A322850 Number of times 2^k for k < n-1 appears as a substring within 2^n.

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

Offset

0,8

Links

Table of n, a(n) for n=0..85.

Formula

a(n) >= A322849(n), for n >= 4.

Example

n =  0, a(n) = 0, 2^n =     1 - no solutions;
n =  1, a(n) = 0, 2^n =     2 - no solutions;
n =  2, a(n) = 0, 2^n =     4 - no solutions;
n =  3, a(n) = 0, 2^n =     8 - no solutions;
n =  4, a(n) = 1, 2^n =    16 - solution is 1;
n =  5, a(n) = 1, 2^n =    32 - solution is 2;
n =  6, a(n) = 1, 2^n =    64 - solution is 4;
n =  7, a(n) = 3, 2^n =   128 - solutions are 1,2,8;
n = 14, a(n) = 4, 2^n = 16384 - solutions are 1,4,8,16;
n = 15, a(n) = 3, 2^n = 32768 - solutions are 2,8,32;
n = 16, a(n) = 0, 2^n = 65536 - no solutions.

Mathematica

Array[If[# < 4, Total@ DigitCount[2^#, 10, 2^Range[0, Min[# - 1, 3]]], Total@ DigitCount[2^#, 10, {1, 2, 4, 8}]] &, 85, 0] (* Michael De Vlieger, Dec 31 2018 *)

Prog

(Python 3.7)
import re
results = []
start_n = 0
N = 100
compare_list = []   # Store powers up to n-1
current_num = int(pow(2, start_n-1))    # Calculate (n-1) power. Convert to integer for better precision
for n in range(start_n, N):
    if n == 0:
        current_num = 1
    else:
        current_num += current_num
    current_string = str(current_num)
    count = 0
    for test_str in compare_list:
        count += len(re.findall(test_str, current_string))
    compare_list.append(current_string)
    results.append(count)
    a=1
print(results)
(PARI) isp2(n) = (n==1) || (n==2) || (ispower(n, , &k) && (k==2));
a(n) = {my(d=digits(2^n), nb = 0); for (i=1, #d-1, for (j=1, #d-i+1, my(nd = vector(i, k, d[j+k-1])); if (nd[1] != 0, nb += isp2(fromdigits(nd))); ); ); nb; } \\ Michel Marcus, Dec 30 2018

Crossrefs

Cf. A065712 (1), A065710 (2), A065715 (4), A065719 (8).

Cf. A322849.

Sequence in context: A026181 A327239 A322849 * A337566 A106601 A325533

Adjacent sequences: A322847 A322848 A322849 * A322851 A322852 A322853

Keyword

base,nonn

Author

Gaitz Soponski, Dec 28 2018

Status

approved