A037308 -id:A037308 - cp's OEIS frontend

A135126 Numbers such that the digital sums in bases 3, 4, 5 and 6 all are equal.

1, 2, 188, 668, 908, 1388, 1628, 2170, 2171, 2830, 2831, 3908, 4330, 4331, 6490, 6491, 8650, 8651, 10390, 10391, 10629, 12792, 12793, 12794, 17110, 17111, 17290, 17291, 25930, 25931, 36312, 36313, 36314, 37812, 37813, 37814, 41532, 41533, 41534

Offset: 1

Hieronymus Fischer, Dec 31 2007

			a(3)=188, since ds_3(188)=ds_4(188)=ds_5(188)=ds_6(188)=8, where ds_x=digital sum base x.

G. C. Greubel, Table of n, a(n) for n = 1..1400

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

Select[Range[3000], Total[IntegerDigits[#, 3]] == Total[IntegerDigits[#, 4]] ==  Total[IntegerDigits[#, 5]] == Total[IntegerDigits[#, 6]] &] (* G. C. Greubel, Sep 27 2016 *)

A135128 Numbers such that the digital sums in bases 2, 3, 5 and 10 all are equal.

Original entry on oeis.org

1, 12250, 12251, 23230, 23231, 32410, 32411, 45010, 45011, 51130, 51131, 52030, 52031, 54010, 54011, 100053, 100090, 100091, 100305, 102250, 102251, 107002, 107003, 110134, 110170, 110171, 110350, 110351, 110460, 110461, 113050, 113051

Offset: 1

Hieronymus Fischer, Dec 31 2007

			a(2)=12250 since ds_2(12250 )=ds_3(12250 )=ds_5(12250 )=ds_10(12250 )=10, where ds_x=digital sum base x.

G. C. Greubel, Table of n, a(n) for n = 1..700

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

Select[Range[32000], Total[IntegerDigits[#, 2]] == Total[IntegerDigits[#, 3]] == Total[IntegerDigits[#, 5]] == Total[IntegerDigits[#, 10]] &] (* G. C. Greubel, Sep 28 2016 *)
Select[Range[120000],Length[Union[Total/@IntegerDigits[#,{2,3,5,10}]]]==1&] (* Harvey P. Dale, Mar 30 2024 *)

A135129 Numbers such that the digital sums in bases 3, 4, 5, 6 and 7 all are equal.

Original entry on oeis.org

1, 2, 1388, 2170, 2171, 2830, 2831, 10390, 10391, 12792, 12793, 12794, 17110, 17111, 17290, 17291, 36312, 36313, 36314, 37814, 41532, 41533, 41534, 50892, 50893, 50894, 52216, 52217, 52395, 56652, 56653, 56654, 95354, 96432, 96433, 96434

Offset: 1

Hieronymus Fischer, Dec 31 2007

			a(3)=1388, since ds_3(1388)=ds_4(1388)=ds_5(1388)=ds_6(1388)=ds_7(1388), where ds_x=digital sum base x.

G. C. Greubel, Table of n, a(n) for n = 1..1700

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

Select[Range[30000], Total[IntegerDigits[#, 3]] == Total[IntegerDigits[#, 4]] == Total[IntegerDigits[#, 5]] == Total[IntegerDigits[#, 6]] == Total[IntegerDigits[#, 7]] &] (* G. C. Greubel, Sep 28 2016 *)
Select[Range[100000],Length[Union[Table[Total[IntegerDigits[#,b]],{b,3,7}]]]==1&] (* Harvey P. Dale, Oct 25 2020 *)

A152207 Numbers k such that (sum of base-2 digits of k) = (sum of base-10 digits of k) = 10.

Original entry on oeis.org

3007, 3070, 4015, 6013, 6103, 7102, 8110, 10171, 10234, 11071, 11134, 11215, 11251, 11260, 11503, 11710, 12007, 12025, 12142, 12205, 12214, 12250, 13051, 13231, 14014, 15031, 15211, 15310, 16030, 16102, 16120, 16300, 20143, 20206, 20341

Offset: 1

Zak Seidov, Nov 29 2008

			a(1000) = 3002410_10 = 1011011101000000101010_2, with digit sums = 10 in both cases.

Zak Seidov, Table of n, a (n) for n = 1 .. 1000

Cf. A152206, A037308.

isok(n) = (sumdigits(n) == 10) && (hammingweight(n) == 10); \\ Michel Marcus, Oct 15 2013

A212284 First a(n) > 1 whose sum of digits is the same in base 10 as in base n.

Original entry on oeis.org

20, 21, 12, 40, 50, 21, 70, 153, 10, 190, 108, 40, 126, 135, 50, 153, 162, 20, 180, 190, 70, 207, 216, 80, 234, 243, 30, 261, 270, 190, 290, 594, 102, 315, 324, 40, 342, 351, 120, 370, 380, 130, 792, 405, 50, 423, 432, 150, 450, 460, 160, 480, 490, 60, 504

Offset: 2

Stanislav Sykora, May 08 2012

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?

			a(12)=108 because 108 is the first number > 1 such that when written in base 10 and in base 12 (i.e., 90), the sum of the expansion digits is the same, namely 9.

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

Cf. A037308.

A224077 The smallest number n such that (sum of base 2 digits of n) = (sum of base 10 digits of n) for four consecutive integers.

