A010878 -id:A010878 - cp's OEIS frontend

A130487 a(n) = Sum_{k=0..n} (k mod 9) (Partial sums of A010878).

0, 1, 3, 6, 10, 15, 21, 28, 36, 36, 37, 39, 42, 46, 51, 57, 64, 72, 72, 73, 75, 78, 82, 87, 93, 100, 108, 108, 109, 111, 114, 118, 123, 129, 136, 144, 144, 145, 147, 150, 154, 159, 165, 172, 180, 180, 181, 183, 186, 190, 195, 201, 208, 216, 216, 217, 219, 222, 226

Offset: 0

Hieronymus Fischer, May 31 2007

Let A be the Hessenberg n X n matrix defined by A[1,j]=j mod 9, 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 (1,0,0,0,0,0,0,0,1,-1).

Cf. A010872, A010873, A010874, A010875, A010876, A010877, A130481, A130482, A130483, A130484, A130485, A130486.

a:=[0,1,3,6,10,15,21,28,36,36];; for n in [11..71] do a[n]:=a[n-1]+a[n-9]-a[n-10]; od; a; # G. C. Greubel, Aug 31 2019

I:=[0,1,3,6,10,15,21,28,36,36]; [n le 10 select I[n] else Self(n-1) + Self(n-9) - Self(n-10): n in [1..71]]; // G. C. Greubel, Aug 31 2019

seq(coeff(series(x*(1-9*x^8+8*x^9)/((1-x^9)*(1-x)^3), x, n+1), x, n), n = 0 .. 70); # G. C. Greubel, Aug 31 2019

Accumulate[PadRight[{},120,Range[0,8]]] (* Harvey P. Dale, Dec 19 2018 *)
Accumulate[Mod[Range[0,100],9]] (* Harvey P. Dale, Oct 16 2021 *)

a(n) = sum(k=0, n, k % 9); \\ Michel Marcus, Apr 28 2018

def A130487_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x*(1-9*x^8+8*x^9)/((1-x^9)*(1-x)^3)).list()
A130487_list(70) # G. C. Greubel, Aug 31 2019

a(n) = 36*floor(n/9) + A010878(n)*(A010878(n) + 1)/2.
G.f.: (Sum_{k=1..8} k*x^k)/((1-x^9)*(1-x)).
G.f.: x*(1 - 9*x^8 + 8*x^9)/((1-x^9)*(1-x)^3).

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

A052382 Numbers without 0 in the decimal expansion, colloquial 'zeroless numbers'.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 98, 99, 111, 112, 113

Offset: 1

Henry Bottomley, Mar 13 2000

The entries 1 to 79 match the corresponding subsequence of A043095, but then 81, 91-98, 100, 102, etc. are only in one of the two sequences. - R. J. Mathar, Oct 13 2008
Complement of A011540; A168046(a(n)) = 1; A054054(a(n)) > 0; A007602, A038186, A038618, A052041, A052043, and A052045 are subsequences. - Reinhard Zumkeller, Apr 25 2012, Apr 07 2011, Dec 01 2009
a(n) = n written in base 9 where zeros are not allowed but nines are. The nine distinct digits used are 1, 2, 3, ..., 9 instead of 0, 1, 2, ..., 8. To obtain this sequence from the "canonical" base 9 sequence with zeros allowed, just replace any 0 with a 9 and then subtract one from the group of digits situated on the left. For example, 9^3 = 729 (10) (in base 10) = 1000 (9) (in base 9) = 889 (9-{0}) (in base 9 without zeros) because 100 (9) = [9-1]9 = 89 (9-{0}) and thus 1000 (9) = [89-1]9 = 889 (9-{0}). - Robin Garcia, Jan 15 2014
From Hieronymus Fischer, May 28 2014: (Start)
Inversion: Given a term m, the index n such that a(n) = m can be calculated by A052382_inverse(m) = m - sum_{1<=j<=k} floor(m/10^j)*9^(j-1), where k := floor(log_10(m)) [see Prog section for an implementation in Smalltalk].
Example 1: A052382_inverse(137) = 137 - (floor(137/10) + floor(137/100)*9) = 137 - (13*1 + 1*9) = 137 - 22 = 115.
Example 2: A052382_inverse(4321) = 4321 - (floor(4321/10) + floor(4321/100)*9 + floor(4321/1000)*81) = 4321 - (432*1 + 43*9 + 4*81) = 4321 - (432 + 387 + 324) = 3178. (End)
The sum of the reciprocals of these numbers from a(1)=1 to infinity, called the Kempner series, is convergent towards a limit: 23.103447... whose decimal expansion is in A082839. - Bernard Schott, Feb 23 2019
Integer n > 0 is encoded using bijective base-9 numeration, see Wikipedia link below. - Alois P. Heinz, Feb 16 2020

