A131383 -id:A131383 - cp's OEIS frontend

A135100 Numbers which divide their digital sumorial (see A131383).

1, 3, 4, 15, 26, 2573, 17226, 19786, 22083, 58133, 67693, 223657, 376460, 464713, 497068, 2621204, 4553376, 6000136, 7671158, 13975944, 14074903, 52731198, 82594577

Offset: 1

Hieronymus Fischer, Dec 24 2007

			a(5)=26, since 26 divides its digital sumorial, which is A131383(26)=182.

Cf. A007953, A054899, A131383, A131451, A133620, A133900, A134599.

a=1;for(n=2,10^6,if(a%(n-1)==0,print1(n-1","));x=divisors(n);L=numdiv(n);a+=n; for(i=2,L-1,d=x[i];k=n;while(k%d==0,a-=d-1;k\=d))) \\ Robert Gerbicz, May 09 2008

a(12)-a(15) from Robert Gerbicz, May 09 2008

a(16)-a(23) from Hieronymus Fischer, Jul 31 2008

A037123 a(n) = a(n-1) + sum of digits of n.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 46, 48, 51, 55, 60, 66, 73, 81, 90, 100, 102, 105, 109, 114, 120, 127, 135, 144, 154, 165, 168, 172, 177, 183, 190, 198, 207, 217, 228, 240, 244, 249, 255, 262, 270, 279, 289, 300, 312, 325, 330, 336, 343, 351, 360, 370, 381

Offset: 0

Vasiliy Danilov (danilovv(AT)usa.net), Jun 15 1998

Sum of digits of A007908(n). - Franz Vrabec, Oct 22 2007
Also digital sum of A138793(n) for n > 0. - Bruno Berselli, May 27 2011
Sum of the digital sum of i for i from 0 to n. - N. J. A. Sloane, Nov 13 2013

N. Agronomof, Sobre una función numérica, Revista Mat. Hispano-Americana 1 (1926), 267-269.
Maurice d'Ocagne, Sur certaines sommations arithmétiques, J. Sciencias Mathematicas e Astronomicas 7 (1886), 117-128.

David A Corneth, Table of n, a(n) for n = 0..10008
P.-H. Cheo and S.-C. Yien, A problem on the k-adic representation of positive integers, Acta Math. Sinica 5, 433-438 (1955).
J. Coquet, Power sums of digital sums, J. Number Theory 22 (1986), no. 2, 161-176.
H. Delange, Sur la fonction sommatoire de la fonction "somme des chiffres", Enseignement Math. (2) 21 (1975), 31-47.
P. J. Grabner, P. Kirschenhofer, H. Prodinger and R. F. Tichy, On the moments of the sum-of-digits function, Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), Kluwer Acad. Publ., Dordrecht, 1993, 263-271.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
J.-L. Mauclaire and Leo Murata, On q-additive functions, I. Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 6, 274-276.
J.-L. Mauclaire and Leo Murata, On q-additive functions, II. Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 9, 441-444.
H. Riede, Asymptotic estimation of a sum of digits, Fibonacci Q. 36, No. 1, 72-75 (1998).
J. R. Trollope, An explicit expression for binary digital sums, Math. Mag. 41 1968 21-25.

Cf. A004207, A016052, A131383, A131384, A131451.
Cf. also A074784, A231688, A231689.
Partial sums of A007953.

[ n eq 0 select 0 else &+[&+Intseq(k): k in [0..n]]: n in [0..56] ];  // Bruno Berselli, May 27 2011

# From N. J. A. Sloane, Nov 13 2013:
digsum:=proc(n,B) local a; a := convert(n, base, B):
add(a[i], i=1..nops(a)): end;
f:=proc(n,k,B) global digsum; local i;
add( digsum(i,B)^k,i=0..n); end;
lprint([seq(digsum(n,10),n=0..100)]); # A007953
lprint([seq(f(n,1,10),n=0..100)]); #A037123
lprint([seq(f(n,2,10),n=0..100)]); #A074784
lprint([seq(f(n,3,10),n=0..100)]); #A231688
lprint([seq(f(n,4,10),n=0..100)]); #A231689

Table[Plus@@Flatten[IntegerDigits[Range[n]]], {n, 0, 200}] (* Enrique Pérez Herrero, Oct 12 2015 *)
a[0] = 0; a[n_] := a[n - 1] + Plus @@ IntegerDigits@ n; Array[a, 70, 0] (* Robert G. Wilson v, Jul 06 2018 *)

a(n)=n*(n+1)/2-9*sum(k=1,n,sum(i=1,ceil(log(k)/log(10)),floor(k/10^i)))

a(n)={n++;my(t,i,s);c=n;while(c!=0,i++;c\=10);for(j=1,i,d=(n\10^(i-j))%10;t+=(10^(i-j)*(s*d+binomial(d,2)+d*9*(i-j)/2));s+=d);t} \\ David A. Corneth, Aug 16 2013

for $i (0..100){ @j = split "", $i; for (@j){ $sum += $; } print "$sum,"; } __END_ # gamo(AT)telecable.es

a(n) = Sum_{k=0..n} s(k) = Sum_{k=0..n} A007953(k), where s(k) denote the sum of the digits of k in decimal representation. Asymptotic expression: a(n-1) = Sum_{k=0..n-1} s(k) = 4.5*n*log_10(n) + O(n). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002
a(n) = n*(n+1)/2 - 9*Sum_{k=1..n} Sum_{i=1..ceiling(log_10(k))} floor(k/10^i). - Benoit Cloitre, Aug 28 2003
From Hieronymus Fischer, Jul 11 2007: (Start)
G.f.: Sum_{k>=1} ((x^k - x^(k+10^k) - 9x^(10^k))/(1-x^(10^k)))/(1-x)^2.
a(n) = (1/2)*((n+1)*(n - 18*Sum_{k>=1} floor(n/10^k)) + 9*Sum_{k>=1} (1 + floor(n/10^k))*floor(n/10^k)*10^k).
a(n) = (1/2)*((n+1)*(2*A007953(n)-n) + 9*Sum_{k>=1} (1+floor(n/10^k))*floor(n/10^k)*10^k). (End)
a(n) = A007953(A053064(n)). - Reinhard Zumkeller, Oct 10 2008
From Wojciech Raszka, Jun 14 2019: (Start)
a(10^k - 1) = 10*a(10^(k - 1) - 1) + 45*10^(k - 1) for k > 0.
a(n) = a(n mod m) + MSD*a(m - 1) + (MSD*(MSD - 1)/2)*m + MSD*((n mod m) + 1), where m = 10^(A055642(n) - 1), MSD = A000030(n). (End)

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002

A138530 Triangle read by rows: T(n,k) = sum of digits of n in base k representation, 1<=k<=n.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 1, 2, 1, 5, 2, 3, 2, 1, 6, 2, 2, 3, 2, 1, 7, 3, 3, 4, 3, 2, 1, 8, 1, 4, 2, 4, 3, 2, 1, 9, 2, 1, 3, 5, 4, 3, 2, 1, 10, 2, 2, 4, 2, 5, 4, 3, 2, 1, 11, 3, 3, 5, 3, 6, 5, 4, 3, 2, 1, 12, 2, 2, 3, 4, 2, 6, 5, 4, 3, 2, 1, 13, 3, 3, 4, 5, 3, 7, 6, 5, 4, 3, 2, 1, 14, 3, 4, 5, 6, 4, 2, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Reinhard Zumkeller, Mar 26 2008

A131383(n) = sum of n-th row;
A000027(n) = T(n,1);
A000120(n) = T(n,2) for n>1;
A053735(n) = T(n,3) for n>2;
A053737(n) = T(n,4) for n>3;
A053824(n) = T(n,5) for n>4;
A053827(n) = T(n,6) for n>5;
A053828(n) = T(n,7) for n>6;
A053829(n) = T(n,8) for n>7;
A053830(n) = T(n,9) for n>8;
A007953(n) = T(n,10) for n>9;
A053831(n) = T(n,11) for n>10;
A053832(n) = T(n,12) for n>11;
A053833(n) = T(n,13) for n>12;
A053834(n) = T(n,14) for n>13;
A053835(n) = T(n,15) for n>14;
A053836(n) = T(n,16) for n>15;
A007395(n) = T(n,n-1) for n>1;
A000012(n) = T(n,n).

