A134599 -id:A134599 - cp's OEIS frontend

A037308 Numbers whose base-2 and base-10 expansions have the same digit sum.

0, 1, 20, 21, 122, 123, 202, 203, 222, 223, 230, 231, 302, 303, 410, 411, 502, 503, 1130, 1131, 1150, 1151, 1202, 1203, 1212, 1213, 1230, 1231, 1300, 1301, 1402, 1403, 1502, 1503, 1510, 1511, 2006, 2007, 2032, 2033, 2102, 2103, 2200, 2201, 3006, 3007, 3012

Offset: 1

Clark Kimberling

n is in the sequence iff n+(-1)^n is in the sequence. [Robert Israel, Mar 25 2013]

			122 is a member, since digital-sum_2(122) = 5 = digital-sum_10(122).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000040, A000120, A007953, A054899, A131451, A133620, A133900, A134599, A135110, A180018.

N:= 10000; # to get all elements up to N
select(x -> (convert(convert(x,base,10),`+`)-convert(convert(x,base,2),`+`)=0), [$0..N]); # Robert Israel, Mar 25 2013

Select[Range[0, 5000], Total[IntegerDigits[#, 2]] == Total[IntegerDigits[#, 10]] &] (* Jean-François Alcover, Mar 07 2016 *)

is(n)=hammingweight(n)==sumdigits(n); \\ Charles R Greathouse IV, Sep 25 2012

def ok(n): return sum(map(int, str(n))) == sum(map(int, bin(n)[2:]))
print(list(filter(ok, range(3013)))) # Michael S. Branicky, Jun 20 2021

[n for n in (0..10000) if sum(n.digits(base=2)) == sum(n.digits(base=10))] # Freddy Barrera, Oct 12 2018

From Reinhard Zumkeller, Aug 06 2010: (Start)
A007953(a(n)) = A000120(a(n));
A180018(a(n)) = 0. (End)

Edited by N. J. A. Sloane Nov 29 2008 at the suggestion of Zak Seidov

A135120 Numbers such that the digital sum base 2 and the digital sum base 3 and the digital sum base 10 all are equal.

Original entry on oeis.org

1, 21, 222, 223, 1230, 1231, 1502, 2200, 2201, 3012, 3013, 10431, 12214, 12215, 12250, 12251, 14102, 15003, 15021, 16011, 20040, 20041, 22130, 23211, 23230, 23231, 24003, 30070, 30071, 30105, 30231, 30321, 31005, 31150, 31151, 31420

Offset: 1

Hieronymus Fischer, Dec 24 2007

			a(2)=21, since ds_2(21)=ds_3(21)=ds_10(21)=3.

G. C. Greubel, Table of n, a(n) for n = 1..1100

Cf. A000040, A007953, A054899, A131451, A133620, A133900, A134599, A135110.

Select[Range[5000], Total[IntegerDigits[#, 2]] == Total[IntegerDigits[#, 3]] ==  Total[IntegerDigits[#, 10]] &] (* G. C. Greubel, Sep 26 2016 *)

is(n)=my(t=sumdigits(n)); t==hammingweight(n) && t==sumdigits(n,3) \\ Charles R Greathouse IV, Sep 26 2016

A135100 Numbers which divide their digital sumorial (see A131383).

Original entry on oeis.org

1, 3, 4, 15, 26, 2573, 17226, 19786, 22083, 58133, 67693, 223657, 376460, 464713, 497068, 2621204, 4553376, 6000136, 7671158, 13975944, 14074903, 52731198, 82594577

Offset: 1

Hieronymus Fischer, Dec 24 2007

			a(5)=26, since 26 divides its digital sumorial, which is A131383(26)=182.

Cf. A007953, A054899, A131383, A131451, A133620, A133900, A134599.

a=1;for(n=2,10^6,if(a%(n-1)==0,print1(n-1","));x=divisors(n);L=numdiv(n);a+=n; for(i=2,L-1,d=x[i];k=n;while(k%d==0,a-=d-1;k\=d))) \\ Robert Gerbicz, May 09 2008

a(12)-a(15) from Robert Gerbicz, May 09 2008

a(16)-a(23) from Hieronymus Fischer, Jul 31 2008

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

A080773 In binary representation: sum of number of 1's in prime factors of n (with repetition).

Original entry on oeis.org

0, 1, 2, 2, 2, 3, 3, 3, 4, 3, 3, 4, 3, 4, 4, 4, 2, 5, 3, 4, 5, 4, 4, 5, 4, 4, 6, 5, 4, 5, 5, 5, 5, 3, 5, 6, 3, 4, 5, 5, 3, 6, 4, 5, 6, 5, 5, 6, 6, 5, 4, 5, 4, 7, 5, 6, 5, 5, 5, 6, 5, 6, 7, 6, 5, 6, 3, 4, 6, 6, 4, 7, 3, 4, 6, 5, 6, 6, 5, 6, 8, 4, 4, 7, 4, 5, 6, 6, 4, 7, 6, 6, 7, 6, 5, 7, 3, 7, 7, 6, 4, 5, 5, 6, 7