			For k >= 0, a(10^k) = (1, 11, 121, 1331, 14641, 162151, 1783661, 19731371, ...) = A325203(k). - _Hieronymus Fischer_, May 30 2012 and Jun 06 2012; edited by _M. F. Hasler_, Jan 13 2020

Paul Halmos, "Problems for Mathematicians, Young and Old", Dolciani Mathematical Expositions, 1991, p. 258.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
K. Mahler, On the generating function of the integers with a missing digit, J. Indian Math. Soc. 15A (1951), 34-40.
Eric Weisstein's World of Mathematics, Kempner Series.
Eric Weisstein's World of Mathematics, Zerofree
Wikipedia, Bijective numeration
Index entries for 10-automatic sequences.

Cf. A004719, A052040, different from A067251.
Cf. A055640, A046034, A007931, A007932, A084544, A084545.
Column k=9 of A214676.
Cf. A011540 (complement), A043489, A054054, A168046.
Cf. A052383 (without 1), A052404 (without 2), A052405 (without 3), A052406 (without 4), A052413 (without 5), A052414 (without 6), A052419 (without 7), A052421 (without 8), A007095 (without 9).
Zeroless numbers in some other bases <= 10: A000042 (base 2), A032924 (base 3), A023705 (base 4), A248910 (base 6), A255805 (base 8), A255808 (base 9).
Cf. A082839 (sum of reciprocals).
Cf. A038618 (subset of primes)
Subsequences: A007602, A038186, A052041, A052043, A052045, A059405, A066484, A099542.

a052382 n = a052382_list !! (n-1)
a052382_list = iterate f 1 where
f x = 1 + if r < 9 then x else 10 * f x' where (x', r) = divMod x 10
-- Reinhard Zumkeller, Mar 08 2015, Apr 07 2011

[ n: n in [1..114] | not 0 in Intseq(n) ]; // Bruno Berselli, May 28 2011

a:= proc(n) local d, l, m; m:= n; l:= NULL;
      while m>0 do d:= irem(m, 9, 'm');
        if d=0 then d:=9; m:= m-1 fi;
        l:= d, l
      od; parse(cat(l))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 11 2015
is_zeroless := n -> not is(0 in convert(n, base, 10)):
select(is_zeroless, [seq(1..113)]);  # Peter Luschny, Jun 20 2025

A052382 = Select[Range[100], DigitCount[#, 10, 0] == 0 &] (* Alonso del Arte, Mar 10 2011 *)

select( {is_A052382(n)=n&&vecmin(digits(n))}, [0..111]) \\ actually: is_A052382 = (bool) A054054. - M. F. Hasler, Jan 23 2013, edited Jan 13 2020

a(n) = for (w=0, oo, if (n >= 9^w, n -= 9^w, return ((10^w-1)/9 + fromdigits(digits(n, 9))))) \\ Rémy Sigrist, Jul 26 2017

apply( {A052382(n,L=logint(n,9))=fromdigits(digits(n-9^L>>3,9))+10^L\9}, [1..100])
next_A052382(n, d=digits(n+=1))={for(i=1, #d, d[i]|| return(n-n%(d=10^(#d-i+1))+d\9)); n} \\ least a(k) > n. Used in A038618.
( {A052382_vec(n,M=1)=M--;vector(n, i, M=next_A052382(M))} )(99) \\ n terms >= M
\\ See OEIS Wiki page (cf. LINKS) for more programs. - M. F. Hasler, Jan 11 2020

A052382 = [n for n in range(1,10**5) if not str(n).count('0')]
# Chai Wah Wu, Aug 26 2014

from sympy import integer_log
def A052382(n):
    m = integer_log(k:=(n<<3)+1,9)[0]
    return sum((1+(k-9**m)//(9**j<<3)%9)*10**j for j in range(m)) # Chai Wah Wu, Jun 27 2025

