A010875 -id:A010875 - cp's OEIS frontend

A130484 a(n) = Sum_{k=0..n} (k mod 6) (Partial sums of A010875).

0, 1, 3, 6, 10, 15, 15, 16, 18, 21, 25, 30, 30, 31, 33, 36, 40, 45, 45, 46, 48, 51, 55, 60, 60, 61, 63, 66, 70, 75, 75, 76, 78, 81, 85, 90, 90, 91, 93, 96, 100, 105, 105, 106, 108, 111, 115, 120, 120, 121, 123, 126, 130, 135, 135, 136, 138, 141, 145, 150, 150, 151, 153

Offset: 0

Hieronymus Fischer, May 31 2007

Let A be the Hessenberg n X n matrix defined by A[1,j] = j mod 6, A[i,i]=1, A[i,i-1]=-1. Then, for n >= 1, a(n)=det(A). - Milan Janjic, Jan 24 2010

Shawn A. Broyles, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Cf. A010872, A010873, A010874, A010876, A010877, A130481, A130482, A130483, A130485.

a:=[0,1,3,6,10,15,15];; for n in [8..71] do a[n]:=a[n-1]+a[n-6]-a[n-7]; od; a; # G. C. Greubel, Aug 31 2019

I:=[0,1,3,6,10,15,15]; [n le 7 select I[n] else Self(n-1) + Self(n-6) - Self(n-7): n in [1..71]]; // G. C. Greubel, Aug 31 2019

seq(coeff(series(x*(1-6*x^5+5*x^6)/((1-x^6)*(1-x)^3), x, n+1), x, n), n = 0 .. 70); # G. C. Greubel, Aug 31 2019

Accumulate[Mod[Range[0,70],6]] (* or *) Accumulate[PadRight[ {},70, Range[0,5]]] (* Harvey P. Dale, Jul 12 2016 *)

a(n) = sum(k=0, n, k % 6); \\ Michel Marcus, Apr 28 2018

a(n)=n\6*15 + binomial(n%6+1,2) \\ Charles R Greathouse IV, Jan 24 2022

def A130484_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x*(1-6*x^5+5*x^6)/((1-x^6)*(1-x)^3)).list()
A130484_list(70) # G. C. Greubel, Aug 31 2019

a(n) = 15*floor(n/6) + A010875(n)*(A010875(n) + 1)/2.

G.f.: (Sum_{k=1..5} k*x^k)/((1-x^6)*(1-x)) = x*(1 - 6*x^5 + 5*x^6)/((1-x^6)*(1-x)^3).

A319717 Filter sequence combining the largest proper divisor of n (A032742) with modulo 6 residue of the smallest prime factor, A010875(A020639(n)), and a single bit A319710(n) telling whether the smallest prime factor is unitary.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 5, 11, 7, 12, 13, 14, 5, 15, 7, 16, 17, 18, 5, 19, 20, 21, 22, 23, 5, 24, 7, 25, 26, 27, 28, 29, 7, 30, 31, 32, 5, 33, 7, 34, 35, 36, 5, 37, 38, 39, 40, 41, 5, 42, 43, 44, 45, 46, 5, 47, 7, 48, 49, 50, 51, 52, 7, 53, 54, 55, 5, 56, 7, 57, 58, 59, 60, 61, 7, 62, 63, 64, 5, 65, 66, 67, 68, 69, 5, 70, 71, 72, 73, 74, 75, 76, 7, 77, 78, 79, 5, 80, 7, 81, 82, 83, 5, 84, 7, 85, 86, 87, 5, 88, 89, 90, 91, 92, 93, 94, 95

