A037123 -id:A037123 - cp's OEIS frontend

A075543 a(n) = a(n-1) + digit sum(n + 2) with a(0) = 2.

2, 5, 9, 14, 20, 27, 35, 44, 45, 47, 50, 54, 59, 65, 72, 80, 89, 99, 101, 104, 108, 113, 119, 126, 134, 143, 153, 164, 167, 171, 176, 182, 189, 197, 206, 216, 227, 239, 243, 248, 254, 261, 269, 278, 288, 299, 311, 324, 329, 335, 342, 350, 359, 369, 380, 392

Offset: 0

Jon Perry, Oct 11 2002

			a(1) = a(0) + digit sum(1 + 2) = 2 + 3 = 5.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A007953, A037123.

Block[{k = 2}, NestList[# + DigitSum[++k] &, 2, 100]] (* Paolo Xausa, May 24 2024 *)

c=0; for (n=2,100,c=c+sumdigits(n); print1(c,","))

Definition corrected by Georg Fischer, May 22 2024

A138376 a(n+1) = abs[ a(n) + (-1)^(n+1) * Sum_of_digits_of(n+1)], with a(0)=0.

Original entry on oeis.org

0, 1, 3, 0, 4, 1, 7, 0, 8, 1, 2, 0, 3, 1, 6, 0, 7, 1, 10, 0, 2, 1, 5, 0, 6, 1, 9, 0, 10, 1, 4, 0, 5, 1, 8, 0, 9, 1, 12, 0, 4, 1, 7, 0, 8, 1, 11, 0, 12, 1, 6, 0, 7, 1, 10, 0, 11, 1, 14, 0, 6, 1, 9, 0, 10, 1, 13, 0, 14, 1, 8, 0, 9, 1, 12, 0, 13, 1, 16, 0, 8, 1, 11, 0, 12, 1, 15, 0, 16, 1, 10, 0, 11, 1, 14

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, May 08 2008

a(4*k)=0, with k>=1

a(4*k-2)=1, with k>=1

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A037123.

P:=proc(n) local a,i,k,w; a:=0; print(a); for i from 1 by 1 to n do w:=0; k:=i; while k>0 do w:=w+k-(trunc(k/10)*10); k:=trunc(k/10); od; a:=abs(a+(-1)^i*w); print(a); od; end: P(100);

nxt[{n_,a_}]:={n+1,Abs[a+(-1)^(n+1) Total[IntegerDigits[n+1]]]}; NestList[nxt,{0,0},100][[All,2]] (* Harvey P. Dale, Jan 04 2019 *)

Definition corrected by N. J. A. Sloane, Jan 04 2019

A271627 Numbers n such that the sum of the digits of the numbers from 1 to n is a prime.

Original entry on oeis.org

2, 16, 22, 25, 61, 118, 133, 193, 217, 226, 232, 262, 265, 286, 310, 337, 358, 397, 433, 445, 466, 496, 508, 538, 553, 565, 580, 613, 652, 697, 718, 733, 745, 757, 781, 790, 856, 901, 958, 985, 988, 1021, 1093, 1186, 1201, 1210, 1258, 1273, 1285, 1297, 1312, 1321

Offset: 1

Paolo P. Lava, Apr 11 2016

			1 + 2 = 3 that is a prime; 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 1 + 0 + 1 + 1 + 1 + 2 + 1 + 3 + 1 + 4 + 1 + 5 + 1 + 6 = 73 that is a prime.

Paolo P. Lava, Table of n, a(n) for n = 1..10000

Cf. A000040, A037123.

with(numtheory): P:=proc(q) local a,b,c,k,n; a:=0;
for n from 1 to q do b:=0; c:=n; for k from 1 to ilog10(n)+1 do b:=b+(c mod 10); c:=trunc(c/10); od; a:=a+b;
if isprime(a) then print(n); fi; od; end: P(10^4);

Select[Range@ 1350, PrimeQ@ Total@ Map[Total@ IntegerDigits@ # &, Range@ #] &] (* Michael De Vlieger, Apr 11 2016 *)

isok(n) = isprime(sum(k=1, n, sumdigits(k))); \\ Michel Marcus, Apr 11 2016

A271628 Primes that are the sum of the digits of the numbers from 1 to n, for some n.