A052382
"Answers the n-th term of A052382, where n is the receiver."
^self zerofree: 10
A052382_inverse
"Answers that index n which satisfy A052382(n) = m, where m is the receiver.”
^self zerofree_inverse: 10
zerofree: base
"Answers the n-th zerofree number in base base, where n is the receiver. Valid for base > 2.
Usage: n zerofree: b [b = 10 for this sequence]
Answer: a(n)"
| n m s c bi ci d |
n := self.
c := base - 1.
m := (base - 2) * n + 1 integerFloorLog: c.
d := n - (((c raisedToInteger: m) - 1)//(base - 2)).
bi := 1.
ci := 1.
s := 0.
1 to: m
do:
[:i |
s := (d // ci \\ c + 1) * bi + s.
bi := base * bi.
ci := c * ci].
^s
zerofree_inverse: base
"Answers the index n such that the n-th zerofree number in base base is = m, where m is the receiver. Valid for base > 2.
Usage: m zerofree_inverse: b [b = 10 for this sequence]
Answer: n"
| m p q s |
m := self.
s := 0.
p := base.
q := 1.
[p < m] whileTrue:
[s := m // p * q + s.
p := base * p.
q := (base - 1) * q].
^m - s
"by Hieronymus Fischer, May 28 2014"

seq 0 1000 | grep -v 0; # Joerg Arndt, May 29 2011

a(n+1) = f(a(n)) with f(x) = 1 + if x mod 10 < 9 then x else 10*f([x/10]). - Reinhard Zumkeller, Nov 15 2009
From Hieronymus Fischer, Apr 30, May 30, Jun 08 2012, Feb 17 2019: (Start)
a(n) = Sum_{j=0..m-1} (1 + b(j) mod 9)*10^j, where m = floor(log_9(8*n + 1)), b(j) = floor((8*n + 1 - 9^m)/(8*9^j)).
Also: a(n) = Sum_{j=0..m-1} (1 + A010878(b(j)))*10^j.
a(9*n + k) = 10*a(n) + k, k=1..9.
Special values:
a(k*(9^n - 1)/8) = k*(10^n - 1)/9, k=1..9.
a((17*9^n - 9)/8) = 2*10^n - 1.
a((9^n - 1)/8 - 1) = 10^(n-1) - 1, n > 1.
Inequalities:
a(n) <= (1/9)*((8*n+1)^(1/log_10(9)) - 1), equality holds for n=(9^k-1)/8, k>0.
a(n) > (1/10)*((8*n+1)^(1/log_10(9)) - 1), n > 0.
Lower and upper limits:
lim inf a(n)/10^log_9(8*n) = 1/10, for n -> infinity.
lim inf a(n)/n^(1/log_10(9)) = 8^(1/log_10(9))/10, for n -> infinity.
lim sup a(n)/10^log_9(8*n) = 1/9, for n -> infinity.
lim sup a(n)/n^(1/log_10(9)) = 8^(1/log_10(9))/9, for n -> infinity.
G.f.: g(x) = (x^(1/8)*(1-x))^(-1) Sum_{j>=0} 10^j*z(j)^(9/8)*(1 - 10z(j)^9 + 9z(j)^10)/((1-z(j))(1-z(j)^9)), where z(j) = x^9^j.
Also: g(x) = (1/(1-x)) Sum_{j>=0} (1 - 10(x^9^j)^9 + 9(x^9^j)^10)*x^9^j*f_j(x)/(1-x^9^j), where f_j(x) = 10^j*x^((9^j-1)/8)/(1-(x^9^j)^9). Here, the f_j obey the recurrence f_0(x) = 1/(1-x^9), f_(j+1)(x) = 10x*f_j(x^9).
Also: g(x) = (1/(1-x))*((Sum{k=0..8} h_(9,k)(x)) - 9*h_(9,9)(x)), where h_(9,k)(x) = Sum_{j>=0} 10^j*x^((9^(j+1)-1)/8)*x^(k*9^j)/(1-x^9^(j+1)).
Generic formulas for analogous sequences with numbers expressed in base p and only using the digits 1, 2, 3, ... d, where 1 < d < p:
a(n) = Sum_{j=0..m-1} (1 + b(j) mod d)*p^j, where m = floor(log_d((d-1)*n+1)), b(j) = floor(((d-1)*n+1-d^m)/((d-1)*d^j)).
Special values:
a(k*(d^n-1)/(d-1)) = k*(10^n-1)/9, k=1..d.
a(d*((2d-1)*d^(n-1)-1)/(d-1)) = ((d+9)*10^n-d)/9 = 10^n + d*(10^n-1)/9.
a((d^n-1)/(d-1)-1) = d*(10^(n-1)-1)/9, n > 1.
Inequalities:
a(n) <= (10^log_d((d-1)*n+1)-1)/9, equality holds for n = (d^k-1)/(d-1), k > 0.
a(n) > (d/10)*(10^log_d((d-1)*n+1)-1)/9, n > 0.
Lower and upper limits:
lim inf a(n)/10^log_d((d-1)*n) = d/90, for n -> infinity.
lim sup a(n)/10^log_d((d-1)*n) = 1/9, for n -> infinity.
G.f.: g(x) = (1/(1-x)) Sum_{j>=0} (1 - (d+1)(x^d^j)^d + d(x^d^j)^(d+1))*x^d^j*f_j(x)/(1-x^d^j), where f_j(x) = p^j*x^((d^j-1)/(d-1))/(1-(x^d^j)^d). Here, the f_j obey the recursion f_0(x) = 1/(1-x^d), f_(j+1)(x) = px*f_j(x^d).
(End)
A052382 = { n | A054054(n) > 0 }. - M. F. Hasler, Jan 23 2013
From Hieronymus Fischer, Feb 20 2019: (Start)
Sum_{n>=1} (-1)^(n+1)/a(n) = 0.696899720...
Sum_{n>=1} 1/a(n)^2 = 1.6269683705819...
Sum_{n>=1} 1/a(n) = 23.1034479... = A082839. This so-called Kempner series converges very slowly. For the calculation of the sum, it is helpful to use the following fraction of partial sums, which converges rapidly:
lim_{n->infinity} (Sum_{k=p(n)..p(n+1)-1} 1/a(k)) / (Sum_{k=p(n-1)..p(n)-1} 1/a(k)) = 9/10, where p(n) = (9^n-1)/8, n > 1.
(End)

