A010888 -id:A010888 - cp's OEIS frontend

A007612 a(n+1) = a(n) + digital root (A010888) of a(n).

1, 2, 4, 8, 16, 23, 28, 29, 31, 35, 43, 50, 55, 56, 58, 62, 70, 77, 82, 83, 85, 89, 97, 104, 109, 110, 112, 116, 124, 131, 136, 137, 139, 143, 151, 158, 163, 164, 166, 170, 178, 185, 190, 191, 193, 197, 205, 212, 217, 218, 220, 224, 232, 239, 244, 245, 247, 251

Offset: 1

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

Take m, a natural number. If m == 1 (mod 6), then for every n a(m)*a(n) is in A007612. - Ivan N. Ianakiev, May 08 2013

J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 65.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Digital Root.

Cf. A004207, A010888, A029898, A064806.

a007612 n = a007612_list !! (n-1)
a007612_list = iterate a064806 1  -- Reinhard Zumkeller, Apr 13 2013

A007612 := proc(n) option remember: if(n=1)then return 1: fi: return procname(n-1) + ((procname(n-1)-1) mod 9) + 1: end: seq(A007612(n), n=1..100); # Nathaniel Johnston, May 04 2011

dr[n_]:=NestWhile[Total[IntegerDigits[#]]&,n,#>9&]; NestList[#+dr[#]&, 1,60] (* Harvey P. Dale, Sep 24 2011 *)
NestList[#+Mod[#,9]&,1,60] (* Harvey P. Dale, Sep 14 2016 *)

first(n)=my(v=vector(n)); v[1]=1; for(k=2,n, v[k]=v[k-1]+v[k-1]%9); v \\ Charles R Greathouse IV, Jun 25 2017

a(n)=n\6*27 + [-4,1,2,4,8,16][n%6+1] \\ Charles R Greathouse IV, Jun 25 2017

a(1) = 1, a(n+1) = a(n) + a(n) mod 9. - Reinhard Zumkeller, Mar 23 2003
First differences are [1,2,4,8,7,5] repeated. - M. F. Hasler, Sep 15 2009; corrected by John Keith, Aug 17 2022
n == 1, 2, 4, 8, 16, or 23 (mod 27). - Dean Hickerson, Mar 25 2003
Limit_{n->oo} a(n)/n = 9/2; A029898(n) = a(n+1) - a(n) = A010888(a(n)). - Reinhard Zumkeller, Feb 27 2006
a(6n+1)=27n+1, a(6n+2)=27n+2, a(6n+3)=27n+4, a(6n+4)=27n+8, a(6n+5)=27n+16, a(6n+6)=27n+23. - Franklin T. Adams-Watters, Mar 13 2006
G.f.: (1+4*x^4+3*x^3+x^2)/((x+1)*(x^2-x+1)*(x-1)^2). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
a(n+1) = A064806(a(n)). - Reinhard Zumkeller, Apr 13 2013

A064807 Numbers which are divisible by their digital root (A010888).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 19, 20, 21, 24, 27, 28, 30, 36, 37, 38, 39, 40, 42, 45, 46, 48, 50, 54, 55, 56, 57, 60, 63, 64, 66, 70, 72, 73, 74, 75, 76, 78, 80, 81, 82, 84, 90, 91, 92, 93, 95, 96, 99, 100, 102, 108, 109, 110, 111, 112, 114, 117, 118

Offset: 1

Reinhard Zumkeller, Oct 21 2001

All numbers 9m, m > 0, belong to this sequence.
All numbers 6m, m > 0, belong to this sequence. - Christian Schulz, Oct 30 2013
All numbers 280m, m > 0, belong to this sequence. Only 6, 9, 280, and their multiples have this property. - Charles R Greathouse IV, Dec 26 2013
Conjecture: All k-multiply perfect numbers belong to this sequence. - Ivan N. Ianakiev, May 10 2016
The asymptotic density of this sequence is 1321/2520 = 0.524206... (see A074947 and A074949 for the values in other base representations). - Amiram Eldar, Nov 24 2022
The even perfect numbers are a subsequence. It is an open question whether the odd perfect numbers are a subsequence; this would involve ruling out 148 residue classes mod 2520 as OPNs. - Charles R Greathouse IV, Jan 03 2023

			48: 4 + 8 = 12 -> 1 + 2 = 3. 48 = 3 * 16 therefore 48 = a(28).

Ray Chandler, Table of n, a(n) for n = 1..4196 (first 1000 terms from Harry J. Smith)
Index entries for linear recurrences with constant coefficients, order 1322.

Cf. A010888, A074947, A074949.

a064807 n = a064807_list !! (n-1)
a064807_list = filter (\x -> x `mod` a010888 x == 0) [1..]
-- Reinhard Zumkeller, Jan 03 2014

A064807 := proc(n) option remember: local k: if(n=1)then return 1:fi: for k from procname(n-1)+1 do if(k mod (((k-1) mod 9) + 1) = 0)then return k: fi: od: end: seq(A064807(n),n=1..100); # Nathaniel Johnston, May 05 2011

Select[Range[125], Divisible[#, Mod[# - 1, 9] + 1] &] (* Alonso del Arte, Nov 01 2013 *)

is(n)=n%((n-1)%9+1)==0 \\ Charles R Greathouse IV, Dec 26 2013

a(n) = a(n-1321) + 2520. - Charles R Greathouse IV, Dec 26 2013
2520n/1321 - 10 < a(n) <= 2520n/1321. (In fact, if you exclude n = 10 mod 1321, you can replace 10 with 9.) - Charles R Greathouse IV, Jan 03 2023
a(n) = a(n-1) + a(n-1321) - a(n-1322). - Charles R Greathouse IV, Apr 20 2023

Offset changed from 0 to 1 by Harry J. Smith, Sep 26 2009

A147706 Number of partitions of n into parts having distinct digital roots (A010888).

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 11, 15, 16, 20, 25, 28, 32, 39, 46, 50, 62, 66, 78, 93, 101, 112, 132, 150, 161, 192, 202, 232, 268, 287, 312, 361, 400, 425, 497, 516, 582, 658, 698, 748, 858, 932, 982, 1135, 1164, 1296, 1443, 1519, 1610, 1845, 1968, 2059, 2360, 2395

Offset: 1

Reinhard Zumkeller, Nov 11 2008

a(n) <= A000009(n).

Likely a duplicate of A114098. [From R. J. Mathar, Dec 13 2008]

			A000009(16) = 32, in which the following 4 partitions
contain parts with common digital roots:
12 + 3 + 1, 11 + 3 + 2, 10 + 5 + 1 and 10 + 3 + 2 + 1,
therefore a(16) = 32 - 4 = 28.

A114102, A116371, A116372, A116373, A116374, A116375, A116376, A116377, A116378, A114099.

A139413 Triangle read by rows: row n gives the numbers A010888(n*k) for k = 1..n.

Original entry on oeis.org

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

Offset: 1

Philippe Lallouet (philip.lallouet(AT)orange.fr), Apr 19 2008

			Triangle begins:
1
2 4
3 6 9
4 8 3 7
5 1 6 2 7
6 3 9 6 3 9
7 5 3 1 8 6 4
8 7 6 5 4 3 2 1
9 9 9 9 9 9 9 9 9
...

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A007953 (column 1), A180592 - A180599 (columns 2 - 9), A010888, A038194.

A010888:=proc(n) return ((n-1) mod 9)+1; end:
for n from 1 to 9 do for k from 1 to n do printf("%d ",A010888(n*k)); od: printf("\n"): od: # Nathaniel Johnston, Apr 21 2011

Table[FixedPoint[Total@ IntegerDigits[#] &, n k], {n, 13}, {k, n}] // Flatten (* Michael De Vlieger, Oct 31 2021 *)

More terms and several terms corrected by Nathaniel Johnston, Apr 21 2011

A130856 The digital root (A010888) of the Catalan numbers A000108.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Aug 21 2007

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

Cf. A000108, A010888, A130851.

A130856 := proc(n) option remember: if(n=0)then return 1:fi: return ((add(procname(k)*procname(n-1-k),k=0..n-1)-1) mod 9) + 1 end: seq(A130856(n),n=0..100); # Nathaniel Johnston, May 05 2011

droot[n_]:=NestWhile[Total[IntegerDigits[#]]&,n,#>9&]; droot/@CatalanNumber[ Range[ 0,100]] (* Harvey P. Dale, Apr 09 2022 *)

a(n) = (binomial(2*n,n)/(n+1)-1)%9 + 1; \\ Michel Marcus, Nov 28 2022

Name corrected by Nathaniel Johnston, May 05 2011

A381491 a(n) = A010888(A381487(n)).

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Feb 25 2025

			a(12) = 2 because the digital root of A381487(12) = 128 is 2.

Michel Marcus, Table of n, a(n) for n = 1..3347

Cf. A010888, A381487, A381492.

A010888[n_]:=n - 9*Floor[(n-1)/9]; kmax=5*10^6; a={0,1}; For[k=2, k<=kmax, k++, If[A010888[k]!=1, If[IntegerQ[Log[A010888[k],k]], AppendTo[a,A010888[k]]]]]; a

dr(n) = if(n, (n-1)%9+1); \\ A010888
lista(nn) = my(list = List()); listput(list, 0); listput(list, 1); for (n=2, 9, for (k=1, logint(nn, n), if (dr(n^k) == n, listput(list, n^k)););); apply(x->dr(x), vecsort(Vec(list))); \\ Michel Marcus, Feb 27 2025

More terms from Michel Marcus, Feb 27 2025

A082944 a(n) = concatenate(n, A010888(2*n), reverse(n)), where A010888 = digital root.

Original entry on oeis.org

121, 242, 363, 484, 515, 636, 757, 878, 999, 10201, 11411, 12621, 13831, 14141, 15351, 16561, 17771, 18981, 19291, 20402, 21612, 22822, 23132, 24342, 25552, 26762, 27972, 28282, 29492, 30603

Offset: 1

Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 06 2003

The central digit A010888(2*n) cyclically repeats the nine digits (2, 4, 6, 8, 1, 3, 5, 7, 9) in this order.

The sum of the digits of n plus the digits of the reverse of n is always simply 2 times the sum of the digits of n. - Harvey P. Dale, Aug 12 2019

			For a(17): 1+7+7+1 = 16. 1+6 = 7. Thus the center digit is 7 and a(17) = 17771.
For a(23): 2+3+3+2 = 10. 1+0 = 1. Thus the center digit is 1 and a(23) = 23132.

Amarnath Murthy

Cf. A010888 (digital root), A004086 (read n backwards: trailing zeros are lost - but not in this sequence here).

nxt[n_]:=Module[{idn=IntegerDigits[n],b,c},b=NestWhile[Total[IntegerDigits[ # ]]&, 2 Total[idn],#>9&];c=Reverse[idn];FromDigits[Join[idn,{b},c]]]; Table[ nxt[n],{n,20}] (* Harvey P. Dale, Aug 12 2019 *)

A082944(n)=fromdigits(concat([digits(n), (2*n-1)%9+1, Vecrev(digits(n))])) \\ Cf. A010888. - R. J. Cano, Jan 05 2022

Definition, comment and example corrected by M. F. Hasler, Jan 05 2022

A110365 a(1)=2, a(n+1) = a(n)*A010888(a(n)).

Original entry on oeis.org

2, 4, 16, 112, 448, 3136, 12544, 87808, 351232, 2458624, 9834496, 68841472, 275365888, 1927561216, 7710244864, 53971714048, 215886856192, 1511207993344, 6044831973376, 42313823813632, 169255295254528, 1184787066781696, 4739148267126784, 33174037869887488

Offset: 1

Amarnath Murthy, Jul 24 2005

From a(2) onwards, the digital root follows the pattern alternately 4,7,4,7,4,7,...

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,28).

Cf. A010888, A047892.

k = 2; Do[Print[k]; k *= Mod[Plus @@ IntegerDigits[k], 9], {n, 1, 30}] (* Ryan Propper, Oct 13 2005 *)
LinearRecurrence[{0,28},{2,4,16},30] (* Harvey P. Dale, Mar 17 2019 *)

Vec(2*x*(1+2*x-20*x^2)/(1-28*x^2) + O(x^50)) \\ Colin Barker, May 05 2016

a(1) = 2, a(2) = 4, a(3) = 16. a(2*n) = 4*a(2*n-1), a(2*n+1) = 7*a(2*n) for n > 1.
From Colin Barker, May 05 2016: (Start)
a(n) = 2^(-1+n)*(7^(1/2*(-3+n))*(2-2*(-1)^n + sqrt(7) + (-1)^n*sqrt(7))) for n > 1.
a(n) = 2^n*7^(n/2-1) for n > 1 and even.
a(n) = 2^(n+1)*7^((n-3)/2) for n > 1 and odd.
a(n) = 28*a(n-2) for n > 3.
G.f.: 2*x*(1+2*x-20*x^2) / (1-28*x^2).
(End)
E.g.f.: (-7 + 70*x + 7*cosh(2*Sqrt(7)*x) + 2*sqrt(7)*sinh(2*sqrt(7)*x))/49. - Ilya Gutkovskiy, May 05 2016

More terms from Ryan Propper, Oct 13 2005

Name clarified by Robert Israel, May 05 2016

A361203 a(n) = n*A010888(n).

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 10, 22, 36, 52, 70, 90, 112, 136, 162, 19, 40, 63, 88, 115, 144, 175, 208, 243, 28, 58, 90, 124, 160, 198, 238, 280, 324, 37, 76, 117, 160, 205, 252, 301, 352, 405, 46, 94, 144, 196, 250, 306, 364, 424, 486, 55, 112, 171, 232

Offset: 0

Stefano Spezia, Apr 20 2023

Every run of increasing terms ends with a positive multiple of 81, and except for the first run, it starts with a term of A017173 which is a fixed point for this sequence (see 4th formula).

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,-1).

Cf. A010888, A017173, A057147.

a[n_]:=n(n - 9*Floor[(n-1)/9]); Join[{0},Array[a,58]]

def A361203(n): return n*(1 + (n - 1) % 9) # Chai Wah Wu, Apr 23 2023

G.f.: x*(1 + 4*x + 9*x^2 + 16*x^3 + 25*x^4 + 36*x^5 + 49*x^6 + 64*x^7 + 81*x^8 + 8*x^9 + 14*x^10 + 18*x^11 + 20*x^12 + 20*x^13 + 18*x^14 + 14*x^15 + 8*x^16)/((1 - x)^2*(1 + x + x^2)^2*(1 + x^3 + x^6)^2).
a(n) = 2*a(n-9) - a(n-18) for n > 17.
a(n) = n*(n - 9*floor((n-1)/9)) for n > 0.
a(A017173(n)) = A017173(n).

A073637 Digital root (cf. A010888) of prime(n)^3.

Original entry on oeis.org

8, 9, 8, 1, 8, 1, 8, 1, 8, 8, 1, 1, 8, 1, 8, 8, 8, 1, 1, 8, 1, 1, 8, 8, 1, 8, 1, 8, 1, 8, 1, 8, 8, 1, 8, 1, 1, 1, 8, 8, 8, 1, 8, 1, 8, 1, 1, 1, 8, 1, 8, 8, 1, 8, 8, 8, 8, 1, 1, 8, 1, 8, 1, 8, 1, 8, 1, 1, 8, 1, 8, 8, 1, 1, 1, 8, 8, 1, 8, 1, 8, 1, 8, 1, 1, 8, 8, 1, 8, 1, 8, 8, 1, 8, 1, 8, 8, 8, 1, 1

Offset: 1

Zak Seidov, Sep 01 2002, Oct 23 2009

Apart from a(2)=9 all other terms are either 1 or 8.

			a(3)=8 because p(3)=5 and 5^3=125 -> sum-of-digits = 8. a(4)=1 because p(3)=7 and 7^3=343 -> sum-of-digits = 10 -> sum-of-digits = 1.

Cf. A010888, A021596, A056992, A010888, A038194, A166923.

n=3; su[x_] := Sum[IntegerDigits[x][[i]], {i, Length[IntegerDigits[x]]}]; Table[su[su[su[su[Prime[x]^n]]]], {x, 100}]
Table[If[(m9=Mod[Prime[n]^3,9])==0,9,m9],{n,200}]

Edited by N. J. A. Sloane, Oct 29 2009

cp's OEIS Frontend

A007612 a(n+1) = a(n) + digital root (A010888) of a(n).

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A064807 Numbers which are divisible by their digital root (A010888).

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A147706 Number of partitions of n into parts having distinct digital roots (A010888).

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A139413 Triangle read by rows: row n gives the numbers A010888(n*k) for k = 1..n.

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A130856 The digital root (A010888) of the Catalan numbers A000108.

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A381491 a(n) = A010888(A381487(n)).

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A082944 a(n) = concatenate(n, A010888(2*n), reverse(n)), where A010888 = digital root.

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