A191322 -id:A191322 - cp's OEIS frontend

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

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)

A043000 Number of digits in all base-b representations of n, for 2 <= b <= n.

Original entry on oeis.org

2, 4, 7, 9, 11, 13, 16, 19, 21, 23, 25, 27, 29, 31, 35, 37, 39, 41, 43, 45, 47, 49, 51, 54, 56, 59, 61, 63, 65, 67, 70, 72, 74, 76, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126

Offset: 2

Clark Kimberling

From A.H.M. Smeets, Dec 14 2019: (Start)
a(n)-a(n-1) >= 2 due to the fact that n = 10_n, so there is an increment of at least 2. If n can be written as a perfect power m^s, an additional +1 comes to it for the representation of n in each base m.
For instance, for n = 729 we have 729 = 3^6 = 9^3 = 27^2, so there is an additional increment of 3. For n = 1296 we have 1296 = 6^4 = 36^2, so there is an additional increment of 2. For n = 4096 we have 4096 = 2^12 = 4^6 = 8^4 = 16^3= 64^2, so there is an additional increment of 5. (End)

			5 = 101_2 = 12_3 = 11_4 = 10_5. Thus a(5) = 3+2+2+2 = 9.

A.H.M. Smeets, Table of n, a(n) for n = 2..20000
Vaclav Kotesovec, Plot of a(n)/(2*n) for n = 2..1000000

Cf. A001597, A043306, A068953, A191322, A253642, A255165.

[&+[Floor(Log(i,i*n)):k in [2..n]]:n in [1..70]]; // Marius A. Burtea, Nov 13 2019

A043000 := proc(n) add( nops(convert(n,base,b)),b=2..n) ; end proc: # R. J. Mathar, Jun 04 2011

Table[Total[IntegerLength[n,Range[2,n]]],{n,2,60}] (* Harvey P. Dale, Apr 23 2019 *)

a(n)=sum(b=2,n,#digits(n,b)) \\ Jeppe Stig Nielsen, Dec 14 2019

a(n)= n-1 +sum(b=2,n,logint(n,b)) \\ Jeppe Stig Nielsen, Dec 14 2019

a(n) = {2*n-2+sum(i=2, logint(n, 2), sqrtnint(n, i)-1)} \\ David A. Corneth, Dec 31 2019

first(n) = my(res = vector(n)); res[1] = 2; for(i = 2, n, inc = numdiv(gcd(factor(i+1)[,2]))+1; res[i] = res[i-1]+inc); res \\ David A. Corneth, Dec 31 2019

def count(n,b):
    c = 0
    while n > 0:
        n, c = n//b, c+1
    return c
n = 0
while n < 50:
    n = n+1
    a, b = 0, 1
    while b < n:
        b = b+1
        a = a + count(n,b)
    print(n,a) # A.H.M. Smeets, Dec 14 2019

a(n) = Sum_{i=2..n} floor(log_i(i*n)); a(n) ~ 2*n. - Vladimir Shevelev, Jun 03 2011 [corrected by Vaclav Kotesovec, Apr 05 2021]
a(n) = A070939(n) + A081604(n) + A110591(n) + ... + 1. - R. J. Mathar, Jun 04 2011
From Ridouane Oudra, Nov 13 2019: (Start)
a(n) = Sum_{i=1..n-1} floor(n^(1/i));
a(n) = n - 1 + Sum_{i=1..floor(log_2(n))} floor(n^(1/i) - 1);
a(n) = n - 1 + A255165(n). (End)
If n is in A001597 then a(A001597(m)) - a(A001597(m)-1) = 2 + A253642(m), otherwise a(n) - a(n-1) = 2. - A.H.M. Smeets, Dec 14 2019

A191350 The number of bases not exceeding n+1 in which the expansion of n (i) has only digits <=9 and (ii) represents a prime if digits are concatenated/reinterpreted as decimals.

Original entry on oeis.org

1, 2, 1, 3, 1, 3, 2, 2, 3, 3, 2, 4, 2, 4, 4, 5, 3, 7, 4, 6, 6, 8, 4, 7, 5, 6, 6, 8, 4, 9, 4, 9, 7, 7, 4, 11, 5, 9, 6, 8, 4, 13, 4, 8, 7, 10, 5, 10, 5, 8, 7, 9, 4, 14, 5, 8, 8, 11, 4, 12, 4, 10, 8, 8, 5, 15, 6, 8, 6, 13, 4, 14, 5, 10, 6, 8, 6, 17, 5, 8, 7, 12, 6, 13, 5, 11, 8, 11, 4, 15, 5

Offset: 2

Vladimir Shevelev, May 31 2011

			In bases 6, 8, 12 and 14 the digits of n=15 are 15_6=23, 15_8=17, 15_12=13, and 15_14=11. Since in other bases<=16 the expansions of 15 converted to decimal are not primes, a(15)=4.

Charles R Greathouse IV, Table of n, a(n) for n = 2..10000

Cf. A191291, A191322.

a(n)=my(m,t,k,i);sum(b=2,n+1,k=n;m=0;i=0;while(k,t=k%b;if(t>9,m=0;break);m+=10^i*t;i++;k\=b);isprime(m)) \\ Charles R Greathouse IV, Jun 01 2011

a(16)-a(91) from Charles R Greathouse IV, Jun 01 2011

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

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A043000 Number of digits in all base-b representations of n, for 2 <= b <= n.

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A191350 The number of bases not exceeding n+1 in which the expansion of n (i) has only digits <=9 and (ii) represents a prime if digits are concatenated/reinterpreted as decimals.

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