A037301 -id:A037301 - cp's OEIS frontend

A178620 Sum of binary digits ( = sum of ternary digits ) of terms in A037301.

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

Offset: 0

Zak Seidov, May 31 2010

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A037301 (sum of base 2 digits of n) = (sum of base 3 digits of n).

s = {}; Do[id3 = IntegerDigits[n, 3]; id2 = IntegerDigits[n, 2];
If[(t = Total[id2]) == Total[id3], AppendTo[s, t]], {n, 0, 10^3}]; s

A212222 Integers whose sum of digits in base b is the same for every prime b up to 11.

Original entry on oeis.org

0, 1, 1386, 1387, 485353, 981435, 4423035, 14187855, 19652536, 19652537, 19654636, 19654637, 23059876, 23059877, 23063359, 23837177, 25009516, 25009517, 41185278, 41185279, 41409018, 41409019, 49650315, 50262556, 50262557, 58622956, 58622957, 58623315

Offset: 1

Stanislav Sykora, May 06 2012

This sequence is a subsequence of A135127 for bases 2,3,5,7, which is a subsequence of A135121 for bases 2,3,5, which is a subsequence of A037301 for bases 2,3.

Problem: Is the sequence finite?

			1386 = 10101101010_2 = 1220100_3 = 21021_5 = 4020_7 = 1050_11. In each of these bases, the sum of the digits is 6.

Thomas König, Table of n, a(n) for n = 1..21843 (terms from n = 1..206 by Stanislav Sykora).
Thomas König, Fortran program for calculating terms

Cf. A135127, A135121, A037301.

Offset corrected by Thomas König, Aug 16 2020

Name edited by Michel Marcus, Aug 17 2020

A180017 Difference of sums of digits of n in ternary and in binary.

Original entry on oeis.org

0, 0, 1, -1, 1, 1, 0, 0, 3, -1, 0, 0, 0, 0, 1, -1, 3, 3, 0, 0, 2, 0, 1, 1, 2, 2, 3, -3, -1, -1, -2, -2, 3, 1, 2, 2, 0, 0, 1, -1, 2, 2, 1, 1, 3, -1, 0, 0, 2, 2, 3, 1, 3, 3, -2, -2, 1, -1, 0, 0, 0, 0, 1, -3, 3, 3, 2, 2, 4, 2, 3, 3, 2, 2, 3, 1, 3, 3, 2, 2, 6, -2, -1, -1, -1, -1, 0, -2, 1, 1, -2, -2, 0

Offset: 0

Reinhard Zumkeller, Aug 06 2010

This sequence is positive on average, since 1/log(3) > 1/log(4). Do all integers appear infinitely often? - Charles R Greathouse IV, Feb 07 2013

			For n = 7 = 21_3 = 111_2, a(n) = (2+1) - (1+1+1) = 0.
For n = 8 = 22_3 = 1000_2, a(n) = (2+2) - (1+0+0+0) = 3.
For n = 9 = 100_3 = 1001_2, a(n) = (1+0+0) - (1+0+0+1) = -1.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A180018, A180019, A007088, A007089.

Table[Total[IntegerDigits[n,3]]-Total[IntegerDigits[n,2]],{n,0,100}] (* Harvey P. Dale, Dec 08 2015 *)

a(n) = sumdigits(n,3) - sumdigits(n,2); \\ Michel Marcus, Nov 12 2023

a(n) = A053735(n) - A000120(n);
a(A037301(n)) = 0;
a(A000244(n)) = 1 - A000120(A000244(n));
a(A000079(n)) = A053735(A000079(n)) - 1;
a(A024023(n)) = 2*n - A000120(A024023(n)); a(A000225(n)) = A053735(A000225(n)) - n.
a(n) = A011371(n) - 2*A054861(n). - Henry Bottomley, Feb 16 2024

A317725 a(n) is the smallest k > 1 whose sum of digits is the same in all the bases from 2 to n.

Original entry on oeis.org

2, 6, 21, 23162843828305

Offset: 2

Giovanni Resta, Aug 05 2018

No other terms below 4^60 ~= 1.3*10^36. The sequence is likely finite and complete. - Max Alekseyev, Aug 29 2023

			a(3) = A037301(3).
a(4) = A135122(2).
a(5) = A317777(2) = 23162843828305 in base 2, 3, 4, and 5 is equal to: 101010001000100000101000101000001000001010001, 10001000100100102010101011001, 11101010011011001001101, and 11014000001010001210, respectively. The sum of the digits is 13 in the 4 cases.

Cf. A037301, A135122, A317777.

isok(k, n) = #Set(vector(n-1, b, b++; sumdigits(k, b))) == 1;
a(n) = my(k=2); while (!isok(k, n), k++); k; \\ Michel Marcus, Sep 02 2023

a(2) = 2 prepended by Max Alekseyev, Aug 29 2023

A330904 Numbers m such that the number of 1's in the binary expansion of m equals the sum of the balanced ternary trits of m.

Original entry on oeis.org

0, 1, 10, 12, 13, 34, 36, 37, 66, 67, 120, 121, 192, 193, 202, 264, 265, 272, 273, 282, 283, 354, 355, 360, 361, 514, 516, 517, 520, 526, 544, 576, 577, 688, 840, 841, 848, 849, 904, 928, 1026, 1027, 1028, 1029, 1032, 1033, 1038, 1039, 1062, 1063, 1074, 1075

Offset: 1

Thomas König, May 02 2020

If a(n) mod 6 = 0, then a(n+1) = a(n)+1.

a(41) = 1026, a(42) = 1027, a(43) = 1028 and a(44) = 1029 is the first time that four consecutive numbers appear in a(n). Conjecture: There is no occurrence of five or more consecutive numbers in a(n). Tested by exhaustive search up to 3^30. - Thomas König, Jul 19 2020

			34_10 = 11T1_bt = 10010_2, the sum of the digits is 1+1-1+1 = 2 for balanced ternary and 1+1 = 2 for base 2, so 34 is a term.