Typos in formula section corrected by Hieronymus Fischer, May 30 2012

Name clarified by Peter Luschny, Jun 20 2025

A037074 Numbers that are the product of a pair of twin primes.

Original entry on oeis.org

15, 35, 143, 323, 899, 1763, 3599, 5183, 10403, 11663, 19043, 22499, 32399, 36863, 39203, 51983, 57599, 72899, 79523, 97343, 121103, 176399, 186623, 213443, 272483, 324899, 359999, 381923, 412163, 435599, 656099, 675683, 685583, 736163

Offset: 1

Felice Russo

Each entry is the product of p and p+2 where both p and p+2 are prime, i.e., the product of the lesser and greater of a twin prime pair.
Except for the first term, all entries have digital root 8. - Lekraj Beedassy, Jun 11 2004
The above statement follows from p > 3 => (p,p+2) = (6k-1,6k+1) => p*(p+2) = 36k^2 - 1 == 8 (mod 9), and A010888 === A010878 (mod 9). - M. F. Hasler, Jan 11 2013
Albert A. Mullin states that m is a product of twin primes iff phi(m)*sigma(m) = (m-3)*(m+1), where phi(m) = A000010(m) and sigma(m) = A000203(m). Of course, for a product of distinct primes p*q we know sigma(p*q) = (p+1)*(q+1) and if p, q, are twin primes, say q = p + 2, then sigma(p*q) = (p+1)*(q+1) = (p+1)*(p+3). - Jonathan Vos Post, Feb 21 2006
Also the area of twin prime rectangles. A twin prime rectangle is a rectangle whose sides are components of twin prime pairs. E.g., the twin prime pair (3,5) produces a 3 X 5 unit rectangle which has area 15 square units. - Cino Hilliard, Jul 28 2006
Except for 15, a product of twin primes is of the form 36k^2 - 1 (cf. A136017, A002822). - Artur Jasinski, Dec 12 2007
A072965(a(n)) = 1; A072965(m) mod A037074(n) > 0 for all m. - Reinhard Zumkeller, Jan 29 2008
The number of terms less than 10^(2n) is A007508(n). - Robert G. Wilson v, Feb 08 2012
If m is the product of twin primes, then sigma(m) = m + 1 + 2*sqrt(m + 1), phi(m) = m + 1 - 2*sqrt(m + 1). pmin(m) = sqrt(m + 1) - 1, pmax(m) = sqrt(m + 1) + 1. - Wesley Ivan Hurt, Jan 06 2013
Semiprimes of the form 4*k^2 - 1. - Vincenzo Librandi, Apr 13 2013

