A037308 -id:A037308 - cp's OEIS frontend

A152206 a(n) = sum of base-2 digits of A037308(n) = sum of base-10 digits of A037308(n).

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

Offset: 0

Zak Seidov, Nov 29 2008

Cf. A037308, Sum of base-2 digits of n = sum of base-10 digits of n.

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

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.

A135122 Numbers such that the digital sum base 2 and the digital sum base 3 and the digital sum base 4 all are equal.

Original entry on oeis.org

1, 21, 261, 273, 17748, 17749, 20820, 20821, 65620, 65621, 70740, 70741, 83268, 83269, 86292, 86293, 1066068, 1066069, 1070420, 1135701, 1135893, 1135953, 5326161, 5330001, 5330241, 5330260, 5330261, 5506389, 5525829, 5526801, 5571909, 5574933, 5592321

Offset: 1

Hieronymus Fischer, Dec 31 2007

			a(2)=21, since ds_2(21)=ds_3(21)=ds_10(21)=3, where ds_x=digital sum base x.

Giovanni Resta, Table of n, a(n) for n = 1..10000

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

Select[Range[5600000],Length[Union[Table[Total[IntegerDigits[#,n]],{n,2,4}]]]==1&] (* Harvey P. Dale, Aug 14 2013 *)

isok(n) = my(sd2=sumdigits(n, 2)); (sd2==sumdigits(n, 3)) && (sd2==sumdigits(n, 4)); \\ Michel Marcus, Aug 08 2018

a(31)-a(33) from Giovanni Resta, Aug 06 2018

A108580 Numbers whose sum of bits when written in binary > sum of decimal digits.

Original entry on oeis.org

10, 11, 30, 31, 100, 101, 102, 103, 110, 111, 120, 121, 200, 201, 210, 211, 220, 221, 300, 301, 310, 311, 500, 501, 510, 511, 1000, 1001, 1002, 1003, 1004, 1005, 1006, 1007, 1010, 1011, 1012, 1013, 1014, 1015, 1020, 1021, 1022, 1023, 1100, 1101, 1102, 1103

Offset: 1

John L. Drost, Jul 25 2005

The sequence is infinite because 10^n, n = 1,2,3, .... are terms. - Marius A. Burtea, Sep 07 2019

			1103 is on the list since 1103 is 10001001111 (sum=6), 1+1+0+3=5.

Marius A. Burtea, Table of n, a(n) for n = 1..5000

Subsequence of A325483.

Cf. A000120, A007953, A037308.

[n:n in [1..1200]| &+Intseq(n,2) gt &+Intseq(n,10)];  // Marius A. Burtea, Sep 07 2019

Select[Range[1103], Total@IntegerDigits[#, 2] > Total@IntegerDigits[#, 10] &] (* Amiram Eldar, Sep 07 2019 *)

isok(n) = hammingweight(n) > sumdigits(n); \\ Michel Marcus, Sep 07 2019

def ok(n): return sum(map(int, str(n))) < bin(n).count('1')
print(list(filter(ok, range(1104)))) # Michael S. Branicky, Oct 11 2021

a(36) added by Marius A. Burtea, Sep 07 2019

A325483 Numbers whose sum of their decimal digits is less than or equal to the sum of the digits of their binary representation.

Original entry on oeis.org

0, 1, 10, 11, 20, 21, 30, 31, 100, 101, 102, 103, 110, 111, 120, 121, 122, 123, 200, 201, 202, 203, 210, 211, 220, 221, 222, 223, 230, 231, 300, 301, 302, 303, 310, 311, 410, 411, 500, 501, 502, 503, 510, 511, 1000, 1001, 1002, 1003, 1004, 1005, 1006, 1007

Offset: 1

Frederick John Bernet III, Sep 06 2019

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Supersequence of A037308 and A108580.

Cf. A000120, A007953.

q:= n-> (f-> is(f(n, 10)<=f(n, 2)))((x, b)
          -> add(i, i=convert(x, base, b))):
select(q, [$0..1500])[];  # Alois P. Heinz, Sep 06 2019

Select[Range[0,1007], Total[IntegerDigits[#]]<=Total[IntegerDigits[#,2]]&] (* Metin Sariyar, Sep 14 2019 *)

isok(n) = sumdigits(n, 10) <= sumdigits(n, 2); \\ Michel Marcus, Sep 07 2019

x=0
#Adjust the inequality below to generate more numbers of the sequence
while(x<100):
    x = x+1
    Number = int(bin(x)[2:])
    Bin_Sum = 0
    while(Number > 0):
        Reminder = Number % 10
        Bin_Sum = Bin_Sum + Reminder
        Number = Number //10
    Number = x
    Sum = 0
    while(Number > 0):
        Reminder = Number % 10
        Sum = Sum + Reminder
        Number = Number //10
    if (Sum <= Bin_Sum):
        print(x)

def ok(n): return sum(map(int, str(n))) <= bin(n).count('1')
print(list(filter(ok, range(1008)))) # Michael S. Branicky, Oct 11 2021

{ A037308 } union { A108580 }.