Offset: 1

Reinhard Zumkeller, Mar 10 2003

			a(20) = a(2*2*5) = a('10' * '10' * '101') = 1+1+2 = 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000120, A007088, A134599 (base 3).

sn1[n_]:=Total[Flatten[IntegerDigits[#,2]&/@Flatten[Table[#[[1]],{#[[2]]}] &/@ FactorInteger[n]]]]; Join[{0},Rest[Array[sn1,110]]] (* Harvey P. Dale, Nov 19 2013 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, hammingweight(f[k,1])*f[k,2]); \\ Michel Marcus, Aug 28 2019

Totally additive with a(p) = A000120(p). - Amiram Eldar, Jul 30 2025

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

A135101 Digital sum (base the n-th prime) of n^n.

Original entry on oeis.org

1, 2, 3, 10, 15, 24, 55, 46, 71, 116, 101, 180, 213, 196, 205, 276, 307, 444, 337, 610, 621, 646, 687, 808, 985, 876, 921, 996, 1049, 1184, 1417, 1576, 1665, 1576, 2127, 1836, 2377, 1660, 2201, 2088, 2731, 2844, 2847, 2944, 3317, 3232, 3503, 3294, 3165

Offset: 1

Hieronymus Fischer, Dec 24 2007

			a(2) = ds_prime(2)(2^2) = ds_3(4) = 1+1 = 2;
a(10) = ds_prime(5)(5^5) = ds_11(3125) = 2+3+9+1 = 15.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000040, A000045, A007953, A054899, A075771, A131451, A133620, A133900, A134599.

Table[Total[IntegerDigits[n^n, Prime[n]]], {n, 50}] (* G. C. Greubel, Sep 23 2016 *)

a(n) = vecsum(digits(n^n, prime(n))); \\ Michel Marcus, Sep 24 2016

a(n) = ds_prime(n)(n^n), where ds_prime(n) = digital sum base the n-th prime.

a(n) = n^n - (prime(n)-1)*Sum{k>0} ( floor(n^n/prime(n)^k) ).

A135102 Digital sum (base the n-th prime) of Fibonacci(n).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 3, 12, 27, 29, 36, 33, 41, 58, 51, 31, 64, 89, 45, 74, 83, 39, 168, 145, 193, 170, 129, 149, 104, 211, 289, 274, 175, 257, 252, 125, 161, 318, 347, 447, 316, 317, 285, 450, 107, 253, 648, 363, 301, 498, 409, 773, 522, 429, 515, 782, 649, 641

Offset: 1

Hieronymus Fischer, Dec 24 2007

			a(2) = ds_prime(2)(Fib(2)) = ds_3(1) = 1;
a(10) = ds_prime(10)(55) = ds_29(55) = 1+26 = 27.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000040, A000045, A007953, A054899, A075771, A131451, A133620, A133900, A134599.

Table[Total[IntegerDigits[Fibonacci[n],Prime[n]]],{n,60}] (* Harvey P. Dale, May 05 2013 *)

a(n) = vecsum(digits(fibonacci(n), prime(n))); \\ Michel Marcus, Sep 24 2016

a(n) = ds_prime(n)(Fib(n)), where ds_prime(n) = digital sum base the n-th prime.

a(n) = Fibonacci(n) - (prime(n)-1)*Sum{k>0} ( floor(Fibonacci(n)/prime(n)^k) ).

A135103 Digital sum (base the n-th prime) of n^3.

Original entry on oeis.org

1, 4, 3, 4, 5, 12, 7, 26, 25, 20, 41, 36, 37, 56, 63, 40, 41, 72, 61, 90, 117, 118, 113, 96, 73, 76, 99, 116, 89, 120, 181, 138, 169, 112, 251, 156, 109, 116, 57, 188, 35, 108, 87, 128, 181, 118, 83, 258, 129, 284, 179, 188, 317, 214, 231, 338, 273, 442, 311, 400, 253

Offset: 1

Hieronymus Fischer, Dec 24 2007

			a(2)=ds_prime(2)(2^3)=ds_3(8)=2+2=4; a(6)=ds_prime(6)(6^3)=ds_13(216)=1+3+8=12.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A007953, A054899, A075771, A131451, A133620, A133900, A134599.

Table[Total[IntegerDigits[n^3,Prime[n]]],{n,70}] (* Harvey P. Dale, Oct 21 2011 *)

a(n)=ds_prime(n)(n^3), where ds_prime(n)=digital sum base the n-th prime.

a(n)=n^3-(prime(n)-1)*sum{k>0, floor(n^3/prime(n)^k)}.

cp's OEIS Frontend

A037308 Numbers whose base-2 and base-10 expansions have the same digit sum.

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A135120 Numbers such that the digital sum base 2 and the digital sum base 3 and the digital sum base 10 all are equal.

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A135100 Numbers which divide their digital sumorial (see A131383).

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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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A080773 In binary representation: sum of number of 1's in prime factors of n (with repetition).

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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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A135101 Digital sum (base the n-th prime) of n^n.

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