Original entry on oeis.org

3013118, 12101118, 100503038, 100600318, 110231038, 123000318, 131013118, 134213118, 211013118, 222021118, 301002238, 310213118, 331000318, 332101118, 411031038, 501020158, 501112318, 510021118, 511301118, 520005118, 700101118, 1001003518, 1001500158

Offset: 1

Reiner Moewald, Mar 30 2013

If A037308 is split up into maximal subsets of consecutive integers, the size of these subsets is either two or four.

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A037308.

A305493 A binary encoding of the decimal representation of a number: for any number n >= 0, consider its decimal representation and replace each 9 with "111111111" and each other digit d with a "0" followed by d "1"s and interpret the result as a binary string.

Original entry on oeis.org

0, 1, 3, 7, 15, 31, 63, 127, 255, 511, 2, 5, 11, 23, 47, 95, 191, 383, 767, 1023, 6, 13, 27, 55, 111, 223, 447, 895, 1791, 2047, 14, 29, 59, 119, 239, 479, 959, 1919, 3839, 4095, 30, 61, 123, 247, 495, 991, 1983, 3967, 7935, 8191, 62, 125, 251, 503, 1007, 2015

Offset: 0

Rémy Sigrist, Jun 02 2018

This sequence is a permutation of the nonnegative numbers.
The inverse sequence, say b, satisfies b(n) = A001202(n+1) for n = 0..1022, but b(1023) = 19 whereas A001202(1024) = 10.
The first known fixed points are: 0, 1, 65010; they all belong to A037308.
This encoding can be applied to any base b > 1 (when b = 2, we obtain the identity function) as well as to the factorial base and to the primorial base.

			For n = 1972:
- the digit 1 is replaced by "01",
- the digit 9 is replaced by "111111111",
- the digit 7 is replaced by "01111111",
- the digit 2 is replaced by "011",
- hence we obtain the binary string "0111111111101111111011",
- and a(1972) = 2096123.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Colored logarithmic scatterplot of the first 1000000 terms (where the color is function of the digital sum minus the number of nonleading digits different from 9)
Index entries for sequences related to decimal expansion of n
Index entries for sequences that are permutations of the natural numbers

Cf. A000120, A001202, A002275, A002450, A007953, A037308, A051885.

tb=Table[n->PadRight[{0},n+1,1],{n,9}]/.PadRight[{0},10,1]-> PadRight[ {},9,1]; Table[FromDigits[IntegerDigits[n]/.tb//Flatten,2],{n,0,60}] (* Harvey P. Dale, Jul 31 2018 *)

a(n, base=10) = my (b=[], d=digits(n, base)); for (i=1, #d, if (d[i]!=base-1, b=concat(b, 0)); b=concat(b, vector(d[i], k, 1))); fromdigits(b, 2)
/* inverse */ b(n, base=10) = my (v=0, p=1); while (n, my (d = min(valuation(n+1, 2), base-1)); v += p * d; n \= 2^min(base-1, 1+d); p *= base); v

A000120(a(n)) = A007953(n).
a(A051885(k)) = 2^k - 1 for any k >= 0.
a(A002275(k)) = A002450(k) for any k >= 0.
a(10 * n) = 2 * a(n).

A281299 Primes p whose binary representation p_2 is the decimal representation of a prime q; and also the sum of the decimal digits of p equals the sum of the digits of p_2.

Original entry on oeis.org

5011, 7001, 11251, 22501, 32303, 32411, 90031, 101107, 104123, 108011, 111323, 121343, 122131, 124001, 125101, 141023, 224011, 233021, 235003, 241141, 321203, 324011, 421303, 432031, 442201, 510331, 511213, 520411, 801011, 1000183, 1000541, 1001191, 1005223, 1006231

Offset: 1

K. D. Bajpai, Jan 19 2017

Intersection of A037308 and A065720.

			a(1) = 5011 is a prime;
5011_2 = 1001110010011_10 is a prime;
5 + 0 + 1 + 1 = 7;
1 + 0 + 0 + 1 + 1 + 1 + 0 + 0 + 1 + 0 + 0 + 1 + 1 = 7; both the digit sums are equal.

Cf. A000040, A033548, A037308, A065720, A089971.

Select[Prime[Range[1000000]], PrimeQ[FromDigits[IntegerDigits[#, 2]]] && Plus @@ IntegerDigits[#] == Plus @@ IntegerDigits[FromDigits[IntegerDigits[#, 2]]] &]

eva(n) = subst(Pol(n), x, 10)
is(n) = ispseudoprime(n) && ispseudoprime(eva(binary(n))) && sumdigits(n)==sumdigits(eva(binary(n))) \\ Felix Fröhlich, Jan 19 2017

cp's OEIS Frontend

A135126 Numbers such that the digital sums in bases 3, 4, 5 and 6 all are equal.

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A135128 Numbers such that the digital sums in bases 2, 3, 5 and 10 all are equal.

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A135129 Numbers such that the digital sums in bases 3, 4, 5, 6 and 7 all are equal.

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A152207 Numbers k such that (sum of base-2 digits of k) = (sum of base-10 digits of k) = 10.

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A212284 First a(n) > 1 whose sum of digits is the same in base 10 as in base n.

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A224077 The smallest number n such that (sum of base 2 digits of n) = (sum of base 10 digits of n) for four consecutive integers.

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A305493 A binary encoding of the decimal representation of a number: for any number n >= 0, consider its decimal representation and replace each 9 with "111111111" and each other digit d with a "0" followed by d "1"s and interpret the result as a binary string.

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A281299 Primes p whose binary representation p_2 is the decimal representation of a prime q; and also the sum of the decimal digits of p equals the sum of the digits of p_2.

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