Offset: 1

Antti Karttunen, Oct 04 2018

nonn

Restricted growth sequence transform of triple [A010875(A020639(n)), A032742(n), A319710(n)] (with a separate value allotted for a(1)), or equally, of ordered pair [A319716(n), A319710(n)].
In addition to A319716, this filter sequence also records in the value of a(n) also the fact whether the smallest prime factor of n is unitary or not. This information is enough to determine the modulo 6 residues of all the divisors of n, thus sequences like A002324 are essentially functions of this sequence. Moreover, a lot of other information is immediately (and unavoidably) present, for example the exact prime signature of n, including also the relative order of exponents.
Any such filtering sequence can be perceived also in terms of what information it leaves out from a(n) that would be needed to reconstruct whole n from each a(n). If the whole n could be reconstructed from a(n) each time, then sequence a would be injective, and would be useless for filtering, because then it would match with any sequence. In this filter, what is left out is only the exact identity of the smallest prime factor, although its residue class mod 6 is retained. However, when the smallest prime factor is 2 or 3, this can be seen from that residue value, so for any number x in A047229, both A020639(x) and A032742(x) are known, and as x = A020639(x)*A032742(x), it means such numbers must occur in their own singleton equivalence classes.
Likewise, for any n in A283050, even if not divisible by 2 or 3, when we have A319710(n) stored in the triple as 1, this immediately gives away the exact identity of the smallest prime factor, which is equal to A014673(n) = A020639(A032742(n)) in these cases.
Thus there is a substantial subset of N (containing at least the union of A047229 and A283050) which is actually in the "blind sector" of this filter, "where anything goes", as this sequence obtains only unique values in that subdomain.
There is a related filter sequence A319996, which operates by "cleaving n from its high end" (by storing the residue class of the largest prime factor, A006530, instead of the smallest, together with n/A006530(n)), which has its own blind spots, but fortunately, they do not fully coincide with the blind spots of this filter. Naturally, any sequence like A002324 should match both to this sequence and A319996.
For all i, j:
  a(i) = a(j) => A002324(i) = A002324(j),
  a(i) = a(j) => A067029(i) = A067029(j),
  a(i) = a(j) => A071178(i) = A071178(j),
  a(i) = a(j) => A077462(i) = A077462(j) => A101296(i) = A101296(j),
  a(i) = a(j) => A319716(i) = A319716(j) => A319690(i) = A319690(j).

			For n = 65 = 5*13 and 143 = 11*13, the smallest prime factor is of the form 6k+5,  doesn't occur more than once in the factorization, and the largest proper divisor is the same number (13) in both cases, thus a(65) = a(143) (= 51, a running count value allotted by rgs-transform for this equivalence class).
For n = 1805 (5*19^2), 3971 (11*19^2), 6137 (17*19^2), it's like above, but the largest proper divisor is in all three cases 361 = 19^2, thus a(1805) = a(3971) = a(6137) (= 1405).
Note that such nontrivial equivalence classes may only contain numbers that are 5-rough, A007310, with no prime factors 2 or 3, and also, they may not contain numbers from A283050. See the comments section.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A101296, A319707, A319716, A319996.
Cf. also A320004 (analogous sequence for modulo 4 residues).
Differs from A319707 for the first time at n=143, where a(143) = 51, differs from A319716 for the first time at n=121, where a(121) = 95.

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A286476(n) = if(1==n,n,(6*A032742(n) + (n % 6)));
A319710(n) = ((n>1)&&(factor(n)[1,2]>1));
v319717 = rgs_transform(vector(up_to,n,[A286476(n),A319710(n)]));
A319717(n) = v319717[n];

A319716 Filter sequence combining the largest proper divisor of n (A032742) with modulo 6 residue of the smallest prime factor, A010875(A020639(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 5, 11, 7, 12, 13, 14, 5, 15, 7, 16, 17, 18, 5, 19, 20, 21, 22, 23, 5, 24, 7, 25, 26, 27, 28, 29, 7, 30, 31, 32, 5, 33, 7, 34, 35, 36, 5, 37, 38, 39, 40, 41, 5, 42, 43, 44, 45, 46, 5, 47, 7, 48, 49, 50, 51, 52, 7, 53, 54, 55, 5, 56, 7, 57, 58, 59, 60, 61, 7, 62, 63, 64, 5, 65, 66, 67, 68, 69, 5, 70, 71, 72, 73, 74, 75, 76, 7, 77, 78, 79, 5, 80, 7, 81, 82, 83, 5, 84, 7, 85, 86, 87, 5, 88, 89, 90, 91, 92, 93, 94, 43