Thomas König, Table of n, a(n) for n = 1..20000
Thomas König, C program
Thomas König, Fortran program to test the conjecture that there are at most four consecutive numbers in sequence

Aside from the first term, subsequence of A174659.

Cf. A000120, A037301, A065363.

bt(n)= if (n==0, return (0)); my(d=digits(n, 3), c=1); while(c, if(d[1]==2, d=concat(0, d)); c=0; for(i=2, #d, if(d[i]==2, d[i]=-1; d[i-1]+=1; c=1))); vecsum(d); \\ A065363
isok(m) = bt(m) == hammingweight(m); \\ Michel Marcus, Jun 07 2020

Integers m such that A065363(m) = A000120(m).

Offset corrected by Thomas König, Jul 09 2020

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

A317777 Numbers whose digital sums in bases 2, 3, 4, and 5 are all equal.

Original entry on oeis.org

1, 23162843828305, 5722224662500629, 25185954575304707081301, 407805072367801818857674005, 1705412607407578552438012746487125, 1705412607426764386750185803694405, 1705412607426764386750185803695125, 1705412607426764386750186877112645, 1705412607431411795338502226662661

Offset: 1

Giovanni Resta, Aug 06 2018

Comment from Max Alekseyev, Dec 11 2024 (Start)
The beginning of the sequence suggests that the terms may be rather sparse.
However, as terms become larger, they tend to appear in unexpected constellations such as:
   89675268900935540640454882129695060
   89675268900935540640454882129695061
   89675268900935540640454933652788500
   89675268900935540640454933652788501
   89675273848747799396737823490250005
or
   113343080811409868866589092414267671876
   113343080811409868866589092414267671877
   113343080811409868867743138448975156500
   113343080811409868867743138448975156501
   113343080811409868867760032719746187600
   113343080811409868867760032719746187601
   113343080811409868867760239410949735765
   113343080811409868867761158326773437508
   113343080811409868867761158326773437509
I can explain when two terms differ by 1, where the first one happens to end with a digit < b-1 for each base b in {2,3,4,5}, but the above constellations are a total mystery.
(End)

			23162843828305 in bases 2, 3, 4, and 5 is equal to 101010001000100000101000101000001000001010001, 10001000100100102010101011001, 11101010011011001001101, and 11014000001010001210, respectively. The sum of the digits is 13 in all four cases.

Max Alekseyev, Table of n, a(n) for n = 1..28

Subsequence of A037301 and A135122.

Cf. A317725.

Terms a(5) onward from Max Alekseyev, Sep 09 2023

A334765 Numbers m such that the numbers of 1's in the binary expansion of m equals the negative sum of balanced ternary trits of m.

Original entry on oeis.org

0, 41, 68, 131, 132, 368, 384, 528, 1095, 1098, 1100, 1106, 1112, 1122, 1124, 1152, 1176, 1346, 1824, 2561, 3282, 3284, 3336, 3344, 3392, 3524, 4098, 4101, 4104, 4112, 4118, 4128, 4172, 4352, 4496, 4739, 4740, 5504, 6224, 9856, 9857, 9869, 9896, 9923, 9924

Offset: 1

Thomas König, May 10 2020

a(116) = 32770, a(117) = 32771 and a(118) = 32772 is the first time that three consecutive numbers appear in this sequence. Conjecture: There is no occurrence of four or more consecutive numbers. Tested by exhaustive search up to a(n) = 3^26. - Thomas König, Jul 19 2020

			41_10 = 1TTTT_bt = 101001_2, the sum of the digits is 1-1-1-1-1 = -3 for balanced ternary and 1+1+1 = 3 for base 2, so 41 is a term.

Thomas König, Table of n, a(n) for n = 1..30000

Aside from the first term, subsequence of A174657.

Cf. A000120, A037301, A065363, A330904.

Select[Range[0, 10^4], -Total@ If[First@ # == 0, Rest@ #, #] &[Prepend[IntegerDigits[#, 3], 0] //. {x___, y_, k_ /; k > 1, z___} :> {x, y + 1, -1, z}] == DigitCount[#, 2, 1] &] (* Michael De Vlieger, Jul 08 2020 *)

bt(n)= if (n==0, return (0)); my(d=digits(n, 3), c=1); while(c, if(d[1]==2, d=concat(0, d)); c=0; for(i=2, #d, if(d[i]==2, d[i]=-1; d[i-1]+=1; c=1))); vecsum(d); \\ A065363
isok(m) = bt(m) + hammingweight(m) == 0; \\ Michel Marcus, Jun 07 2020

Integers m such that -A065363(m) = A000120(m).

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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A178620 Sum of binary digits ( = sum of ternary digits ) of terms in A037301.

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A212222 Integers whose sum of digits in base b is the same for every prime b up to 11.

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A180017 Difference of sums of digits of n in ternary and in binary.

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A317725 a(n) is the smallest k > 1 whose sum of digits is the same in all the bases from 2 to n.

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A330904 Numbers m such that the number of 1's in the binary expansion of m equals the sum of the balanced ternary trits of m.

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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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A317777 Numbers whose digital sums in bases 2, 3, 4, and 5 are all equal.

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