A014837 -id:A014837 - cp's OEIS frontend

A014842 Difference between A014837 and A014841.

2, 0, 5, 2, 9, 3, 7, 6, 15, 2, 21, 14, 19, 9, 25, 7, 31, 12, 27, 28, 42, 10, 38, 34, 35, 22, 55, 16, 59, 27, 49, 48, 54, 10, 71, 52, 61, 30, 82, 34, 88, 56, 66, 75, 103, 27, 88, 59, 84, 64, 112, 46, 97, 56, 105, 96, 130, 28, 138, 114, 108, 70, 118, 66, 146, 94, 121, 86

Offset: 3

Olivier Gérard

nonn

A043306 Sum of all digits in all base-b representations for n, for 2 <= b <= n.

Original entry on oeis.org

1, 3, 4, 8, 10, 16, 17, 21, 25, 35, 34, 46, 52, 60, 58, 74, 73, 91, 92, 104, 114, 136, 128, 144, 156, 168, 171, 199, 193, 223, 221, 241, 257, 281, 261, 297, 315, 339, 333, 373, 367, 409, 416, 430, 452, 498, 472, 508, 515, 547, 556, 608, 598, 638, 634, 670, 698, 756, 717, 777

Offset: 2

Clark Kimberling

			5 = 101_2 = 12_3 = 11_4 = 10_5. Thus a(5) = 2 + 3 + 2 + 1 = 8.

Alois P. Heinz, Table of n, a(n) for n = 2..1000
Robin Fissum, Digit sums and the number of prime factors of the factorial n!=1.2...n, Bachelor's project in BMAT, Norwegian University of Science and Technology, 2020.
Jan-Christoph Puchta and Jürgen Spilker, Altes und Neues zur Quersumme, Math. Semesterber., Vol. 49, No. 2 (2002), pp. 209-226; alternative link.
Vladimir Shevelev, Compact integers and factorials, Acta Arith., Vol. 126, No. 3 (2007), pp. 195-236 (see p. 205).

Cf. A014837, A043000, A068953, A191251, A191322, A191350, A309531.

Row sums of A240236.

Table[Sum[Total[First[RealDigits[n, i]]], {i, 2, n}], {n, 2, 80}] (* Carl Najafi, Aug 16 2011 *)

a(n) = sum(i=2, n, vecsum(digits(n, i))); \\ Michel Marcus, Jan 03 2017

a(n) = sum(b=2, n, sumdigits(n, b)); \\ Michel Marcus, Aug 18 2017

from sympy.ntheory.digits import digits
def a(n): return sum(sum(digits(n, b)[1:]) for b in range(2, n+1))
print([a(n) for n in range(2, 62)]) # Michael S. Branicky, Apr 04 2022

From Vladimir Shevelev, Jun 03 2011: (Start)
a(n) = (n-1)*n - Sum_{i=2..n} (i-1)*Sum_{r>=1} floor(n/i^r).
a(n) <= (n-1)^2*log(n+1)/log(n).
Problem: find a better upper estimate. (End)
From Amiram Eldar, Apr 16 2021: (Start)
a(n) = A014837(n) + 1.
a(n) ~ (1-Pi^2/12)*n^2 + O(n^(3/2)) (Fissum, 2020). (End)

A014836 Sum modulo n of all the digits of n in every base from 2 to n-1.

Original entry on oeis.org

2, 3, 2, 3, 1, 0, 2, 4, 1, 9, 6, 9, 14, 9, 5, 0, 14, 11, 19, 3, 20, 7, 18, 25, 5, 2, 24, 12, 5, 28, 9, 18, 0, 8, 0, 10, 26, 12, 3, 30, 21, 19, 24, 37, 27, 39, 17, 14, 36, 35, 24, 3, 32, 17, 42, 1, 47, 56, 44, 0, 11, 50, 21, 2, 55, 55, 17, 5, 61, 69, 55, 3, 14, 14, 59, 38, 22, 62

Offset: 3

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 3..10000

Cf. A014834, A014837, A116987.

Table[ Mod[Plus@@Table[Plus@@IntegerDigits[n,k],{k,2,n-1}],n],{n,3,100}]

a(n)=A014837(n) mod n. [From R. J. Mathar, Aug 10 2008]

A014841 Sum modulo the base of all the digits of n in every base from 2 to n-1.

Original entry on oeis.org

0, 3, 2, 7, 6, 13, 13, 18, 19, 31, 24, 37, 40, 48, 48, 65, 59, 79, 76, 85, 93, 117, 105, 121, 132, 148, 143, 176, 163, 193, 191, 208, 226, 250, 225, 262, 277, 302, 290, 332, 320, 359, 363, 376, 394, 444, 419, 455, 462, 491, 495, 551, 540, 577, 564, 601, 625

Offset: 3

Olivier Gérard

Stefano Spezia, Table of n, a(n) for n = 3..10000

Cf. A014836, A014837, A014838, A014839, A014840, A014842, A014843.

Table[Sum[Mod[Total[IntegerDigits[n,i]], i], {i,2,n-1}], {n,3,59}] (* Stefano Spezia, Sep 06 2022 *)

a(n) = sum(b=2, n-1, sumdigits(n, b) % b); \\ Michel Marcus, Sep 06 2022

from sympy.ntheory import digits
def a(n): return sum(sum(digits(n, b)[1:])%b for b in range(2, n))
print([a(n) for n in range(3, 60)]) # Michael S. Branicky, Sep 06 2022

A014843 Sum modulo n of the sum modulo n of all the digits of n in every base from 2 to n-1.

Original entry on oeis.org

2, 2, 2, 3, 1, 7, 2, 4, 1, 9, 6, 9, 14, 8, 5, 0, 14, 11, 19, 3, 20, 7, 18, 25, 5, 2, 24, 12, 5, 27, 9, 18, 0, 8, 0, 10, 26, 12, 3, 30, 21, 19, 24, 37, 27, 39, 17, 14, 36, 35, 24, 3, 32, 17, 42, 1, 47, 56, 44, 0, 11, 49, 21, 2, 55, 55, 17, 5, 61, 69, 55, 3, 14, 14, 59, 38, 22, 62

Offset: 3

Olivier Gérard

This is a variant of A014836 and A014837. Cf. A116987.

A131383 Total digital sum of n: sum of the digital sums of n for all the bases 1 to n (a 'digital sumorial').

Original entry on oeis.org

1, 3, 6, 8, 13, 16, 23, 25, 30, 35, 46, 46, 59, 66, 75, 74, 91, 91, 110, 112, 125, 136, 159, 152, 169, 182, 195, 199, 228, 223, 254, 253, 274, 291, 316, 297, 334, 353, 378, 373, 414, 409, 452, 460, 475, 498, 545, 520, 557, 565, 598, 608, 661, 652, 693, 690

Offset: 1

Hieronymus Fischer, Jul 05 2007, Jul 15 2007, Jan 07 2009

Sums of rows of the triangle in A138530. - Reinhard Zumkeller, Mar 26 2008

			5 = 11111(base 1) = 101(base 2) = 12(base 3) = 11(base 4) = 10(base 5). Thus a(5) = ds_1(5)+ds_2(5)+ds_3(5)+ds_4(5)+ds_5(5) = 5+2+3+2+1 = 13.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Digit Sum

Cf. A131384, A007953.

Table[n + Total@ Map[Total@ IntegerDigits[n, #] &, Range[2, n]], {n, 56}] (* Michael De Vlieger, Jan 03 2017 *)

a(n)=sum(i=2,n+1,vecsum(digits(n,i))); \\ R. J. Cano, Jan 03 2017

a(n) = n^2-sum{k>0, sum{2<=p<=n, (p-1)*floor(n/p^k)}}.

a(n) = n^2-sum{2<=p<=n, (p-1)*sum{0

a(n) = n^2-A024916(n)+A006218(n)-sum{k>1, sum{2<=p<=n, (p-1)*floor(n/p^k)}}.

a(n) = A004125(n)+A006218(n)-sum{k>1, sum{2<=p<=n, (p-1)*floor(n/p^k)}}.