Offset: 1

Antti Karttunen, Oct 04 2018

nonn

Restricted growth sequence transform of A286475, or equally, of A286476.
In each a(n) there is enough information to determine the modulo 6 residues of all the prime factors of n (when counted with multiplicity), thus sequences like A319690 and A319691 (which is the characteristic function of A004611) are essentially functions of this sequence. However, to determine that for all divisors of n, more information is needed. See A319717.
For all i, j:
  A319707(i) = A319707(j) => A319717(i) = A319717(j) => a(i) = a(j),
  a(i) = a(j) => A319690(i) = A319690(i) => A319691(i) = A319691(j).

			For n = 55 = 5*11 and 121 = 11*11, 55 = 121 = 1 mod 6 and 11 is their common largest proper divisor, thus they are allotted the same number by the restricted growth sequence transform, that is a(55) = a(121) = 43 (which is the number allotted). Note that such nontrivial equivalence classes may only contain numbers that are 5-rough, A007310, with no prime factors 2 or 3.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A010875, A032742, A286475, A286476, A319707, A319717.
Cf. A007528 (positions of 5's), A002476 (positions of 7's).
Cf. also A319714.
Differs from A319707 and A319717 for the first time at n=121.

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A286476(n) = if(1==n,n,(6*A032742(n) + (n % 6)));
v319716 = rgs_transform(vector(up_to,n,A286476(n)));
A319716(n) = v319716[n];

A010888 Digital root of n (repeatedly add the digits of n until a single digit is reached).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

This is sometimes also called the additive digital root of n.
n mod 9 (A010878) is a very similar sequence.
Partial sums are given by A130487(n-1) + n (for n > 0). - Hieronymus Fischer, Jun 08 2007
Decimal expansion of 13717421/111111111 is 0.123456789123456789123456789... with period 9. - Eric Desbiaux, May 19 2008
Decimal expansion of 13717421 / 1111111110 = 0.0[123456789] (periodic) - Daniel Forgues, Feb 27 2017
a(A005117(n)) < 9. - Reinhard Zumkeller, Mar 30 2010
My friend Jahangeer Kholdi has found that 19 is the smallest prime p such that for each number n, a(p*n) = a(n). In fact we have: a(m*n) = a(a(m)*a(n)) so all numbers with digital root 1 (numbers of the form 9k + 1) have this property. See comment lines of A017173. Also we have a(m+n) = a(a(m) + a(n)). - Farideh Firoozbakht, Jul 23 2010

			The digits of 37 are 3 and 7, and 3 + 7 = 10. And the digits of 10 are 1 and 0, and 1 + 0 = 1, so a(37) = 1.

Martin Gardner, Mathematics, Magic and Mystery, 1956.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Digitaddition
Eric Weisstein's World of Mathematics, Digital Root
Wikipedia, Vedic square
Index entries for Colombian or self numbers and related sequences
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1).

Cf. A007953, A007954, A031347, A113217, A113218, A010878 (n mod 9), A010872, A010873, A010874, A010875, A010876, A010877, A010879, A004526, A002264, A002265, A002266, A017173, A031286 (additive persistence of n), (multiplicative digital root of n), A031346 (multiplicative persistence of n).

a010888 = until (< 10) a007953
-- Reinhard Zumkeller, Oct 17 2011, May 12 2011

[n eq 0 select 0 else 1+(n-1) mod 9: n in [0..110]]; // Bruno Berselli, Mar 18 2016

A010888 := n->if n=0 then 0 else ((n-1) mod 9) + 1; fi; # N. J. A. Sloane, Feb 20 2013

