A034887 -id:A034887 - cp's OEIS frontend

A364606 Numbers k such that the average digit of 2^k is an integer.

0, 1, 2, 3, 6, 13, 16, 26, 46, 51, 56, 73, 122, 141, 166, 313, 383

Offset: 1

Jon E. Schoenfield, Jul 29 2023

			2^26 = 67108864 is an 8-digit number; its average digit is (6+7+1+0+8+8+6+4)/8 = 40/8 = 5, an integer, so 26 is a term.

Cf. A000079, A001370, A034887, A061383.

q:= n-> (l-> irem(add(i, i=l), nops(l))=0)(convert(2^n, base, 10)):
select(q, [$0..400])[];  # Alois P. Heinz, Jul 29 2023

Select[Range[0, 2^12], IntegerQ@ Mean@ IntegerDigits[2^#] &] (* Michael De Vlieger, Jul 29 2023 *)

isok(k) = my(d=digits(2^k)); !(vecsum(d) % #d); \\ Michel Marcus, Jul 29 2023

from itertools import count, islice
from gmpy2 import mpz, digits
def A364606_gen(startvalue=0): # generator of terms >= startvalue
    m = mpz(1)<A364606_list = list(islice(A364606_gen(),10)) # Chai Wah Wu, Jul 31 2023

{ k : A001370(k) mod A034887(k) = 0 }.

A379865 Number of base 10 digits of 2^(p-1)*(2^p-1) where p = prime(n).

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 10, 12, 14, 18, 19, 22, 25, 26, 28, 32, 36, 37, 41, 43, 44, 48, 50, 54, 59, 61, 62, 65, 66, 68, 77, 79, 83, 84, 90, 91, 95, 98, 101, 104, 108, 109, 115, 116, 119, 120, 127, 134, 137, 138, 140, 144, 145, 151, 155, 159, 162, 163, 167, 169, 171, 177

Offset: 1

Darío Clavijo, Jan 04 2025

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

Cf. A034887, A055642, A060286.

A379865[n_] := IntegerLength[2^(# - 1)*(2^# - 1)] & [Prime[n]];
Array[A379865, 100] (* Paolo Xausa, Jan 08 2025 *)

from sympy import prime
def a(n):
  p = prime(n)
  return len(str((1 << (p-1)) * ((1 << p) - 1)))
print([a(n) for n in range(1,63)])

a(n) = A055642(A060286(n)).

A071423 a(n) = a(n-1) + number of decimal digits of 2^n. Number of decimal digits of concatenation of first n powers of 2.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 12, 15, 18, 22, 26, 30, 34, 39, 44, 49, 55, 61, 67, 74, 81, 88, 95, 103, 111, 119, 128, 137, 146, 156, 166, 176, 186, 197, 208, 219, 231, 243, 255, 268, 281, 294, 307, 321, 335, 349, 364, 379, 394, 410, 426, 442, 458, 475, 492, 509, 527, 545

Offset: 1

Labos Elemer, May 27 2002

Cf. A058183.

Do[s=s+Length[IntegerDigits[2^n]]; Print[s], {n, 1, 128}]
nxt[{n_,a_}]:={n+1,a+IntegerLength[2^(n+1)]}; NestList[nxt,{1,1},60][[All,2]] (* Harvey P. Dale, Nov 11 2022 *)

a(n) = a(n-1)+A034887(n). [R. J. Mathar, Sep 11 2009]

a(n) = 0.5 log 2/log 10 * n^2 + O(n)

An incorrect g.f. was deleted by N. J. A. Sloane, Sep 13 2009

Formula from Charles R Greathouse IV, Apr 28 2010

A329734 a(n) = (10^(1 + floor((n-1) * log_10 2)) + 1)^n, n >= 0.

Original entry on oeis.org

1, 11, 121, 1331, 14641, 10510100501, 1061520150601, 107213535210701, 1008028056070056028008001, 1009036084126126084036009001, 1010045120210252210120045010001, 100110055016503300462046203300165005500110001, 1001200660220049507920924079204950220006600120001

Offset: 0

Daniel Forgues, Nov 19 2019

a(n), n >= 1, gives all the binomial coefficients (without overlap, since binomial(n,k) < 2^(n-1) for n >= 3) of the n-th row of Pascal's triangle (A007318). The k-th "digit", 0 <= k <= n, of the base 10^(d(2^(n-1))) representation of a(n) gives binomial(n,k), albeit with leading zeros: binomial(n,k) = (a(n) div (10^(d(2^(n-1))))^k) mod 10^(d(2^(n-1))), where div means integer division.

A092846 uses number of digits of binomial(n, floor(n/2)) instead [optimal], but requires the evaluation of central binomial coefficients. a(n) uses number of digits of 2^(n-1) [not optimal], but does not require the evaluation of any binomial coefficient.

			a(9) = (10^3 + 1)^9 = 1009036084126126084036009001 = 1:009:036:084:126:126:084:036:009:001 (base 10^3, since 2^8 has 3 decimal digits).

OEIS Wiki, Pascal's triangle rows.

Cf. A007318, A092846 (uses number of digits of central binomial coefficients).

d[n_] := Floor[Log[10, 10*n]]; a[n_] := (10^(d[2^(n - 1)]) + 1)^n; Array[a, 12] (* Amiram Eldar, Nov 23 2019 *)

a(0) = 1; a(n) = (10^(d(2^(n-1))) + 1)^n, n >= 1, where d(2^(n-1)) = 1 + floor((n-1) * log_10 2) = A034887(n-1) is the number of [decimal] digits of 2^(n-1).

A364615 Numbers k such that the average of the decimal digits of 2^k is closer to 9/2 (the expected average for random digits) than for any smaller power of 2.

Original entry on oeis.org

0, 1, 2, 8, 14, 20, 29, 47, 62, 80, 113, 134, 182, 206, 281, 287, 299, 326, 419, 500, 560, 620, 638, 674, 833, 911, 1271, 1289, 1376, 1418, 1583, 1670, 1814, 2273, 2753, 3365, 3794, 4127, 4160, 4202, 4280, 4292, 4538, 4553, 4646, 4805, 4952, 4979, 5105, 5276

Offset: 1

Pontus von Brömssen and Jon E. Schoenfield, Jul 29 2023

The average of the digits of 2^k is never exactly 9/2, because the sum of digits cannot be divisible by 3.

Conjecture: for each term k > 1, digitsum(2^k) - (9/2)*number_of_digits(2^k) = 1/2 if k is odd, -1/2 if k is even. - Jon E. Schoenfield, Jul 30 2023

			   k |   2^k | average of digits | distance from 9/2 | new minimum?
  ---+-------+-------------------+-------------------+-------------
   0 |     1 |         1         |        7/2        |     yes
   1 |     2 |         2         |        5/2        |     yes
   2 |     4 |         4         |        1/2        |     yes
   3 |     8 |         8         |        7/2        |
   4 |    16 |        7/2        |         1         |
   5 |    32 |        5/2        |         2         |
   6 |    64 |         5         |        1/2        |
   7 |   128 |       11/3        |        5/6        |
   8 |   256 |       13/3        |        1/6        |     yes
   9 |   512 |        8/3        |       11/6        |
  10 |  1024 |        7/4        |       11/4        |
  11 |  2048 |        7/2        |         1         |
  12 |  4096 |       19/4        |        1/4        |
  13 |  8192 |         5         |        1/2        |
  14 | 16384 |       22/5        |        1/10       |     yes

Cf. A000079, A001370, A034887, A364606.

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A364606 Numbers k such that the average digit of 2^k is an integer.

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A379865 Number of base 10 digits of 2^(p-1)*(2^p-1) where p = prime(n).

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A071423 a(n) = a(n-1) + number of decimal digits of 2^n. Number of decimal digits of concatenation of first n powers of 2.

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A329734 a(n) = (10^(1 + floor((n-1) * log_10 2)) + 1)^n, n >= 0.

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A364615 Numbers k such that the average of the decimal digits of 2^k is closer to 9/2 (the expected average for random digits) than for any smaller power of 2.

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