			a(2)=35 because 5*7=35, that is (5,7) is the 2nd pair of twin primes.

Albert A. Mullin, "Bicomposites, twin primes and arithmetic progression", Abstract 04T-11-48, Abstracts of AMS, Vol. 25, No. 4, 2004, p. 795.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A001359, A006512, A014574, A136017, A074480 (multiplicative closure), A209328.
Cf. A071700 (subsequence).
Cf. A075369.

a037074 = subtract 1 . a075369  -- Reinhard Zumkeller, Feb 10 2015
-- Reinhard Zumkeller, Feb 10 2015, Aug 14 2011

[p*(p+2): p in PrimesUpTo(1000) | IsPrime(p+2)];  // Bruno Berselli, Jul 08 2011

IsSemiprime:=func; [s: n in [1..500] | IsSemiprime(s) where s is 4*n^2-1]; // Vincenzo Librandi, Apr 13 2013

ZL:=[]: for p from 1 to 863 do if (isprime(p) and isprime(p+2) ) then ZL:=[op(ZL),(p*(p+2))]; fi; od; print(ZL); # Zerinvary Lajos, Mar 07 2007
for i from 1 to 150 do if ithprime(i+1) = ithprime(i) + 2 then print({ithprime(i)*ithprime(i+1)}); fi; od; # Zerinvary Lajos, Mar 19 2007

s = Select[ Prime@ Range@170, PrimeQ[ # + 2] &]; s(s + 2) (* Robert G. Wilson v, Feb 21 2006 *)
(* For checking large numbers, the following code is better. For instance, we could use the fQ function to determine that 229031718473564142083 is in this sequence. *) fQ[n_] := Block[{fi = FactorInteger[n]}, Last@# & /@ fi == {1, 1} && Differences[ First@# & /@ fi] == {2}]; Select[ Range[750000], fQ] (* Robert G. Wilson v, Feb 08 2012 *)
Times@@@Select[Partition[Prime[Range[500]],2,1],Last[#]-First[#]==2&] (* Harvey P. Dale, Oct 16 2012 *)

g(n) = for(x=1,n,if(prime(x+1)-prime(x)==2,print1(prime(x)*prime(x+1)","))) \\ Cino Hilliard, Jul 28 2006

a(n) = A001359(n)*A006512(n). A000010(a(n))*A000203(a(n)) = (a(n)-3)*(a(n)+1). - Jonathan Vos Post, Feb 21 2006
a(n) = (A014574(n))^2 - 1. a(n+1) = (6*A002822(n))^2 - 1. - Lekraj Beedassy, Sep 02 2006
a(n) = A075369(n) - 1. - Reinhard Zumkeller, Feb 10 2015
Sum_{n>=1} 1/a(n) = A209328. - Amiram Eldar, Nov 20 2020
A000010(a(n)) == 0 (mod 8). - Darío Clavijo, Oct 26 2022

More terms from Erich Friedman

A052223 Numbers whose sum of digits is 9.