Join[{0}, Array[Mod[ # - 1, 9] + 1 &, 104]] (* Robert G. Wilson v, Jan 04 2006 *)
Join[Range[0, 1], Table[n - 9 Floor[(n - 1) / 9], {n, 2, 100}]] (* José de Jesús Camacho Medina, Nov 10 2014 *) (* Corrected by Vincenzo Librandi, Nov 11 2014 *)
Join[{0},LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 1},{1, 2, 3, 4, 5, 6, 7, 8, 9},104]] (* Ray Chandler, Aug 26 2015 *)
Table[FixedPoint[Total[IntegerDigits[#, 10]] &, n], {n, 0, 104}] (* IWABUCHI Yu(u)ki, Jun 03 2016 *)

A010888(n)=if(n,(n-1)%9+1) \\ M. F. Hasler, Jan 04 2011

def A010888(n):
    return 1 + (n - 1) % 9 if n else 0 # Chai Wah Wu, Aug 23 2014, Apr 23 2023

0 :: List.fill(10)(1 to 9).flatten // Alonso del Arte, Feb 01 2020

If n = 0 then a(n) = 0; otherwise a(n) = (n reduced mod 9), but if the answer is 0 change it to 9.
Equivalently, if n = 0 then a(n) = 0, otherwise a(n) = (n - 1 reduced mod 9) + 1.
If the initial 0 term is ignored, the sequence is periodic with period 9.
From Hieronymus Fischer, Jun 08 2007: (Start)
a(n) = A010878(n-1) + 1 (for n > 0).
G.f.: g(x) = x*(Sum_{k = 0..8}(k+1)*x^k)/(1 - x^9). Also: g(x) = x(9x^10 - 10x^9 + 1)/((1 - x^9)(1 - x)^2). (End)
a(n) = n - 9*floor((n-1)/9), for n > 0. - José de Jesús Camacho Medina, Nov 10 2014

A130481 a(n) = Sum_{k=0..n} (k mod 3) (i.e., partial sums of A010872).

Original entry on oeis.org

0, 1, 3, 3, 4, 6, 6, 7, 9, 9, 10, 12, 12, 13, 15, 15, 16, 18, 18, 19, 21, 21, 22, 24, 24, 25, 27, 27, 28, 30, 30, 31, 33, 33, 34, 36, 36, 37, 39, 39, 40, 42, 42, 43, 45, 45, 46, 48, 48, 49, 51, 51, 52, 54, 54, 55, 57, 57, 58, 60, 60, 61, 63, 63, 64, 66, 66, 67, 69, 69, 70, 72, 72

Offset: 0

Hieronymus Fischer, May 29 2007

Essentially the same as A092200. - R. J. Mathar, Jun 13 2008
Let A be the Hessenberg n X n matrix defined by: A[1,j]=j mod 3, A[i,i]:=1, A[i,i-1]=-1. Then, for n>=1, a(n)=det(A). - Milan Janjic, Jan 24 2010
2-adic valuation of A104537(n+1). - Gerry Martens, Jul 14 2015
Conjecture: a(n) is the exponent of the largest power of 2 that divides all the entries of the matrix {{3,1},{1,-1}}^n. - Greg Dresden, Sep 09 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A010872, A010873, A010874, A010875, A010876, A010877, A130482, A130483, A130484.

Cf. A092200, A104537.

List([0..80], n-> Int((n+1)/3) + Int(2*(n+1)/3)); # G. C. Greubel, Aug 31 2019

[Floor((n+1)/3) + Floor(2*(n+1)/3): n in [0..80]]; // G. C. Greubel, Aug 31 2019

seq(coeff(series(x*(1+2*x)/((1-x^3)*(1-x)), x, n+1), x, n), n = 0..80); # G. C. Greubel, Aug 31 2019

a[n_]:= Floor[(n+1)/3] + Floor[2(n+1)/3]; Table[a[n], {n, 0, 80}] (* Clark Kimberling, May 28 2012 *)
a[n_]:= IntegerExponent[A104537[n + 1], 2];
Table[a[n], {n, 0, 80}]  (* Gerry Martens, Jul 14 2015 *)
CoefficientList[Series[x(1+2x)/((1-x^3)(1-x)), {x, 0, 80}], x] (* Stefano Spezia, Sep 09 2018 *)
LinearRecurrence[{1,0,1,-1},{0,1,3,3},100] (* Harvey P. Dale, Jun 14 2021 *)

main(size)=my(n,k);vector(size,n,sum(k=0,n,k%3)) \\ Anders Hellström, Jul 14 2015

first(n)=my(s); concat(0, vector(n,k,s+=k%3)) \\ Charles R Greathouse IV, Jul 14 2015

a(n)=n\3*3+[0,1,3][n%3+1] \\ Charles R Greathouse IV, Jul 14 2015

def A130481_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x*(1+2*x)/((1-x^3)*(1-x))).list()
A130481_list(80) # G. C. Greubel, Aug 31 2019

a(n) = 3*floor(n/3) + A010872(n)*(A010872(n) + 1)/2.
G.f.: x*(1 + 2*x)/((1-x^3)*(1-x)).
a(n) = n + 1 - (Fibonacci(n+1) mod 2). - Gary Detlefs, Mar 13 2011
a(n) = floor((n+1)/3) + floor(2*(n+1)/3). - Clark Kimberling, May 28 2010
a(n) = n when n+1 is not a multiple of 3, and a(n) = n+1 when n+1 is a multiple of 3. - Dennis P. Walsh, Aug 06 2012
a(n) = n + 1 - sign((n+1) mod 3). - Wesley Ivan Hurt, Sep 25 2017
a(n) = n + (1-cos(2*(n+2)*Pi/3))/3 + sin(2*(n+2)*Pi/3)/sqrt(3). - Wesley Ivan Hurt, Sep 27 2017
a(n) = n + 1 - (n+1)^2 mod 3. - Ammar Khatab, Aug 14 2020
E.g.f.: ((1 + 3*x)*cosh(x) - (cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))*(cosh(x/2) - sinh(x/2)) + (1 + 3*x)*sinh(x))/3. - Stefano Spezia, May 28 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(3*sqrt(3)) + log(2)/3. - Amiram Eldar, Sep 17 2022

A130482 a(n) = Sum_{k=0..n} (k mod 4) (Partial sums of A010873).

Original entry on oeis.org

0, 1, 3, 6, 6, 7, 9, 12, 12, 13, 15, 18, 18, 19, 21, 24, 24, 25, 27, 30, 30, 31, 33, 36, 36, 37, 39, 42, 42, 43, 45, 48, 48, 49, 51, 54, 54, 55, 57, 60, 60, 61, 63, 66, 66, 67, 69, 72, 72, 73, 75, 78, 78, 79, 81, 84, 84, 85, 87, 90, 90, 91, 93, 96, 96, 97, 99, 102, 102, 103, 105

Offset: 0

Hieronymus Fischer, May 29 2007

Let A be the Hessenberg n X n matrix defined by: A[1,j]=j mod 4, A[i,i]:=1, A[i,i-1]=-1. Then, for n>=1, a(n)=det(A). - Milan Janjic, Jan 24 2010

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A010872, A010874, A010875, A010876, A010877. A130481, A130483, A130484, A130485.

a:=[0,1,3,6,6];; for n in [6..71] do a[n]:=a[n-1]+a[n-4]-a[n-5]; od; a; # G. C. Greubel, Aug 31 2019

I:=[0,1,3,6,6]; [n le 5 select I[n] else Self(n-1) + Self(n-4) - Self(n-5): n in [1..71]]; // G. C. Greubel, Aug 31 2019

a:=n->add(chrem( [n,j], [1,4] ),j=1..n):seq(a(n), n=0..70); # Zerinvary Lajos, Apr 07 2009

