A178142 -id:A178142 - cp's OEIS frontend

A171182 Period 6: repeat [0, 1, 1, 1, 0, 2].

0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1

Offset: 1

Juri-Stepan Gerasimov, Dec 04 2009, Dec 07 2009

The number of divisors d of n of the form d=2 or 3. - Vladimir Shevelev, May 21 2010
a(n) = s(n+6), where s(k) is the number of partitions of k into distinct parts such that max(p) = 2 + min(p) for k >= 1, and (s(0)..s(6)) = (0,0,0,0,1,0,2). - Clark Kimberling, Apr 15 2014
Number of r X s integer-sided rectangles such that r < s, r + s = 2n, r | s and (s - r)/2 | s. - Wesley Ivan Hurt, Apr 24 2020
Number of positive integer solutions, (r,s,t), of the equation r^2 + t*s^2 = (n + 6)^2, where r + s = n + 6 and t < r <= s. For example, when n=6 we have the two solutions (4,8,2) and (6,6,3) since 4^2 + 2*8^2 = 12^2 and 6^2 + 3*6^2 = 12^2. - Wesley Ivan Hurt, Oct 04 2020

Vladimir Shevelev, A recursion for divisor function over divisors belonging to a prescribed finite sequence of positive integers and a solution of the Lahiri problem for divisor function sigma_x(n), arXiv:0903.1743 [math.NT], 2009. [From _Vladimir Shevelev_, May 21 2010]
Index entries for linear recurrences with constant coefficients, signature (-1,0,1,1).

Cf. A178142. - Vladimir Shevelev, May 21 2010
Cf. A115357.
Cf. A001221, A059841, A065331, A079978, A089128, A169611.
Number of distinct prime factors <= p: this sequence (p=3), A178146 (p=5), A210679 (p=7).

&cat[[0, 1, 1, 1, 0, 2]^^20]; // Wesley Ivan Hurt, Jun 19 2016

A171182:=n->(1+floor((n-3)^2/2)) mod 3: seq(A171182(n), n=1..100); # Wesley Ivan Hurt, Aug 27 2014

PadRight[{},110,{0,1,1,1,0,2}] (* Harvey P. Dale, Jan 28 2013 *)
LinearRecurrence[{-1, 0, 1, 1},{0, 1, 1, 1},105] (* Ray Chandler, Aug 26 2015 *)

a(n) = A115357(n-2) for n>1. - R. J. Mathar, Dec 09 2009
a(2) = 1, a(3) = 1, a(5) = 0, otherwise a(n) = a(n-2) + a(n-3) - a(n-5), where we put a(n) = 0, if n<0. - Vladimir Shevelev, May 21 2010
a(n) = floor(((n+1) mod 6)/3) + 2*floor(((n+5) mod 6)/5). - Gary Detlefs, Feb 15 2014
From Wesley Ivan Hurt, Aug 27 2014: (Start)
G.f.: (2+2*x+x^2)/(1+x-x^3-x^4).
a(n) + a(n-1) = a(n-3) + a(n-4) for n>4.
a(n) = (1 + floor((n-3)^2/2)) mod 3. (End)
a(n) = (5 + 3*cos(n*Pi) + 4*cos(2*n*Pi/3))/6. - Wesley Ivan Hurt, Jun 19 2016
From Amiram Eldar, Sep 16 2023: (Start)
Additive with a(p^e) = 1 if p <= 3, and 0 otherwise.
a(n) = A059841(n) + A079978(n).
a(n) = A001221(A089128(n)).
a(n) = A001221(A065331(n)). (End)

Edited by Charles R Greathouse IV, Mar 23 2010

A178143 Sum of squares d^2 over the divisors d=2 and/or d=3 of n.

Original entry on oeis.org

0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4

Offset: 1

Vladimir Shevelev, May 21 2010

Period 6: repeat [0, 4, 9, 4, 0, 13]. - Wesley Ivan Hurt, Jul 05 2016

			a(1)=0, a(2)=2^2=4 since 2|2, a(3)=3^2=9 since 3|3, a(4)=2^2=4 since 2|4.