Original entry on oeis.org

9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 108, 117, 126, 135, 144, 153, 162, 171, 180, 207, 216, 225, 234, 243, 252, 261, 270, 306, 315, 324, 333, 342, 351, 360, 405, 414, 423, 432, 441, 450, 504, 513, 522, 531, 540, 603, 612, 621, 630, 702, 711, 720, 801, 810

Offset: 1

Henry Bottomley, Feb 01 2000

Any term of this sequence with an 11 appended cannot have 11 as prime factor. See A075154. [Lekraj Beedassy, Sep 27 2009]
A007953(a(n)) = 9; number of repdigits = #{9,333,1^9} = A242627(9) = 3. - Reinhard Zumkeller, Jul 17 2014
A010872(a(n)) = A010878(a(n)) = 0. - Ilya Gutkovskiy, Jun 04 2016

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)

Cf. A007953.
Row n=9 of A245062.
Cf. A011557 (1), A052216 (2), A052217 (3), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052224 (10), A166311 (11), A235151 (12), A143164 (13), A235225(14), A235226 (15), A235227 (16), A166370 (17), A235228 (18), A166459 (19), A235229 (20).
Cf. A242614, A242627.

a052223 n = a052223_list !! (n-1)
a052223_list = filter ((== 9) . a007953) [0..]
-- Reinhard Zumkeller, Jul 17 2014

[n: n in [1..1500] | &+Intseq(n) eq 9 ]; // Vincenzo Librandi, Mar 08 2013

Select[Range[1500], Total[IntegerDigits[#]] == 9 &] (* Vincenzo Librandi, Mar 08 2013 *)

More terms from Larry Reeves (Larryr(AT)acm.org), Sep 05 2000

Offset changed by Bruno Berselli, Mar 07 2013

A070366 a(n) = 5^n mod 9.

Original entry on oeis.org

1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7

Offset: 0

N. J. A. Sloane, May 12 2002

Period 6: repeat [1, 5, 7, 8, 4, 2].
Also the digital root of 5^n. - Cino Hilliard, Dec 31 2004
Digital root of the powers of any number congruent to 5 mod 9. - Alonso del Arte, Jan 26 2014

Cecil Balmond, Number 9: The Search for the Sigma Code. Munich, New York: Prestel (1998): 203.

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

Cf. Digital roots of powers of c mod 9: c = 2, A153130; c = 4, A100402; c = 7, A070403; c = 8, A010689.

Cf. A000351, A007953, A010878, A010888.

[Modexp(5, n, 9): n in [0..100]]; // Wesley Ivan Hurt, Jun 28 2016

A070366:=n->power(5,n) mod 9: seq(A070366(n), n=0..100); # Wesley Ivan Hurt, Jun 28 2016

PowerMod[5, Range[0, 120], 9] (* Harvey P. Dale, Mar 27 2011 *)
Table[Mod[5^n, 9], {n, 0, 100}] (* G. C. Greubel, Mar 05 2016 *)

a(n)=lift(Mod(5,9)^n); \\ Charles R Greathouse IV, Sep 24 2015

From R. J. Mathar, Apr 20 2010: (Start)
a(n) = a(n-1) - a(n-3) + a(n-4) for n>3.
G.f.: ( 1+4*x+2*x^2+2*x^3 ) / ( (1-x)*(1+x)*(x^2-x+1) ). (End)
a(n) = 1/2^n (mod 9), n >= 0. - Wolfdieter Lang, Feb 18 2014
a(n) = A010878(A000351(n)). - Michel Marcus, Feb 20 2014
From G. C. Greubel, Mar 05 2016: (Start)
a(n) = a(n-6) for n>5.
E.g.f.: (1/2)*(9*exp(x) - exp(-x) + 2*sqrt(3)*exp(x/2)*sin(sqrt(3)*x/2) - 6*exp(x/2)*cos(sqrt(3)*x/2)). (End)
a(n) = (9 - cos(n*Pi) - 6*cos(n*Pi/3) + 2*sqrt(3)*sin(n*Pi/3))/2. - Wesley Ivan Hurt, Jun 28 2016
a(n) = 2^((-n) mod 6) mod 9. - Joe Slater, Mar 23 2017
a(n) = A007953(5*a(n-1)) = A010888(5*a(n-1)). - Stefano Spezia, Mar 20 2025

A168182 Characteristic function of numbers that are not multiples of 9.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Nov 30 2009

			G.f. = x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^10 + x^11 + x^12 + x^13 + ...

Antti Karttunen, Table of n, a(n) for n = 0..999
Michael Somos, Rational Function Multiplicative Coefficients
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1).

Cf. A168185, A145568, A168184, A168181, A109720, A097325, A011558, A166486, A011655, A000035, A267142, A033441.