Table[(6*n +(1-(-1)^n)*(1+2*I^(n+1)))/4, {n,0,70}] (* G. C. Greubel, Aug 31 2019 *)
LinearRecurrence[{1,0,0,1,-1},{0,1,3,6,6},80] (* Harvey P. Dale, Feb 16 2024 *)

a(n) = (1 - (-1)^n - (2*I)*(-I)^n + (2*I)*I^n + 6*n) / 4 \\ Colin Barker, Oct 15 2015

def A130482_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x*(1+2*x+3*x^2)/((1-x^4)*(1-x))).list()
A130482_list(70) # G. C. Greubel, Aug 31 2019

a(n) = 6*floor(n/4) + A010873(n)*(A010873(n)+1)/2.
G.f.: x*(1 + 2*x + 3*x^2)/((1-x^4)*(1-x)).
a(n) = (1 - (-1)^n - (2*i)*(-i)^n + (2*i)*i^n + 6*n) / 4 where i = sqrt(-1). - Colin Barker, Oct 15 2015
a(n) = 3*n/2 + (n mod 2)* ( (n-1) mod 4 ) - (n mod 2)/2. - Ammar Khatab, Aug 27 2020
E.g.f.: (3*x*exp(x) - 2*sin(x) + sinh(x))/2. - Stefano Spezia, Apr 22 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(4*sqrt(3)) + log(3)/4. - Amiram Eldar, Sep 17 2022

A130483 a(n) = Sum_{k=0..n} (k mod 5) (Partial sums of A010874).

Original entry on oeis.org

0, 1, 3, 6, 10, 10, 11, 13, 16, 20, 20, 21, 23, 26, 30, 30, 31, 33, 36, 40, 40, 41, 43, 46, 50, 50, 51, 53, 56, 60, 60, 61, 63, 66, 70, 70, 71, 73, 76, 80, 80, 81, 83, 86, 90, 90, 91, 93, 96, 100, 100, 101, 103, 106, 110, 110, 111, 113, 116, 120, 120, 121, 123, 126, 130, 130

Offset: 0

Hieronymus Fischer, May 29 2007

Let A be the Hessenberg n X n matrix defined by: A[1,j]=j mod 5, A[i,i]=1, A[i,i-1]=-1. Then, for n>=1, a(n)=det(A). - Milan Janjic, Jan 24 2010

Shawn A. Broyles, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A010872, A010873, A010875, A010876, A010877, A130481, A130482, A130484, A130485.

a:=[0,1,3,6,10,10];; for n in [7..71] do a[n]:=a[n-1]+a[n-5]-a[n-6]; od; a; # G. C. Greubel, Aug 31 2019

I:=[0,1,3,6,10,10]; [n le 6 select I[n] else Self(n-1) + Self(n-5) - Self(n-6): n in [1..71]]; // G. C. Greubel, Aug 31 2019

seq(coeff(series(x*(1+2*x+3*x^2+4*x^3)/((1-x^5)*(1-x)), x, n+1), x, n), n = 0..70); # G. C. Greubel, Aug 31 2019

Accumulate[Mod[Range[0,70],5]] (* or *) Accumulate[PadRight[{},70,{0,1,2,3,4}]] (* Harvey P. Dale, Nov 11 2016 *)

a(n) = sum(k=0, n, k % 5); \\ Michel Marcus, Apr 28 2018

def A130483_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x*(1+2*x+3*x^2+4*x^3)/((1-x^5)*(1-x))).list()
A130483_list(70) # G. C. Greubel, Aug 31 2019

a(n) = 10*floor(n/5) + A010874(n)*(A010874(n)+1)/2.
G.f.: x*(1 + 2*x + 3*x^2 + 4*x^3)/((1-x^5)*(1-x)).
From Wesley Ivan Hurt, Jul 23 2016: (Start)
a(n) = a(n-5) - a(n-6) for n>5; a(n) = a(n-5) + 10 for n>4.
a(n) = 10 + Sum_{k=1..4} k*floor((n-k)/5). (End)
a(n) = ((n mod 5)^2 - 3*(n mod 5) + 4*n)/2. - Ammar Khatab, Aug 13 2020

A084301 a(n) = sigma(n) mod 6.