Vladimir Shevelev,A recursion for divisor function over divisors belonging to a prescribed finite sequence of positive integers and a solution of the Lahiri problem for divisor function sigma_x(n), arXiv:0903.1743 [math.NT], 2009.
Index entries for linear recurrences with constant coefficients, signature (-1,0,1,1).

Cf. A010675, A021115, A098002, A178142.

&cat [[0, 4, 9, 4, 0, 13]^^20]; // Wesley Ivan Hurt, Jul 05 2016

seq(op([0, 4, 9, 4, 0, 13]), n=1..30); # Wesley Ivan Hurt, Jul 05 2016

PadRight[{}, 100, {0, 4, 9, 4, 0, 13}] (* Wesley Ivan Hurt, Jul 05 2016 *)

a(n)=[13,0,4,9,4,0][n%6+1] \\ Charles R Greathouse IV, May 21 2013

a(n) = Sum_{d|n, d=2 and/or d=3} d^2.
a(n) = -a(n-1) + a(n-3) + a(n-4) for n>4.
G.f.: x*(4+13*x+13*x^2) / ( (1-x)*(1+x)*(1+x+x^2) ).
a(n+6) = a(n).
a(n) = A010675(n) + A021115(n). [R. J. Mathar, May 28 2010]
a(n) = 4 * (1 + floor(n/2) - ceiling(n/2)) + 9 * (1 + floor(n/3) - ceiling(n/3)). - Wesley Ivan Hurt, May 20 2013
a(n) = 5 + 2*cos(n*Pi) + 6*cos(2*n*Pi/3). - Wesley Ivan Hurt, Jul 05 2016

Replaced recurrence by a shorter one; added keyword:less - R. J. Mathar, May 28 2010

A178144 Sum of divisors d of n which are d=2, 3 or 5.

Original entry on oeis.org

0, 2, 3, 2, 5, 5, 0, 2, 3, 7, 0, 5, 0, 2, 8, 2, 0, 5, 0, 7, 3, 2, 0, 5, 5, 2, 3, 2, 0, 10, 0, 2, 3, 2, 5, 5, 0, 2, 3, 7, 0, 5, 0, 2, 8, 2, 0, 5, 0, 7, 3, 2, 0, 5, 5, 2, 3, 2, 0, 10, 0, 2, 3, 2, 5, 5, 0, 2, 3, 7, 0, 5, 0, 2, 8, 2, 0, 5, 0, 7, 3, 2, 0, 5, 5, 2, 3, 2, 0, 10, 0, 2, 3, 2, 5, 5, 0, 2, 3, 7, 0, 5, 0, 2, 8

Offset: 1

Vladimir Shevelev, May 21 2010

The sequence is periodic with period length 30.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Vladimir Shevelev, A recursion for divisor function over divisors belonging to a prescribed finite sequence of positive integers and a solution of the Lahiri problem for divisor function sigma_x(n), arXiv:0903.1743 [math.NT], 2009.
Index entries for linear recurrences with constant coefficients, signature (-2,-2,-1,0,1,2,2,1).

Cf. A000203, A008472, A178143, A171182, A178142.

A178144 := proc(n)
    local a;
    a := 0 ;
    for d in {2,3,5} do
        if (n mod d) = 0 then
            a := a+d ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Jul 23 2012

a[n_] := DivisorSum[n, Boole[MatchQ[#, 2|3|5]]*#&];
Array[a, 105] (* Jean-François Alcover, Nov 24 2017 *)
a[n_] := Sum[d * Boole[Divisible[n, d]], {d, {2, 3, 5}}]; Array[a, 100] (* Amiram Eldar, Dec 20 2024 *)

a(n) = sumdiv(n, d, if ((d==2) || (d==3) || (d==5), d)); \\ Michel Marcus, Nov 24 2017

a(n) = my(d = [2, 3, 5]); sum(k = 1, 3, d[k] * !(n % d[k])); \\ Amiram Eldar, Dec 20 2024