A168182[n_]:=Boole[!Divisible[n,9]]; Array[A168182, 10, 0]
Table[If[Mod[n,9]==0,0,1],{n,0,120}] (* Harvey P. Dale, Mar 05 2025 *)

{a(n) = sign(n%9)}; /* Michael Somos, Mar 22 2011 */

Euler transform of length 9 sequence [1, 0, 0, 0, 0, 0, 0, -1, 1]. - Michael Somos, Mar 22 2011
Moebius transform is length 9 sequence [1, 0, 0, 0, 0, 0, 0, 0, -1]. - Michael Somos, Mar 22 2011
Expansion of x * (1 - x^8) / ((1 - x) * (1 - x^9)) in powers of x. - Michael Somos, Mar 22 2011
Multiplicative with a(p^e) = (if p=3 then 0^(e-1) else 1), p prime and e>0.
a(n) = a(n+9) = a(-n) for all n in Z.
a(n) = A000007(A010878(n)).
a(A168183(n)) = 1. a(A008591(n)) = 0.
A033441(n) = Sum_{k=0..n} a(k)*(n-k).
G.f.: -x*(1+x)*(1+x^2)*(1+x^4) / ( (x-1)*(1+x+x^2)*(x^6+x^3+1) ). - R. J. Mathar, Jan 07 2011
Dirichlet g.f. (1-3^(-2s))*zeta(s). - R. J. Mathar, Mar 06 2011
For the general case: the characteristic function of numbers that are not multiples of m is a(n)=floor((n-1)/m)-floor(n/m)+1, m,n > 0. - Boris Putievskiy, May 08 2013
a(n) = 1 - A267142(n). - Antti Karttunen, Oct 07 2017

A130486 a(n) = Sum_{k=0..n} (k mod 8) (Partial sums of A010877).

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 28, 29, 31, 34, 38, 43, 49, 56, 56, 57, 59, 62, 66, 71, 77, 84, 84, 85, 87, 90, 94, 99, 105, 112, 112, 113, 115, 118, 122, 127, 133, 140, 140, 141, 143, 146, 150, 155, 161, 168, 168, 169, 171, 174, 178, 183, 189, 196, 196, 197, 199, 202, 206

Offset: 0

Hieronymus Fischer, May 31 2007

Let A be the Hessenberg n X n matrix defined by A[1,j] = j mod 8, 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 (1,0,0,0,0,0,0,1,-1).

Cf. A010872, A010873, A010874, A010875, A010876, A010878. A130481, A130482, A130483, A130484, A130485, A130487.

a:=[0,1,3,6,10,15,21,28,28];; for n in [10..71] do a[n]:=a[n-1]+a[n-8]-a[n-9]; od; a; # G. C. Greubel, Aug 31 2019

I:=[0,1,3,6,10,15,21,28,28]; [n le 9 select I[n] else Self(n-1) + Self(n-8) - Self(n-9): n in [1..71]]; // G. C. Greubel, Aug 31 2019

seq(coeff(series(x*(1-8*x^7+7*x^8)/((1-x^8)*(1-x)^3), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Aug 31 2019

Array[28 Floor[#1/8] + #2 (#2 + 1)/2 & @@ {#, Mod[#, 8]} &, 61, 0] (* Michael De Vlieger, Apr 28 2018 *)
Accumulate[PadRight[{},100,Range[0,7]]] (* Harvey P. Dale, Dec 21 2018 *)

a(n) = sum(k=0, n, k % 8); \\ Michel Marcus, Apr 28 2018

def A130486_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x*(1-8*x^7+7*x^8)/((1-x^8)*(1-x)^3)).list()
A130486_list(70) # G. C. Greubel, Aug 31 2019

a(n) = 28*floor(n/8) + A010877(n)*(A010877(n) + 1)/2.
G.f.: (Sum_{k=1..7} k*x^k)/((1-x^8)*(1-x)).
G.f.: x*(1 - 8*x^7 + 7*x^8)/((1-x^8)*(1-x)^3).

cp's OEIS Frontend

A130487 a(n) = Sum_{k=0..n} (k mod 9) (Partial sums of A010878).

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

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A052382 Numbers without 0 in the decimal expansion, colloquial 'zeroless numbers'.

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A037074 Numbers that are the product of a pair of twin primes.

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A052223 Numbers whose sum of digits is 9.

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