Original entry on oeis.org

1, 3, 4, 1, 0, 0, 2, 3, 1, 0, 0, 4, 2, 0, 0, 1, 0, 3, 2, 0, 2, 0, 0, 0, 1, 0, 4, 2, 0, 0, 2, 3, 0, 0, 0, 1, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 4, 3, 3, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 1, 0, 0, 2, 0, 0, 0, 0, 3, 2, 0, 4, 2, 0, 0, 2, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 3, 0, 1, 0, 0, 2, 0, 0

Offset: 1

Labos Elemer, Jun 02 2003

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sums of divisors

Cf. A000203, A010875, A054763, A084299, A084300.
Sequences sigma(n) mod k: A053866 (k=2), A074941 (k=3), A105824 (k=4), A105825 (k=5), A084301 (k=6), A105826 (k=7), A105827 (k=8).
Cf. A074627 (locations of 0), A074628 (locations of 2), A067051 (locations of 3), A074630 (locations of 4), A074384 (locations of 5).

A084301:= n-> (numtheory[sigma](n) mod 6):
seq (A084301(n), n=1..105); # Jani Melik, Jan 26 2011

Table[Mod[DivisorSigma[1,n],6],{n,120}]  (* Harvey P. Dale, Apr 07 2011 *)

a(n)=sigma(n)%6 \\ Charles R Greathouse IV, Mar 09 2014

a(n) = A010875(A000203(n)). - Antti Karttunen, Nov 07 2017

A079979 Characteristic function of multiples of six.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1

Offset: 0

Vladimir Baltic, Feb 17 2003

Period 6: repeat [1, 0, 0, 0, 0, 0].
a(n)=1 if n=6k, a(n)=0 otherwise.
Decimal expansion of 1/999999.
Number of permutations satisfying -k <= p(i)-i <= r and p(i)-i not in I, i=1..n, with k=3, r=3, I={-2,-1,0,1,2}.
Also, number of permutations satisfying -k <= p(i)-i <= r and p(i)-i not in I, i=1..n, with k=1, r=5, I={0,1,2,3,4}.
a(n) is also the number of partitions of n such that each part is six (a(0)=1 because the empty partition has no parts to test equality with six). Hence a(n) is also the number of 2-regular graphs on n vertices with each part having girth exactly six. - Jason Kimberley, Oct 10 2011
This sequence is the Euler transformation of A185016. - Jason Kimberley, Oct 10 2011

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), pp. 755-770. North-Holland, Amsterdam, 1970.

Antti Karttunen, Table of n, a(n) for n = 0..65538
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (April, 2010), 119-135
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).
Index entries for characteristic functions

Cf. A002524-A002529, A010875, A072827, A072850-A072856, A079955-A080014, A097325, A122841.

Characteristic function of multiples of g: A000007 (g=0), A000012 (g=1), A059841 (g=2), A079978 (g=3), A121262 (g=4), A079998 (g=5), this sequence (g=6), A082784 (g=7).

Magma
```
&cat[[1,0^^5]^^30];
```

A079979 := func; [A079979:n in [0..59]];  // Jason Kimberley, Oct 10 2011

PadRight[{},120,{1,0,0,0,0,0}] (* Harvey P. Dale, Feb 19 2013 *)

a(n)=!(n%6) \\ Charles R Greathouse IV, Oct 10 2011

(define (A079979 n) (if (zero? (modulo n 6)) 1 0)) ;; Antti Karttunen, Dec 22 2017

a(n) = a(n-6).
G.f.: 1/(1-x^6).
a(n) = floor((1/2)*cos(n*Pi/3) + 1/2). - Gary Detlefs, May 16 2011
a(n) = floor(n/6) - floor((n-1)/6). - Tani Akinari, Oct 23 2012
a(n) = (((((v^n - w^n)^2)*(2 - (-1)^n)*(w^(2*n) + w^n - 3))^2 - 144)^2)/20736, where w = (-1+i*sqrt(3))/2, v = (1+i*sqrt(3))/2. - Bogart B. Strauss, Sep 20 2013
E.g.f.: (2*cos(sqrt(3)*x/2)*cosh(x/2) + cosh(x))/3. - Vaclav Kotesovec, Feb 15 2015

More terms from Antti Karttunen, Dec 22 2017

A010882 Period 3: repeat [1, 2, 3].

