A180017 -id:A180017 - cp's OEIS frontend

A037301 Numbers whose base-2 and base-3 expansions have the same digit sum.

0, 1, 6, 7, 10, 11, 12, 13, 18, 19, 21, 36, 37, 46, 47, 58, 59, 60, 61, 86, 92, 102, 103, 114, 115, 120, 121, 166, 167, 172, 173, 180, 181, 198, 199, 216, 217, 222, 223, 261, 273, 282, 283, 285, 298, 299, 300, 301, 306, 307, 309, 318

Offset: 1

Clark Kimberling

If Sum_{i=0..k} (binomial(k,i) mod 2) == Sum_{i=0..k} (binomial(k,i) mod 3) then k is in the sequence. (The converse does not hold.) - Benoit Cloitre, Nov 16 2003
Problem: To prove that the sequence is infinite. A generalization: Let s_m(k) denote the sum of digits of k in base m; does the Diophantine equation s_p(k) = s_q(k), where p,q are fixed distinct primes, have infinitely many solutions? - Vladimir Shevelev, Jul 30 2009
Also, numbers k such that the exponent of the largest power of 2 dividing k! is exactly twice the exponent of the largest power of 3 dividing k!. - Ivan Neretin, Mar 08 2015
a(5) = 10, a(6) = 11, a(7) = 12 and a(8) = 13 is the first time that four consecutive terms appear in this sequence. Conjecture: There is no occurrence of five or more consecutive terms of a(n). Tested by exhaustive search up to a(n) = 3^29. - Thomas König, Aug 15 2020

Stanislav Sykora, Table of n, a(n) for n = 1..10000
Vladimir Shevelev, Compact integers and factorials, Acta Arith. 126 (2007), no. 3, 195-236.
Vladimir Shevelev, Binomial predictors, arXiv:0907.3302 [math.NT], 2009.
Lukas Spiegelhofer, Collisions of the binary and ternary sum-of-digits functions, arXiv:2105.11173 [math.NT], 2021. Proves that sequence is infinite.

Cf. A000120, A053735, A180017.

Cf. A001316, A051638, A212222, A330904, A334765.

Select[ Range@ 320, Total@ IntegerDigits[#, 2] == Total@ IntegerDigits[#, 3] &] (* Robert G. Wilson v, Oct 24 2014 *)

is(n)=sumdigits(n,3)==hammingweight(n) \\ Charles R Greathouse IV, May 21 2015

A053735(a(n)) = A000120(a(n)); A180017(a(n)) = 0. - Reinhard Zumkeller, Aug 06 2010

Zero prepended by Zak Seidov, May 31 2010

A180018 Difference of sums of digits of n in decimal and in binary representation.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 4, 4, 7, 7, -1, -1, 1, 1, 2, 2, 6, 6, 7, 7, 0, 0, 1, 1, 4, 4, 5, 5, 7, 7, -1, -1, 4, 4, 5, 5, 7, 7, 8, 8, 2, 2, 3, 3, 5, 5, 6, 6, 10, 10, 2, 2, 4, 4, 5, 5, 8, 8, 9, 9, 2, 2, 3, 3, 9, 9, 10, 10, 12, 12, 4, 4, 7, 7, 8, 8, 10, 10, 11, 11, 6, 6, 7, 7, 9, 9, 10, 10, 13, 13, 5, 5, 7, 7, 8, 8

Offset: 0

Reinhard Zumkeller, Aug 06 2010

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for Colombian or self numbers and related sequences

Cf. A180017, A180019, A007088.
Cf. A007953, A000120, A037308, A011557.
Cf. A000079, A002283, A008591, A000225.