From R. J. Mathar, Jul 23 2012: (Start)
a(n) = -2*a(n-1) -2*a(n-2) -a(n-3) +a(n-5) +2*a(n-6) +2*a(n-7) +a(n-8).
G.f.: ( -x*(2+7*x+12*x^2+17*x^3+22*x^4+10*x^6+20*x^5) ) / ( (x-1)*(1+x)*(1+x+x^2)*(x^4+x^3+x^2+x+1) ). (End)

A178146 a(n) is the number of distinct prime factors <= 5 of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 1, 0, 3, 0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 1, 0, 3, 0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 1, 0, 3, 0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2

Offset: 1

Vladimir Shevelev, May 21 2010

The sequence is periodic with period {0 1 1 1 1 2 0 1 1 2 0 2 0 1 2 1 0 2 0 2 1 1 0 2 1 1 1 1 0 3} of length 30. There are 26 coincidences on the interval [1,30] with A156542.

Vladimir Shevelev, A recursion for divisor function over divisors belonging to a prescribed finite sequence of positive integers and a solution of the Lahiri problem for divisor function sigma_x(n), arXiv:0903.1743 [math.NT], 2009.
Index entries for linear recurrences with constant coefficients, signature (-2,-2,-1,0,1,2,2,1).

Cf. A000005, A001221, A156542, A171182, A178143, A178144, A178142.
Cf. A059841, A079978, A079998, A355582, A356006.
Number of distinct prime factors <= p: A171182 (p=3), this sequence (p=5), A210679 (p=7).

Rest@ CoefficientList[Series[-x^2*(3*x^6 + 6*x^5 + 7*x^4 + 6*x^3 + 5*x^2 + 3*x + 1)/((x - 1)*(x + 1)*(x^2 + x + 1)*(x^4 + x^3 + x^2 + x + 1)), {x, 0, 50}], x] (* G. C. Greubel, May 16 2017 *)
LinearRecurrence[{-2,-2,-1,0,1,2,2,1},{0,1,1,1,1,2,0,1},120] (* Harvey P. Dale, Sep 29 2021 *)
a[n_] := PrimeNu[GCD[n, 30]]; Array[a, 100] (* Amiram Eldar, Sep 16 2023 *)

my(x='x+O('x^50)); concat([0], Vec(-x^2*(3*x^6+6*x^5+7*x^4+6*x^3+5*x^2+3*x+1)/((x-1)*(x+1)*(x^2+x+1)*(x^4+x^3+x^2+x+1)))) \\ G. C. Greubel, May 16 2017

a(n) = omega(gcd(n, 30)); \\ Amiram Eldar, Sep 16 2023

a(n) = a(n-2) + a(n-3) - a(n-7) - a(n-8) + a(n-10), a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 1, a(5) = 1, a(6) = 2, a(7) = 0, a(8) = 1, a(9) = 1, a(10) = 2.
G.f.: -x^2*(3*x^6+6*x^5+7*x^4+6*x^3+5*x^2+3*x+1) / ((x-1)*(x+1)*(x^2+x+1)*(x^4+x^3+x^2+x+1)). - Colin Barker, Mar 13 2013
From Amiram Eldar, Sep 16 2023: (Start)
Additive with a(p^e) = 1 if p <= 5, and 0 otherwise.
a(n) = A059841(n) + A079978(n) + A079998(n).
a(n) = A001221(gcd(n, 30)).
a(n) = A001221(A355582(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 31/30. (End)

Name edited by Amiram Eldar, Sep 16 2023

A171405 Sum of divisors of n, excluding divisors 2 and 3.

Original entry on oeis.org

1, 1, 1, 5, 6, 7, 8, 13, 10, 16, 12, 23, 14, 22, 21, 29, 18, 34, 20, 40, 29, 34, 24, 55, 31, 40, 37, 54, 30, 67, 32, 61, 45, 52, 48, 86, 38, 58, 53, 88, 42, 91, 44, 82, 75, 70, 48, 119, 57, 91, 69, 96, 54, 115, 72, 118, 77, 88, 60, 163, 62, 94, 101, 125, 84, 139, 68, 124, 93, 142

Offset: 1

Juri-Stepan Gerasimov, Dec 07 2009, Apr 20 2010

nonn

Cf. A000203, A178142.

a[n_] := DivisorSigma[1, n] - {0,2,3,2,0,5}[[Mod[n-1, 6] + 1]]; Array[a, 100] (* Amiram Eldar, Aug 03 2024 *)

a(n) = sigma(n) - [0,2,3,2,0,5][(n-1) % 6 + 1]; \\ Amiram Eldar, Aug 03 2024

a(n) = A000203(n) - A178142(n). - Amiram Eldar, Aug 03 2024

Definition and some values corrected by R. J. Mathar, Jun 07 2010

A178147 Sum of squares d^2 of distinct divisors of n, d in {2, 3, 5}.

Original entry on oeis.org

0, 4, 9, 4, 25, 13, 0, 4, 9, 29, 0, 13, 0, 4, 34, 4, 0, 13, 0, 29, 9, 4, 0, 13, 25, 4, 9, 4, 0, 38, 0, 4, 9, 4, 25, 13, 0, 4, 9, 29, 0, 13, 0, 4, 34, 4, 0, 13, 0, 29, 9, 4, 0, 13, 25, 4, 9, 4, 0, 38, 0, 4, 9, 4, 25, 13, 0, 4, 9, 29, 0, 13, 0, 4, 34, 4, 0, 13, 0

Offset: 1

Vladimir Shevelev, May 21 2010, May 23 2010

The sequence is periodic with period {0 4 9 4 25 13 0 4 9 29 0 13 0 4 34 4 0 13 0 29 9 4 0 13 25 4 9 4 0 38} of length 30.
A generalization: let B={b_1,...,b_t} be a set of t positive (not necessarily distinct) integers and m>=0 an integer.
For m>=0, let A(n)=Sum d^m over divisors d of n which are elements of B (with the multiplicities as in B). Calculating directly values of
A(b_i),A(b_i+b_j),A(b_i+b_j+b_k),...,
A(b_1+...+b_t), for the other values of A(n) we have the recursion:
  A(n)=Sum{1<=i<=t}A(n-b_i)- Sum{1<=i

V. Shevelev, A recursion for divisor function over divisors belonging to a prescribed finite sequence of positive integers and a solution of the Lahiri problem for divisor function sigma_x(n), arXiv:0903.1743
Index entries for linear recurrences with constant coefficients, signature (-2,-2,-1,0,1,2,2,1)

Cf. A005063, A098002, A178142-A178144, A178146.

a(n)= a(n-2) +a(n-3) -a(n-7)- a(n-8) +a(n-10), n>10.
By the comment, up to 10 it is sufficient to
calculate directly only values a(2)=4, a(3)=9, a(5)=25, a(7)=0, a(8)=4, a(10)=29.
For other n's we can use the recursion, accepting formally a(n)=0 for n<0. So a(1)=0; a(4)=a(2)+a(1)=4;a(6)=a(4)+a(3)=4+9=13,
a(9)=a(7)+a(6)-a(2)-a(1)=0+13-4+0=9.
a(n) = -2*a(n-1) -2*a(n-2) -a(n-3) +a(n-5) +2*a(n-6) +2*a(n-7) +a(n-8). - R. J. Mathar, Jul 13 2010
G.f. -x^2*(4+17*x+30*x^2+55*x^3+80*x^4+38*x^6+76*x^5) / ( (x-1)*(1+x)*(1+x+x^2)*(x^4+x^3+x^2+x+1) ). - R. J. Mathar, Dec 17 2012

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A171182 Period 6: repeat [0, 1, 1, 1, 0, 2].

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A178143 Sum of squares d^2 over the divisors d=2 and/or d=3 of n.

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A178144 Sum of divisors d of n which are d=2, 3 or 5.

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A178146 a(n) is the number of distinct prime factors <= 5 of n.

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A171405 Sum of divisors of n, excluding divisors 2 and 3.

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A178147 Sum of squares d^2 of distinct divisors of n, d in {2, 3, 5}.

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