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Partial sums are given by A130481(n)+n+1. - Hieronymus Fischer, Jun 08 2007
41/333 = 0.123123123... - Eric Desbiaux, Nov 03 2008
Terms of the simple continued fraction for 3/(sqrt(37)-4). - Paolo P. Lava, Feb 16 2009
This is the lexicographically earliest sequence with no substring of more than 1 term being a palindrome. - Franklin T. Adams-Watters, Nov 24 2013

Index entries for linear recurrences with constant coefficients, signature (0,0,1).
Index entries for sequences that are fixed points of mappings

Cf. A010872, A010873, A010874, A010875, A010876, A004526, A002264, A002265, A002266, A177036 (decimal expansion of (4+sqrt(37))/7), A214090.

Cf. A130481, A180593.

a010882 = (+ 1) . (`mod` 3)
a010882_list = cycle [1,2,3]
-- Reinhard Zumkeller, Mar 20 2013

&cat[[1..3]^^30]; // Vincenzo Librandi, Feb 04 2016

seq(op([1, 2, 3]), n=0..50); # Wesley Ivan Hurt, Jul 05 2016

Nest[ Flatten[ # /. {1 -> {1, 2}, 2 -> {3, 1}, 3 -> {2, 3}}] &, {1}, 7] (* Robert G. Wilson v, Mar 08 2005 *)
PadRight[{},120,{1,2,3}] (* Harvey P. Dale, Apr 09 2018 *)

a(n) = 1 + n%3; \\ Michel Marcus, Feb 04 2016

G.f.: (1+2x+3x^2)/(1-x^3). - Paul Barry, May 25 2003
a(n) = 1 + (n mod 3). - Paolo P. Lava, Nov 21 2006
a(n) = A010872(n) + 1. - Hieronymus Fischer, Jun 08 2007
a(n) = 6 - a(n-1) - a(n-2) for n > 1. - Reinhard Zumkeller, Apr 13 2008
a(n) = n+1-3*floor(n/3) = floor(41*10^(n+1)/333)-floor(41*10^n/333)*10; a(n)-a(n-3)=0 with n>2. - Bruno Berselli, Jun 28 2010
a(n) = A180593(n+1)/3. - Reinhard Zumkeller, Oct 25 2010
a(n) = floor((4*n+3)/3) mod 4. - Gary Detlefs, May 15 2011
a(n) = -cos(2/3*Pi*n)-1/3*3^(1/2)*sin(2/3*Pi*n)+2. - Leonid Bedratyuk, May 13 2012
E.g.f.: 2*(3*exp(3*x/2) - sqrt(3)*cos(Pi/6-sqrt(3)*x/2))*exp(-x/2)/3. - Ilya Gutkovskiy, Jul 05 2016

cp's OEIS Frontend

A130484 a(n) = Sum_{k=0..n} (k mod 6) (Partial sums of A010875).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

A319717 Filter sequence combining the largest proper divisor of n (A032742) with modulo 6 residue of the smallest prime factor, A010875(A020639(n)), and a single bit A319710(n) telling whether the smallest prime factor is unitary.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A319716 Filter sequence combining the largest proper divisor of n (A032742) with modulo 6 residue of the smallest prime factor, A010875(A020639(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A010888 Digital root of n (repeatedly add the digits of n until a single digit is reached).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Python

Scala

Formula

A130481 a(n) = Sum_{k=0..n} (k mod 3) (i.e., partial sums of A010872).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

PARI

PARI

Sage

Formula

A130482 a(n) = Sum_{k=0..n} (k mod 4) (Partial sums of A010873).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs