A088995 -id:A088995 - cp's OEIS frontend

A088935 Numbers n such that leading digits of 2^n and 5^n are equal.

0, 5, 15, 78, 88, 98, 108, 118, 181, 191, 201, 211, 274, 284, 294, 304, 367, 377, 387, 397, 407, 470, 480, 490, 500, 563, 573, 583, 593, 603, 666, 676, 686, 696, 759, 769, 779, 789, 852, 862, 872, 882, 892, 955, 965, 975, 985, 1048, 1058, 1068, 1078, 1088

Offset: 1

Lekraj Beedassy, Dec 01 2003

Write lg = log_10, let {x} denote the fractional part of x. Note that {k*lg(5)} = 1 - {k*lg(2)}, so we have {k > 0 : 2^k, 5^k, 8^k all start with a} = {k: {k*lg(2)} is in I_a}, where I_a = (lg(a), lg(a+1)) intersect (1-lg(a+1), 1-lg(a)). Note that I_3 = (lg(3), 1-lg(3)) and I_a is empty otherwise. As a result, k > 0 is a term if and only if lg(3) < {k*lg(2)} < 1-lg(3). - Jianing Song, Dec 26 2022

			78 is in the sequence since 2^78 = 302231454903657293676544 and 5^78 = 3308722450212110699485634768279851414263248443603515625
98 is in the sequence since 2^98 = 316912650057057350374175801344 and 5^98 = 315544362088404722164691426113114491869282574043609201908111572265625.

David A. Corneth, Table of n, a(n) for n = 1..10000
T. Sillke, Powers of 2 and 5 Puzzle

Cf. A088995.

filter:= n -> convert(2^n,base,10)[-1]=convert(5^n,base,10)[-1]:
select(filter, [$0..1000]); # Robert Israel, Aug 09 2018

Select[ Range[ 1000 ], IntegerDigits[ 2^# ][ [ 1 ] ] == IntegerDigits[ 5^# ][ [ 1 ] ] & ]

is(n)=(digits(2^n)[1]==digits(5^n)[1]);
for(n=0,10^3,if(is(n),print1(n,", "))); \\ Joerg Arndt, Aug 10 2018

def ok(n): return str(2**n)[0] == str(5**n)[0]
print([k for k in range(1100) if ok(k)]) # Michael S. Branicky, Nov 03 2022

Edited by Robert G. Wilson v, Dec 02 2003

A359698 Least k > 0 such that the first n digits of 2^k and 3^k are identical.

Original entry on oeis.org

1, 17, 193, 619, 2016, 91958, 91958, 8186278, 45392361, 977982331, 26450915298, 91600221212, 196425900073, 14810317269038, 44430951807114, 626642721222487, 626642721222487, 102882886570917135, 874191214492184404, 3830977578643912683, 86801197487071715103

Offset: 0

Keith F. Lynch, May 20 2023

			   n    k = a(n)   1st n digits
  --  -----------  -------------
   0            1
   1           17  1...
   2          193  12...
   3          619  217...
   4         2016  7524...
   5        91958  13071...
   6        91958  130719...
   7      8186278  1701547...
   8     45392361  17179395...
   9    977982331  725070476...
  10  26450915298  2919267309...
a(3) = 619 because 2^619 = 2.175...*10^186 and 3^619 = 2.177...*10^295 both begin with the same three digits (in base ten), and this is not true of any smaller positive integer than 619.
a(0) = 1 because 2^1 and 3^1 share zero leading digits.

Zhao Hui Du, Table of n, a(n) for n = 0..1000

Cf. A000079, A000244, A088995.

a(11)-a(20) from Jon E. Schoenfield, May 21 2023

A363683 Square array A(n, k), n, k > 0, read by antidiagonals; A(n, k) is the least e > 0 such that n^e and k^e have the same initial digit, or -1 if no such e exists.

Original entry on oeis.org

1, 4, 4, 9, 1, 9, 2, 17, 17, 2, 3, 7, 1, 7, 3, 4, 5, 8, 8, 5, 4, 5, 4, 9, 1, 9, 4, 5, 8, 2, 3, 11, 11, 3, 2, 8, 16, 5, 11, 6, 1, 6, 11, 5, 16, 1, 17, 7, 4, 9, 9, 4, 7, 17, 1, 1, 4, 17, 10, 6, 1, 6, 10, 17, 4, 1, 1, 4, 9, 14, 5, 15, 15, 5, 14, 9, 4, 1

Offset: 1

Rémy Sigrist, Jun 15 2023

Conjecture: all terms are positive.

			Array A(n, k) begins:
  n\k |  1   2   3   4   5   6   7   8   9  10  11  12
  ----+-----------------------------------------------
    1 |  1   4   9   2   3   4   5   8  16   1   1   1
    2 |  4   1  17   7   5   4   2   5  17   4   4   9
    3 |  9  17   1   8   9   3  11   7  17   9  16   5
    4 |  2   7   8   1  11   6   4  10  14   2   2   2
    5 |  3   5   9  11   1   9   6   5   4   3   3   3
    6 |  4   4   3   6   9   1  15   7  11   4   4  10
    7 |  5   2  11   4   6  15   1  17  18   5   5   4
    8 |  8   5   7  10   5   7  17   1  18   8  28   6
    9 | 16  17  17  14   4  11  18  18   1  16  23   8
   10 |  1   4   9   2   3   4   5   8  16   1   1   1
   11 |  1   4  16   2   3   4   5  28  23   1   1   1
   12 |  1   9   5   2   3  10   4   6   8   1   1   1

Rémy Sigrist, Colored representation of the table for n, k <= 1200

Cf. A000030, A073601, A088995, A098174, A359698.

A(n,k) = { for (e = 1, oo, if (digits(n^e)[1]==digits(k^e)[1], return (e));); }

A(n, k) = A(k, n).
A(10*n, k) = A(n, k).
A(n, n) = 1.
A(n, 1) = A098174(n).

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A088935 Numbers n such that leading digits of 2^n and 5^n are equal.

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A359698 Least k > 0 such that the first n digits of 2^k and 3^k are identical.

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Extensions

A363683 Square array A(n, k), n, k > 0, read by antidiagonals; A(n, k) is the least e > 0 such that n^e and k^e have the same initial digit, or -1 if no such e exists.

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Programs

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