A135123 Numbers such that the digital sum base 2 and the digital sum base 3 and the digital sum base 6 all are equal.

Original entry on oeis.org

1, 12, 13, 114, 115, 366, 367, 477, 687, 864, 865, 876, 877, 1086, 1087, 1305, 1326, 1327, 1386, 1387, 1596, 1597, 1626, 1627, 1656, 1657, 1746, 1747, 1836, 1837, 1956, 1957, 2595, 2607, 2646, 2647, 3276, 3277, 3906, 3907, 3948, 3949, 4068, 4069, 5438

Offset: 1

Hieronymus Fischer, Dec 31 2007

			a(2)=12, since ds_2(12)=ds_3(12)=ds_6(12), where ds_x=digital sum base x.

G. C. Greubel, Table of n, a(n) for n = 1..1200

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

Select[Range[5000], Total[IntegerDigits[#, 2]] == Total[IntegerDigits[#, 3]] == Total[IntegerDigits[#, 6]] &] (* G. C. Greubel, Sep 26 2016 *)

A135124 Numbers such that the digital sums in base 2, base 4 and base 8 are all equal.

Original entry on oeis.org

1, 64, 65, 4096, 4097, 4160, 4161, 262144, 262145, 262208, 262209, 266240, 266241, 266304, 266305, 16777216, 16777217, 16777280, 16777281, 16781312, 16781313, 16781376, 16781377, 17039360, 17039361, 17039424, 17039425, 17043456

Offset: 1

Hieronymus Fischer, Dec 31 2007, Dec 31 2008

Written as base 64 numbers the sequence is 1,10,11,100,101,110,111,1000,1001, ... (cf. A007088)

			a(7)=4161, since ds_2(4161 )=ds_4(4161 )=ds_8(4161 ), where ds_x=digital sum base x.

Michael De Vlieger, Table of n, a(n) for n = 1..8192
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 45.

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

Cf. A000695, A033045.

Select[Range[500000], Total[IntegerDigits[#, 2]] == Total[IntegerDigits[#, 4]] == Total[IntegerDigits[#, 8]] &] (* G. C. Greubel, Sep 26 2016 *)
With[{k = 64}, Rest@ Map[FromDigits[#, k] &, Tuples[{0, 1}, 5]]] (* Michael De Vlieger, Oct 28 2022 *)
Select[Range[171*10^5],Length[Union[Total/@IntegerDigits[#,{2,4,8}]]]==1&] (* Harvey P. Dale, May 14 2025 *)

a(n) = fromdigits(binary(n),64); \\ Kevin Ryde, Apr 02 2025

a(n) = (1/2)*Sum_{k=0..floor(log_2(n))} (1-(-1)^floor(n/2^k))*64^k.
G.f.: (1/(1-x))*Sum_{k>=0} 64^k*x^(2^k)/(1+x^(2^k)).
a(n) = A000695(A033045(n)) = A033045(A000695(n)). - Alan Michael Gómez Calderón, Mar 27 2025

Edited by N. J. A. Sloane, Jan 17 2009

A135125 Numbers such that the digital sum base 2 and the digital sum base 5 and the digital sum base 10 all are equal.

Original entry on oeis.org

1, 1300, 1301, 5010, 5011, 7102, 7103, 10050, 10051, 10235, 11135, 12250, 12251, 14015, 16102, 16103, 20060, 20061, 20206, 20207, 23230, 23231, 32012, 32013, 32302, 32303, 32410, 32411, 44000, 44001, 45010, 45011, 50012, 50013, 50300

Offset: 1

Hieronymus Fischer, Dec 31 2007

			a(2)=1300, since ds_2(1300)=ds_5(1300)=ds_10(1300), where ds_x=digital sum base x.

G. C. Greubel, Table of n, a(n) for n = 1..989

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

Select[Range[10000], Total[IntegerDigits[#, 2]] == Total[IntegerDigits[#, 5]] == Total[IntegerDigits[#, 10]] &] (* G. C. Greubel, Sep 27 2016 *)

cp's OEIS Frontend

A152206 a(n) = sum of base-2 digits of A037308(n) = sum of base-10 digits of A037308(n).

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

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A135122 Numbers such that the digital sum base 2 and the digital sum base 3 and the digital sum base 4 all are equal.

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A108580 Numbers whose sum of bits when written in binary > sum of decimal digits.

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Magma

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A325483 Numbers whose sum of their decimal digits is less than or equal to the sum of the digits of their binary representation.

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Maple

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A135123 Numbers such that the digital sum base 2 and the digital sum base 3 and the digital sum base 6 all are equal.

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