Table[Total[IntegerDigits[n]]-Total[IntegerDigits[n,2]],{n,0,100}] (* Harvey P. Dale, Dec 27 2015 *)

a(n) = sumdigits(n) - hammingweight(n); \\ Michel Marcus, Nov 06 2022

a(n) = A007953(n) - A000120(n);
a(A037308(n)) = 0;
a(A011557(n)) = 1 - A000120(A011557(n));
a(A000079(n)) = A007953(A000079(n)) - 1;
a(A002283(n)) = A008591(n) - A000120(A002283(n));
a(A000225(n)) = A007953(A000225(n)) - n.

A180019 Difference of sums of digits of n in decimal and in ternary representation.

Original entry on oeis.org

0, 0, 0, 2, 2, 2, 4, 4, 4, 8, -1, -1, 1, 1, 1, 3, 3, 3, 7, 7, -2, 0, 0, 0, 2, 2, 2, 8, 8, 8, 1, 1, 1, 3, 3, 3, 7, 7, 7, 9, 0, 0, 2, 2, 2, 6, 6, 6, 8, 8, -1, 1, 1, 1, 7, 7, 7, 9, 9, 9, 2, 2, 2, 6, 6, 6, 8, 8, 8, 10, 1, 1, 5, 5, 5, 7, 7, 7, 9, 9, 0, 8, 8, 8, 10, 10, 10, 12, 12, 12, 7, 7, 7, 9, 9, 9, 11, 11, 11

Offset: 0

Reinhard Zumkeller, Aug 06 2010

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for Colombian or self numbers and related sequences

Cf. A180017, A180018, A007089.
Cf. A007953, A053735, A037315, A011557.
Cf. A000244, A002283, A008591, A024023.

a(n) = sumdigits(n) - sumdigits(n, 3); \\ Michel Marcus, Nov 06 2022

a(n) = A007953(n) - A053735(n);
a(A037315(n)) = 0;
a(A011557(n)) = 1 - A053735(A011557(n));
a(A000244(n)) = A007953(A000244(n)) - 1;
a(A002283(n)) = A008591(n) - A053735(A002283(n));
a(A024023(n)) = A007953(A024023(n)) - 2*n.

A375257 Numbers whose sum of base-2 digits is 1 more than their sum of base-3 digits.

Original entry on oeis.org

3, 9, 15, 28, 29, 39, 45, 57, 82, 83, 84, 85, 94, 95, 99, 110, 118, 119, 123, 135, 162, 163, 165, 174, 175, 183, 207, 219, 248, 297, 303, 315, 324, 325, 334, 335, 342, 343, 363, 382, 383, 406, 407, 411, 423, 435, 441, 447, 459, 488, 494, 496, 497, 502, 503, 506, 508, 509, 543, 570, 571, 573, 603

Offset: 1

Robert Israel, Aug 07 2024

Numbers k such that A000120(k) = A053735(k) + 1.

			a(3) = 15 is a term because 15 = 1111_2 = 120_3 so A000120(15) = 1+1+1+1 = 4 and A053735(15) = 1+2+0 = 3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000120, A037301, A053735, A180017.

filter:= proc(n) convert(convert(n,base,2),`+`) = convert(convert(n,base,3),`+`)+1 end proc:
select(filter, [$1..1000]);

Select[Range[600], Subtract @@ DigitSum[#, {2, 3}] == 1 &] (* Amiram Eldar, Aug 08 2024 *)

isok(k) = sumdigits(k,2) == 1 + sumdigits(k, 3); \\ Michel Marcus, Aug 08 2024

from sympy.ntheory import digits
def ok(n): return sum(digits(n, 2)[1:]) == sum(digits(n, 3)[1:]) + 1
print([k for k in range(604) if ok(k)]) # Michael S. Branicky, Aug 08 2024

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A037301 Numbers whose base-2 and base-3 expansions have the same digit sum.

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A180018 Difference of sums of digits of n in decimal and in binary representation.

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A180019 Difference of sums of digits of n in decimal and in ternary representation.

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A375257 Numbers whose sum of base-2 digits is 1 more than their sum of base-3 digits.

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