			Start of the triangle for n in base k representation:
......................1
....................11....10
......... ........111....11...10
................1111...100...11..10
..............11111...101...12..11..10
............111111...110...20..12..11..10
..........1111111...111...21..13..12..11..10
........11111111..1000...22..20..13..12..11..10
......111111111..1001..100..21..14..13..12..11..10
....1111111111..1010..101..22..20..14..13..12..11..10
..11111111111..1011..102..23..21..15..14..13..12..11..10
111111111111..1100..110..30..22..20..15..14..13..12..11..10,
and the triangle of sums of digits starts:
......................1
.....................2...1
......... ..........3...2...1
...................4...1...2...1
..................5...2...3...2...1
.................6...2...2...3...2...1
................7...3...3...4...3...2...1
...............8...1...4...2...4...3...2...1
..............9...2...1...3...5...4...3...2...1
............10...2...2...4...2...5...4...3...2...1
...........11...3...3...5...3...6...5...4...3...2...1
..........12...2...2...3...4...2...6...5...4...3...2...1.

Alois P. Heinz, Rows n = 1..141, flattened
Eric Weisstein's World of Mathematics, Digit Sum

Cf. A007953. See A240236 for another version.

Cf. A002260.

a138530 n k = a138530_tabl !! (n-1) !! (k-1)
a138530_row n = a138530_tabl !! (n-1)
a138530_tabl = zipWith (map . flip q) [1..] a002260_tabl where
   q 1 n = n
   q b n = if n < b then n else q b n' + d where (n', d) = divMod n b
-- Reinhard Zumkeller, Apr 29 2015

T[n_, k_] := If[k == 1, n, Total[IntegerDigits[n, k]]];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 25 2021 *)

A131384 Product of the digital sums of n for all the bases 2 to n (a 'digital-sum factorial').

Original entry on oeis.org

1, 1, 2, 2, 12, 24, 216, 192, 720, 3840, 97200, 69120, 2721600, 14515200, 130636800, 30965760, 3810240000, 3762339840, 384072192000, 445906944000, 10184771520000, 123605404876800, 12674382336000000, 2542739757465600

Offset: 1

Hieronymus Fischer, Jul 05 2007

Definition changed to run from bases 2 to n instead of 1 to n, in order to reflect the sequence numbers, correctly. - suggested by Michel Marcus and changed by the author, Jul 19 2013

			5 = 11111_1 = 101_2 = 12_3 = 11_4 = 10_5. Thus a(5) = ds_1(5)*ds_2(5)*ds_3(5)*ds_4(5)*ds_5(5) = 5*2*3*2*1 = 60. - _Michel Marcus_, Jul 15 2013

Hieronymus Fischer, Table of n, a(n) for n = 1..200

Cf. A131383.

Table[Times @@ Map[Total@ IntegerDigits[n, #] &, Range[2, n]], {n, 24}] (* Michael De Vlieger, Jan 03 2017 *)

a(n) = prod(k=2, n, sumdigits(n,k)); \\ Michel Marcus, Jun 13 2022

a(n) = Product_{p=2..n} ds_p(n) where ds_p = digital sum base p.
a(n) = Product_{p=2..n} (n-(p-1)*Sum_{k>0} floor(n/p^k)).
[Both formulas corrected in accordance with the changed definition by the author; Jul 19 2013]

cp's OEIS Frontend

A135100 Numbers which divide their digital sumorial (see A131383).

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Extensions

A037123 a(n) = a(n-1) + sum of digits of n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Perl

Formula

Extensions

A138530 Triangle read by rows: T(n,k) = sum of digits of n in base k representation, 1<=k<=n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

A131384 Product of the digital sums of n for all the bases 2 to n (a 'digital-sum factorial').

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula