

Moebius transform is period 3 sequence [2, 2, 4, ...]. - Michael Somos, Sep 20 2005

G.f.: Sum_{k>0} x^k(2+2*x^k+4*x^(2k))/(1-x^(3k)).

From Amiram Eldar, Nov 29 2022: (Start)

Dirichlet g.f.: 2 * zeta(s)^2 * (1 + 1/3^s).

Sum_{k=1..n} a(k) ~ 2*(4*n*log(n) + (8*gamma - 4 - log(3))*n)/3, where gamma is Euler's constant (A001620). (End)