Original entry on oeis.org

3, 73, 109, 127, 433, 1009, 1117, 1801, 2017, 2089, 2143, 2467, 2503, 2791, 3079, 3331, 3583, 4159, 4519, 4663, 4951, 5437, 5581, 5923, 6121, 6301, 6553, 7039, 7561, 8353, 8623, 8821, 9001, 9199, 9631, 9811, 10837, 11719, 12637, 13177, 13249, 13627, 14401, 15391

Offset: 1

Paolo P. Lava, Apr 11 2016

			The sum of the digits of 1 and 2 is a prime: 1 + 2 = 3.
The sum of the digits of the number from 1 to 16 is a prime: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 1 + 0 + 1 + 1 + 1 + 2 + 1 + 3 + 1 + 4 + 1 + 5 + 1 + 6 = 73.

Paolo P. Lava, Table of n, a(n) for n = 1..10000

Cf. A000040, A037123, A271626.

with(numtheory): P:=proc(q) local a,b,c,k,n; a:=0;
for n from 0 to q do b:=0; c:=n; for k from 1 to ilog10(n)+1 do b:=b+(c mod 10); c:=trunc(c/10); od; a:=a+b;
if isprime(a) then print(a); fi; od; end: P(10^6);

Select[Accumulate@ Map[Total@ IntegerDigits@ # &, Range[0, 1200]], PrimeQ] (* Michael De Vlieger, Apr 11 2016 *)

lista(nn) = for (n=1, nn, if (isprime(p=sum(k=1, n, sumdigits(k))), print1(p, ", "))); \\ Michel Marcus, Apr 11 2016

A316312 Numbers k such that the sum of the digits of the numbers 1, 2, 3, ... up to (k - 1) is divisible by k.

Original entry on oeis.org

1, 3, 5, 7, 9, 12, 15, 20, 27, 40, 45, 60, 63, 80, 81, 100, 180, 181, 300, 360, 363, 500, 540, 545, 700, 720, 727, 900, 909, 912, 915, 1137, 1140, 1200, 1500, 1560, 1563, 2000, 2700, 2720, 2727, 4000, 4500, 4540, 4545, 6000, 6300, 6360, 6363, 8000, 8100, 8180

Offset: 1

Debapriyay Mukhopadhyay, Jun 29 2018

Numbers k such that A007953(A007908(k - 1)) is divisible by k. - Felix Fröhlich, Jun 29 2018
From Robert Israel, Jun 29 2018: (Start)
Numbers k such that A037123(k - 1) is divisible by k.
If m is even, then 10^m, 3 * 10^m, 5 * 10^m, 7 * 10^m and 9 * 10^m are included.
If m is odd, then 2 * 10^m, 4 * 10^m, 6 * 10^m, and 8 * 10^m are included. (End)
Is it true that if k is a term then 100 * k is a term?

			For n = 7, sum of the digits of the numbers 1 to 6 is 21, which is divisible by 7.
For n = 12, sum of the digits of the numbers 1 to 11 is 48, which is divisible by 12.
For n = 15, sum of the digits of the numbers 1 to 14 is 60, which is divisible by 15.
16 is not in the sequence because the sum of the digits of the numbers 1 to 15 is 66, which is not divisible by 16.

Henry Bottomley, Table of n, a(n) for n = 1..118

Cf. A007953, A007908, A037123, A110740, A114136.

t:= 0: Res:= NULL:
for n from 1 to 10000 do
  t:= t + convert(convert(n-1,base,10),`+`);
  if (t/n)::integer then Res:= Res, n fi
od:
Res; # Robert Israel, Jun 29 2018

s = 0; Reap[Do[If[Mod[s, n] == 0, Sow[n]]; s += Plus @@ IntegerDigits@n, {n, 10000}]][[2, 1]] (* Giovanni Resta, Jun 29 2018 *)

sumsod(n) = sum(i=1, n, sumdigits(i))
is(n) = sumsod(n-1)%n==0 \\ Felix Fröhlich, Jun 29 2018

upto(n) = my(s=0,res=List()); for(i=0, n, s += vecsum(digits(i)); if(s%(i+1)==0, listput(res, i+1))); res \\ David A. Corneth, Jun 29 2018

More terms from Felix Fröhlich, Jun 29 2018

A071122 a(n) = a(n-1) + sum of decimal digits of 2^n.

Original entry on oeis.org

2, 6, 14, 21, 26, 36, 47, 60, 68, 75, 89, 108, 128, 150, 176, 201, 215, 234, 263, 294, 320, 345, 386, 423, 452, 492, 527, 570, 611, 648, 695, 753, 815, 876, 935, 999, 1055, 1122, 1193, 1254, 1304, 1350, 1406, 1464, 1526, 1596, 1664, 1737, 1802, 1878, 1958

Offset: 1

Labos Elemer, May 27 2002

Cf. A037123, A000788, A051351, A071317, A071121.

s=0; Do[s=s+Apply[Plus, IntegerDigits[2^n]]; Print[s], {n, 1, 128}]

A071123 a(n) = a(n-1) + sum of decimal digits of n!.

Original entry on oeis.org

1, 3, 9, 15, 18, 27, 36, 45, 72, 99, 135, 162, 189, 234, 279, 342, 405, 459, 504, 558, 621, 693, 792, 873, 945, 1026, 1134, 1224, 1350, 1467, 1602, 1710, 1854, 1998, 2142, 2313, 2466, 2574, 2763, 2952, 3096, 3285, 3465, 3681, 3888, 4104, 4329, 4563, 4788

Offset: 1

Labos Elemer, May 27 2002

Cf. A037123, A000788, A051351.

s=0; Do[s=s+Apply[Plus, IntegerDigits[n! ]]; Print[s], {n, 1, 128}]

A099358 a(n) = sum of digits of k^4 as k runs from 1 to n.

Original entry on oeis.org

1, 8, 17, 30, 43, 61, 68, 87, 105, 106, 122, 140, 162, 184, 202, 227, 246, 273, 283, 290, 317, 339, 370, 397, 422, 459, 477, 505, 530, 539, 561, 592, 619, 644, 663, 699, 727, 752, 770, 783, 814, 841, 866, 903, 921, 958, 1001, 1028, 1059, 1072, 1099, 1124, 1161

Offset: 1

Yalcin Aktar, Nov 16 2004

Partial sums of A055565.

			a(3) = sum_digits(1^4) + sum_digits(2^4) + sum_digits(3^4) = 1 + 7 + 9 = 17.

Cf. k^1 in A037123, k^2 in A071317 & k^3 in A071121.

f[n_] := Block[{s = 0, k = 1}, While[k <= n, s = s + Plus @@ IntegerDigits[k^4]; k++ ]; s]; Table[ f[n], {n, 50}] (* Robert G. Wilson v, Nov 18 2004 *)
Accumulate[Table[Total[IntegerDigits[n^4]],{n,60}]] (* Harvey P. Dale, Jun 08 2021 *)

a(n) = a(n-1) + sum of decimal digits of n^4.
a(n) = sum(k=1, n, sum(m=0, floor(log(k^4)), floor(10((k^4)/(10^(((floor(log(k^4))+1))-m)) - floor((k^4)/(10^(((floor(log(k^4))+1))-m))))))).
General formula: a(n)_p = sum(k=1, n, sum(m=0, floor(log(k^p)), floor(10((k^p)/(10^(((floor(log(k^p))+1))-m)) - floor ((k^p)/(10^(((floor(log(k^p))+1))-m))))))). Here a(n)_p is a sum of digits of k^p from k=1 to n.

Edited and extended by Robert G. Wilson v, Nov 18 2004

Existing example replaced with a simpler one by Jon E. Schoenfield, Oct 20 2013

A166352 Table read by rows where row n contains the sorted digits of the integers 0 through n.

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Oct 12 2009

A037123(n) is the sum of the terms in row n.

			Row 10, the sorted digits of 0 through 10, is 0, 0, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Cf. A037123.

Flatten[Table[Sort[Flatten[IntegerDigits/@Range[0,n]]],{n,0,15}]] (* Harvey P. Dale, Nov 30 2015 *)

A272360 Numbers n such that the sum of the digits of the numbers from 1 to n divides the sum of the numbers from 1 to n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 80, 119, 180, 398, 399, 998, 999, 1055, 1809, 2715, 4063, 4529, 11374, 18180, 19199, 27269, 54519, 110549, 113695, 168399, 294999, 454511, 624591, 636349, 639999, 1090719, 1818180, 2350079, 9639999, 17576999, 17914111, 54545436, 61484399, 81818169, 91728090, 466359999, 909091519, 909113679, 909156319, 911363679, 915636319, 999999998, 999999999

Offset: 1

Paolo P. Lava, Apr 27 2016

The ratios for the numbers in the list are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 7, 10, 19, 19, 37, 37, 40, 67, 97, 136, 151, 325, 505, 526, 737, 1363, 2497, 2584, 3789, 6313, 9469, 12507, 12727, 12784, 20451, 33670, 43214, 154481, 280413, 284734, 814111, 905187

The sequence is infinite, since it contains all the numbers of the form 10^(3^k)-1 and 10^(3^k)-2. For these numbers the ratio is (10^(3^k)-1)/(9*3^k), which is an integer because in general x^(3^k)-1 can be factored into (x-1)(x^2+x+1)(x^6+x^3+1)(x^18+x^9+1)...(x^(2*3^(k-1))+x^(3^(k-1))+1) and since here x=10, x-1=9 and each of the following k factors is divisible by 3 because its sum of digits is 3, thus 10^(3^k)-1 is divisible by 9*3^k. - Giovanni Resta, Apr 27 2016

			The sum of the digits of the numbers from 1 to 80 is 648; the sum of the numbers from 1 to 80 is 80*81/2 = 3240 and 3240 / 648 = 5

Cf. A000217, A037123.

P:=proc(q) local b,k; global a,n; a:=0; for n from 1 to q do
b:=n; for k from 1 to ilog10(n)+1 do a:=a+(b mod 10); b:=trunc(b/10); od;
if type(n*(n+1)/(2*a),integer) then print(n); fi; od; end: P(10^9);

With[{nn=10^9},Position[Thread[{Accumulate[Range[nn]], Accumulate[ Table[ Total[ IntegerDigits[n]],{n,nn}]]}],?(Divisible[#[[1]],#[[2]]]&), 1,Heads->False]]//Flatten (* _Harvey P. Dale, Sep 30 2017 *)

list(lim)=my(v=List(),s,t); for(n=1,lim,s+=n; t+=sumdigits(n); if(s%t==0, listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Apr 29 2016

Solutions for A037123(n) | A000217(n).

a(43)-a(52) from Charles R Greathouse IV, Apr 29 2016

cp's OEIS Frontend

A075543 a(n) = a(n-1) + digit sum(n + 2) with a(0) = 2.

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A138376 a(n+1) = abs[ a(n) + (-1)^(n+1) * Sum_of_digits_of(n+1)], with a(0)=0.

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A271627 Numbers n such that the sum of the digits of the numbers from 1 to n is a prime.

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A271628 Primes that are the sum of the digits of the numbers from 1 to n, for some n.

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A316312 Numbers k such that the sum of the digits of the numbers 1, 2, 3, ... up to (k - 1) is divisible by k.

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A071122 a(n) = a(n-1) + sum of decimal digits of 2^n.

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A071123 a(n) = a(n-1) + sum of decimal digits of n!.

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A099358 a(n) = sum of digits of k^4 as k runs from 1 to n.

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