A212222 -id:A212222 - 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

A135121 Numbers such that the digital sum base 2 and the digital sum base 3 and the digital sum base 5 all are equal.

Original entry on oeis.org

0, 1, 6, 7, 10, 11, 60, 61, 180, 181, 285, 300, 301, 575, 687, 754, 826, 827, 882, 883, 900, 901, 910, 911, 1254, 1305, 1311, 1326, 1327, 1335, 1377, 1383, 1386, 1387, 1395, 1431, 1506, 1507, 1532, 1626, 1627, 1650, 1651, 1890, 1891, 1955, 2013, 2036, 2040

Offset: 1

Hieronymus Fischer, Dec 31 2007

			a(2)=6, since ds_2(6)=ds_3(6)=ds_5(6), where ds_x=digital sum base x.

Stanislav Sykora, Table of n, a(n) for n = 1..10000

Cf. A007953, A054899, A131451, A133620, A133900, A134599, A135100, A135110, A135120, A037308, A212222.

Select[Range[0,3000],Length[Union[Total/@IntegerDigits[#,{2,3,5}]]]==1&] (* Harvey P. Dale, Sep 04 2014 *)

Added 0, Stanislav Sykora, May 06 2012

A135127 Numbers such that the digital sums in bases 2, 3, 5 and 7 all are equal.

Original entry on oeis.org

0, 1, 882, 883, 1386, 1387, 2502, 2503, 3453, 7555, 7652, 7665, 7931, 9751, 10101, 12250, 12251, 16893, 17010, 17011, 17515, 17550, 17551, 18285, 20301, 22050, 22051, 24406, 24407, 25053, 27503, 31654, 40930, 40931, 41951, 50878, 50879

Offset: 1

Hieronymus Fischer, Dec 31 2007

			a(2)=882, since ds_2(882 )=ds_3(882 )=ds_5(882 )=ds_7(882 )=6, where ds_x=digital sum base x.

Stanislav Sykora, Table of n, a(n) for n = 1..10000

Cf. A007953, A054899, A131451, A133620, A133900, A134599, A135100, A135110, A135120, A037308, A212222.

Select[Range[0, 32000], Total[IntegerDigits[#, 2]] == Total[IntegerDigits[#, 3]] == Total[IntegerDigits[#, 5]] == Total[IntegerDigits[#, 7]] &] (* G. C. Greubel, Sep 27 2016 *)
Select[Range[0,51000],Length[Union[Total/@IntegerDigits[#,{2,3,5,7}]]] == 1&] (* Harvey P. Dale, Sep 18 2019 *)

Added 0, Stanislav Sykora, May 06 2012

A335839 Integers whose sum of digits in base b is the same for every prime b up to 13.

Original entry on oeis.org

0, 1, 2007986541, 2834822783, 31939595966, 33952616126, 42737313983, 44878987167, 309231463167, 318362221465, 415332522143, 881935644447, 1898245489647, 2077690289610, 2077690289611, 2153926044391, 3998461033469, 4285034622330, 4285034622331, 4294899857375

Offset: 1

Thomas König, Sep 13 2020

This is a subset of A212222 for bases 2, 3, 5, 7, 11, which is a subset of A135127 for bases 2, 3, 5, 7, which is a subset of A135121 for bases 2 ,3, 5, which is a subset of A037301 for bases 2, 3. The third term also occurs in A212223.

			31939595966 is 11101101111101111111000111010111110_2, 10001102220222120211202_3, 1010403014032331_5, 2210331041405_7, 12600084203_11 and 3020180615_13. In these bases, the sum of digits is 26, so 31939595966 is a term.

Thomas König, Table of n, a(n) for n = 1..19840

Cf. A212223, A212222, A135127, A135121, A037301.

def digsum(n,b):
    s = 0
    while n > 0:
        n, d = n//b, n%b
        s = s+d
    return s
p = [2,3,5,7,11,13]
n, a = 0, 0
while n <= 20:
    s2, i = digsum(a,2), 1
    while i < len(p) and digsum(a,p[i]) == s2:
        i = i+1
    if i == len(p):
        print(a, end = ", ")
        n = n+1
    a = a+1 # A.H.M. Smeets, May 16 2021

A212223 a(n) is the least integer greater than 1 whose expansion in prime bases 2 to prime(n) have equal sum of digits.