Asymptotic behavior: a(n) = (1-Pi^2/12)*n^2 + O(n*log(n)) = A004125(n) + A006218(n) + O(n*log(n)).

Lim a(n)/n^2 = 1 - Pi^2/12 for n-->oo.

G.f.: (1/(1-x))*(x(1+x)/(1-x)^2-sum{k>0,sum{j>1,(j-1)*x^(j^k)/(1-x^(j^k))}= }).

Also: (1/(1-x))*(x(1+x)/(1-x)^2-sum{m>1, sum{10,j^(1/k) is an integer, j^(1/k)-1}}*x^m}).

a(n) = n^2-sum{10,sum{1

Recurrence: a(n)=a(n-1)-b(n)+2n-1, where b(n)=sum{1

a(n) = sum{1<=p<=n, ds_p(n)} where ds_p = digital sum base p.

a(n) = A043306(n) + n (that sequence ignores unary) = A014837(n) + n + 1 (that sequence ignores unary and base n in which n is "10"). - Alonso del Arte, Mar 26 2009

A014840 Sum of all the digits of n in every base prime to n from 2 to n-1.

Original entry on oeis.org

2, 2, 7, 2, 15, 10, 15, 8, 34, 12, 45, 22, 31, 32, 73, 28, 90, 40, 57, 50, 135, 46, 118, 74, 117, 70, 198, 58, 222, 120, 139, 122, 192, 92, 296, 152, 216, 136, 372, 112, 408, 202, 235, 208, 497, 176, 442, 224, 338, 260, 607, 202, 454, 276, 416, 330, 755, 194, 776

Offset: 3

Olivier Gérard

Robert Israel, Table of n, a(n) for n = 3..10000

Cf. A014836, A014837, A014838, A014839, A014841, A014842, A014843.

f:= proc(n) local b;
 add(convert(convert(n,base,b),`+`), b = select(t -> igcd(t,n)=1, [$2..n-1]))
end proc:
map(f, [$3..100]); # Robert Israel, Nov 08 2024

Table[Sum[If[CoprimeQ[i, n], Mod[Total[IntegerDigits[n, i]], n], 0], {i, 2, n-1}], {n, 3, 61}] (* Stefano Spezia, Sep 06 2022 *)

a(n) = {s = 0; for (i=2, n-1, if (gcd(i, n) == 1, d = digits(n, i); s += sum(j=1, #d, d[j]););); s;} \\ Michel Marcus, May 30 2014

from math import gcd
from sympy.ntheory import digits
def a(n): return sum(sum(digits(n, b)[1:]) for b in range(2, n) if gcd(b, n) == 1)
print([a(n) for n in range(3, 62)]) # Michael S. Branicky, Sep 06 2022

A014838 Sum of all the digits of n in every prime base from 2 to n-1.

Original entry on oeis.org

2, 3, 5, 6, 9, 11, 11, 10, 14, 16, 21, 21, 21, 23, 29, 32, 39, 42, 42, 39, 47, 52, 53, 49, 52, 53, 62, 66, 76, 83, 82, 76, 77, 82, 93, 87, 85, 90, 102, 107, 120, 123, 129, 120, 134, 144, 147, 153, 150, 151, 166, 176, 178, 185, 181, 168, 184, 194, 211, 199, 207

Offset: 3

Olivier Gérard

Cf. A014836, A014837, A014839, A014840, A014841, A014842, A014843.

Table[Sum[If[PrimeQ[i], Mod[Total[IntegerDigits[n, i]], n], 0], {i, 2, n-1}],{n, 3, 63}] (* Stefano Spezia, Sep 06 2022 *)

a(n) = {s = 0; forprime (i=2, n-1, d = digits(n, i); s += sum(j=1, #d, d[j]);); s;} \\ Michel Marcus, May 30 2014

from sympy.ntheory import digits, isprime
def a(n): return sum(sum(digits(n, b)[1:]) for b in range(2, n) if isprime(b))
print([a(n) for n in range(3, 64)]) # Michael S. Branicky, Sep 06 2022

A014839 Sum of all the digits of n in every prime-power base from 2 to n-1.

Original entry on oeis.org

2, 3, 7, 9, 13, 13, 16, 19, 26, 28, 36, 39, 42, 34, 45, 44, 55, 59, 63, 64, 76, 75, 80, 82, 82, 87, 102, 112, 128, 113, 120, 121, 129, 130, 148, 149, 154, 156, 175, 187, 207, 214, 219, 217, 238, 227, 237, 228, 233, 239, 262, 246, 256, 261, 265, 260, 284, 299

Offset: 3

Olivier Gérard

Cf. A014836, A014837, A014838, A014840, A014841, A014842, A014843.

Table[Sum[If[PrimePowerQ[i], Mod[Total[IntegerDigits[n, i]], n], 0], {i, 2, n-1}], {n, 3, 60}] (* Stefano Spezia, Sep 06 2022 *)

A343481 a(n) is the sum of all digits of n in every prime base 2 <= p <= n.

Original entry on oeis.org

1, 3, 3, 6, 6, 10, 11, 11, 10, 15, 16, 22, 21, 21, 23, 30, 32, 40, 42, 42, 39, 48, 52, 53, 49, 52, 53, 63, 66, 77, 83, 82, 76, 77, 82, 94, 87, 85, 90, 103, 107, 121, 123, 129, 120, 135, 144, 147, 153, 150, 151, 167, 176, 178, 185, 181, 168, 185, 194, 212, 199

Offset: 2

Amiram Eldar, Apr 16 2021

			a(5) = 6 since in the prime bases 2, 3 and 5 the representations of 5 are 101_2, 12_3 and 10_5, respectively, and (1 + 0 + 1) + (1 + 2) + (1 + 0) = 6.

Amiram Eldar, Table of n, a(n) for n = 2..10001
Robin Fissum, Digit sums and the number of prime factors of the factorial n!=1.2...n, Bachelor's project in BMAT, Norwegian University of Science and Technology, 2020; ResearchGate link.

Cf. A014837, A043306, A072691, A073002, A222171.

s[n_, b_] := Plus @@ IntegerDigits[n, b]; ps[n_] := Select[Range[n], PrimeQ]; a[n_] := Sum[s[n, b], {b, ps[n]}]; Array[a, 100, 2]

a(n) = sum(b=2, n, if (isprime(b), sumdigits(n, b))); \\ Michel Marcus, Apr 17 2021

a(n) ~ (1-Pi^2/12)*n^2/log(n) + c*n^2/log(n)^2 + o(n^2/log(n)^2), where c = 1 - Pi^2/24 + zeta'(2)/2 = 1 - A222171 - (1/2)*A073002 = 0.1199923561... (Fissum, 2020).

Showing 1-10 of 10 results.

cp's OEIS Frontend

A014842 Difference between A014837 and A014841.

Views

Author

Keywords

A043306 Sum of all digits in all base-b representations for n, for 2 <= b <= n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Formula

A014836 Sum modulo n of all the digits of n in every base from 2 to n-1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A014841 Sum modulo the base of all the digits of n in every base from 2 to n-1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

A014843 Sum modulo n of the sum modulo n of all the digits of n in every base from 2 to n-1.

Views

Author

Keywords

Comments

A131383 Total digital sum of n: sum of the digital sums of n for all the bases 1 to n (a 'digital sumorial').

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A014840 Sum of all the digits of n in every base prime to n from 2 to n-1.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

A014838 Sum of all the digits of n in every prime base from 2 to n-1.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Python

A014839 Sum of all the digits of n in every prime-power base from 2 to n-1.

Views

Author

Keywords

Crossrefs

Programs