Original entry on oeis.org

2, 6, 6, 882, 1386, 2007986541, 70911040973874056146188543

Offset: 1

Stanislav Sykora, May 10 2012

Case a(1) is trivial since only base prime(1)=2 is involved.
Conjecture: the sequence never terminates.
a(7) > 2.3*10^16, if it exists. - Giovanni Resta, Oct 29 2018
Based on a search for the next term of A345296, a(8) is larger than 2.1*10^28. - Thomas König, Dec 15 2024

			a(5) = 1386 because that number has the same sum of digits in the first 5 prime bases 2, 3, 5, 7, 11 (see A212222 and A000040).

Cf. A000040, A212222, A135127, A135121, A037301, A335839, A345296.

f[n_] := Block[{p = Prime@ Range@ n, k = 2}, While[ Length[ Union[ Total@# & /@ IntegerDigits[k, p]]] != 1, k++]; k] (* Robert G. Wilson v, Oct 24 2014 *)

isok(n, k) = my(s=hammingweight(k)); forprime (b=3, prime(n), if (sumdigits(k, b) != s, return (0))); return (1);
a(n) = my(k=2); while (!isok(n, k), k++); k; \\ Michel Marcus, Jun 08 2021

Name edited by Michel Marcus, Sep 14 2020

a(7) from Thomas König, Jun 08 2021

A212283 First a(n) > 1 whose sum of digits is the same in base 2 as in base n.

Original entry on oeis.org

2, 6, 4, 6, 12, 21, 8, 10, 20, 12, 14, 172, 30, 46, 16, 18, 36, 20, 22, 126, 46, 24, 26, 126, 28, 30, 58, 60, 120, 126, 32, 34, 68, 36, 38, 185, 78, 40, 42, 126, 44, 46, 90, 92, 138, 48, 50, 246, 52, 54, 106, 108, 56, 58, 114, 60, 62, 120, 182, 126, 188, 378

Offset: 2

Stanislav Sykora, May 08 2012

Theoretically, there might exist an n for which there is no solution, in which case a(n) would be set to 0 by convention; however, no such case was found so far. Problem: does it exist?

			Example: a(13) = 172 because 172 is the first number >1 such that its expansions in base 2 (10101100) and in base 13 (103) have the same sum of digits, namely 4.

Stanislav Sykora, Table of n, a(n) for n = 2..10000

Cf. A037301, A212222.

sdn[n_]:=Module[{a=2},While[Total[IntegerDigits[a,2]]!=Total[ IntegerDigits[ a,n]], a++];a]; Array[sdn,70,2] (* Harvey P. Dale, May 29 2013 *)

A345296 Integers whose sum of digits in base b is the same for every prime b up to 17.

Original entry on oeis.org

0, 1, 70911040973874056146188543, 77332999599545910254098143, 139857575920160383360253101

Offset: 1

Thomas König, Jun 13 2021

This is a subset of A335839 for bases 2,3,5,11,13, which is a subset of A212222 for bases 2, 3, 5, 7, 11, which is a subset of A135127 for bases 2, 3, 5, 7, which is a subset of A135121 for bases 2, 3, 5, which is a subset of A037301 for bases 2, 3. The third term also occurs in A212223.

Based on a computer search, the next term is believed to be larger than 2.1e28. - Thomas König, Dec 08 2024

			77332999599545910254098143 = 11111111110111111001100100111011111011111110101111110010001010111101111101011011011111_2 =
1022220111022022121010102021222111100222120112011112120_3 = 10124120314223101043140143200022120033_5 = 3300561310042202241132326120022_7 = 7940063801000011830000282_11 = 1B101304100834600A304201_13 = 120802053643008116067_17. In these bases, the sum of digits is 63, so 77332999599545910254098143 is a term.

Cf. A212223, A212222, A135127, A135121, A037301, A335839.

a(5) from Thomas König, Dec 08 2024

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

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A135121 Numbers such that the digital sum base 2 and the digital sum base 3 and the digital sum base 5 all are equal.

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A135127 Numbers such that the digital sums in bases 2, 3, 5 and 7 all are equal.

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A335839 Integers whose sum of digits in base b is the same for every prime b up to 13.

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A212223 a(n) is the least integer greater than 1 whose expansion in prime bases 2 to prime(n) have equal sum of digits.

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A212283 First a(n) > 1 whose sum of digits is the same in base 2 as in base n.

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A345296 Integers whose sum of digits in base b is the same for every prime b up to 17.

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