A010061 -id:A010061 - cp's OEIS frontend

A350601 Array read by antidiagonals: row n lists even numbers that are "generated" (in Kaprekar's sense) in all bases 2, 4, 6, ..., 2n.

0, 0, 2, 0, 2, 8, 0, 2, 10, 10, 0, 2, 10, 12, 12, 0, 2, 10, 14, 14, 14, 0, 2, 10, 14, 16, 16, 16, 0, 2, 10, 14, 22, 22, 22, 20, 0, 2, 10, 14, 22, 24, 24, 24, 22, 0, 2, 10, 14, 22, 24, 28, 28, 26, 24

Offset: 1

N. J. A. Sloane, Jan 08 2022

Max Alekseyev's PARI "Gen" program (see A010061) is essential for computing the rows. Cf. A349833.

			The initial rows of the array are:
  0, 2,  8, 10, 12, 14, 16, 20, 22, 24, 26, 28, 34, 36, 38, 40, 42, 44, 50, 52,  ... [the even terms of A228082]
  0, 2, 10, 12, 14, 16, 22, 24, 26, 28, 34, 36, 38, 40, 44, 50, 58, 60, 62, 66  ... [A349831]
  0, 2, 10, 14, 16, 22, 24, 28, 34, 36, 38, 44, 50, 58, 60, 62, 66, 68, 72, 74,  ... [A349832]
  0, 2, 10, 14, 22, 24, 28, 36, 38, 44, 50, 58, 60, 62, 66, 68, 74, 76, 82, 84,  ... [A349833]
  0, 2, 10, 14, 22, 24, 28, 36, 38, 44, 50, 58, 60, 62, 66, 68, 74, 76, 82, 84,  ...
  0, 2, 10, 14, 22, 28, 36, ...
  0, 2, 10, 14, 22, 36, ...
  0, 2, 10, 14, 22, 36,...
  0, 2, 10, 14, 22, ...
...
The rows converge to A230624, which is
  0, 2, 10, 14, 22, 38, 62, 94, 158, 206, 318, 382, 478, 606, 766, 958, 1022, ...
The initial antidiagonals are:
  0,
  0, 2,
  0, 2, 8,
  0, 2, 10, 10,
  0, 2, 10, 12, 12,
  0, 2, 10, 14, 14, 14,
  0, 2, 10, 14, 16, 16, 16,
  0, 2, 10, 14, 22, 22, 22, 20,
  0, 2, 10, 14, 22, 24, 24, 24, 22,
  0, 2, 10, 14, 22, 24, 28, 28, 26, 24,
  ...

Index entries for Colombian or self numbers and related sequences

The first few rows of the array are A228082 (even terms only), A349831, A349832, and A349833.

Cf. A003052, A010061, A230624.

[Needs checking and extending]

A227359 Natural numbers that are not of the form (k +- sum of binary digits of k) for any k.

Original entry on oeis.org

6, 13, 21, 30, 37, 48, 51, 80, 83, 111, 121, 133, 144, 147, 175, 185, 192, 207, 217, 226, 233, 242, 245, 248, 250, 272, 275, 303, 313, 320, 335, 345, 354, 361, 370, 373, 376, 378, 387, 399, 409, 418, 425, 434, 437, 440, 442, 457, 466, 469, 472, 474, 481, 488, 490, 497, 505, 507, 528, 531, 559, 569, 576, 591, 601, 610, 617

Offset: 1

Andres M. Torres, Jul 08 2013

This sequence is the intersection of sets A010061 and A055938, where: set A010061 is NONE of ( k + count of set binary bits(k) ), and set A055938 is NONE of ( k - count of set binary bits(k) ), for any k.

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 1, 9, 127, 1362, 12921, 128429, 1261747, 12554142, 125697648, 1257065977, ... . Conjecture: This sequence has an asymptotic density (1/2) * A242403 = 0.126330... . - Amiram Eldar, Oct 02 2022

			Find the list of values not defined by:
V = i +- count of set binary bits(i), for any integer i.
Assume that setbits(n) returns the count of set binary digits of n.
A227359 sample: 6,13,21,30,37,48,51,80,83,111, ...
0 +- setbits(0) = 0     therefore 0 does not make the list
1 +- setbits(1) = 0,2   therefore 0 and 2 do not make the list
2 +- setbits(2) = 1,3   therefore 1 and 3 do not make the list
3 +- setbits(3) = 1,5   therefore 1 and 5 do not make the list
4 +- setbits(4) = 3,5   ...
5 +- setbits(5) = 3,7   therefore 3 and 7 do not make the list
6 +- setbits(6) = 4,8   therefore 4 and 8 do not make the list
7 +- setbits(7) = 4,10  therefore 4 and 10 do not make the list
8 +- setbits(8) = 7,9   therefore 7 and 9 do not make the list
6 and 13 did make the list because there is no solution for
6 = i +- setbits(i),  nor
13 = i +- setbits(i), for any integer i.

Andres M. Torres, Table of n, a(n) for n = 1..1000
Andres M. Torres, Zip file containing Blitz3D code.

Cf. A055938, A010061, A010062, A242403.

Blitz3D
```
;; See link.
```

A227361 If n is even, then a(n) = n + bitsum(n), else a(n) = n - bitsum(n), where bitsum(n) is the count of binary 1's in n, A000120.

Original entry on oeis.org

0, 0, 3, 1, 5, 3, 8, 4, 9, 7, 12, 8, 14, 10, 17, 11, 17, 15, 20, 16, 22, 18, 25, 19, 26, 22, 29, 23, 31, 25, 34, 26, 33, 31, 36, 32, 38, 34, 41, 35, 42, 38, 45, 39, 47, 41, 50, 42, 50, 46, 53, 47, 55, 49, 58, 50, 59, 53, 62, 54, 64, 56, 67, 57, 65, 63, 68, 64, 70, 66, 73, 67, 74, 70, 77, 71, 79, 73, 82, 74, 82, 78, 85, 79, 87, 81, 90, 82, 91, 85

Offset: 0

Andres M. Torres, Jul 08 2013

I gathered together some interesting statistics for this seq A227361:
Within the first 200000001 members of this sequence, only 70 were repeated 4 times, and then only when n > 2 million. None were repeated 5 times.
The first value to become repeated 3 times is 50, occurring at indexes (n=) 46, 48, and 55.
The first value to become repeated 4 times is 2097170, occurring at indexes (n=) 2097150, 2097166, 2097168, and 2097175.
The total count of those only occurring once is 96226727, or about 48.11 %.
Total count of those repeated 2 times is 45055158.
Total count of those repeated 3 times is 4554221, or about 2.28 %.
Total count of those repeated 4 times is 70 (extremely low).
Repeatedly applying this BitStoneA(v) function to values in a recursive (nested) style has shown that only 21 starting values shall become zero. These are those values: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 23, 27. All other values shall cycle forever in small loops.
274877906962 is the smallest number that occurs 5 times. - Donovan Johnson, Jul 27 2013

			a(0) = 0 because 0 is even, so 0 + bitsum(0) = 0.
a(1) = 0 because 1 is odd, so 1 - bitsum(1) = 0.
a(2) = 3 because 2 is even, so 2 + bitsum(2) = 3.
a(3) = 1 because 3 is odd, so 3 - bitsum(3) = 1.

Andres M. Torres, Table of n, a(n) for n = 0..10000

Cf. A000120, A055938, A010061, A010062.

;; Each a(n) is generated simply as follows:  a(n) = BitStoneA(n)
Function BitStoneA(n)
         If  (n Mod 2)              ;; if is odd
                 Return n-bitsum(n)
         Else                       ;; if is even
                 Return n+bitsum(n)
         End If
End Function
;; --- Or, If n is even, then return n+A000120(n), else return n-A000120(n), where A000120(n) = bitsum(n)

Table[n + (-1)^n DigitCount[n, 2, 1], {n, 0, 127}] (* Alonso del Arte, Jul 08 2013 *)

a(n)=n+(-1)^(n%2)*hammingweight(n) \\ Charles R Greathouse IV, Jul 09 2013

a(n) = n + (-1)^n * Sum_{j = 1 .. floor(log_2(n)) + 1} (floor(n/2^j + 1/2) - floor(n/2^j)). - Alonso del Arte, Jul 08 2013, based on one of Hieronymus Fischer's formulas for A000120.

A227408 Set of all n, where n = r(s(n)) = s(r(n)), given that r(n) = n+bitcount(n), s(n) = n-bitcount(n), and bitcount(n) is the count of binary 1's in n.

Original entry on oeis.org

0, 22, 25, 38, 41, 70, 73, 134, 137, 237, 243, 262, 265, 365, 371, 429, 435, 461, 467, 492, 494, 498, 501, 518, 521, 621, 627, 685, 691, 717, 723, 748, 750, 754, 757, 813, 819, 845, 851, 876, 878, 882, 885, 909, 915, 940, 942, 946, 949, 972, 974, 978, 981, 988, 995, 1002, 1009, 1030, 1033, 1133, 1139, 1197, 1203, 1229

Offset: 1

Andres M. Torres, Jul 10 2013

This is a simple sequence where the nesting of functions r(n), and s(n), are grouped in a special way: n = r(s(n)) = s(r(n)), and those three values must be equal.

			0  = r(s(0)) = s(r(0))  = r(0)  = s(0)  = 0.
22 = r(s(22))= s(r(22)) = r(19) = s(25) = 22.
25 = r(s(25))= s(r(25)) = r(22) = s(28) = 25.
38 = r(s(38))= s(r(38)) = r(35) = s(41) = 38.

Andres M. Torres, Table of n, a(n) for n = 1..10000

Cf. A055938, A010061, A010062, A227359, A227361.

npbc(n) = n + hammingweight(n)
nmbc(n) = n - hammingweight(n)
isok(n) = (n == npbc(nmbc(n))) && (n == nmbc(npbc(n))) \\ Michel Marcus, Aug 08 2013

Find all n, such that: n = r(s(n)) = s(r(n)), where r(n) = n+bitcount(n) and s(n) = n-bitcount(n)

Offset changed from 0 to 1 by Michel Marcus, Aug 08 2013

A349832 Even numbers that are "generated" (in Kaprekar's sense) in all three bases 2, 4, and 6.

Original entry on oeis.org

0, 2, 10, 14, 16, 22, 24, 28, 34, 36, 38, 44, 50, 58, 60, 62, 66, 68, 72, 74, 76, 82, 84, 92, 94, 96, 98, 106, 108, 110, 118, 120, 122, 126, 132, 134, 136, 140, 146, 154, 156, 158, 162, 164, 170, 176, 178, 186, 196, 198, 202, 206, 210, 214, 216, 222, 228, 234, 238, 244, 246, 252, 256, 258, 260

Offset: 1

N. J. A. Sloane, Jan 07 2022

Using Max Alekseyev's PARI "Gen" program (see A010061), we run
vector(500,k,length(Gen(k,2))),
vector(500,k,length(Gen(k,4))), and
vector(500,k,length(Gen(k,6)))
to find the numbers that are generated in bases 2, 4, and 6, and then take the even numbers that are common to all three lists.

Cf. A003052, A010061, A228082, A349829, A349830, A349831, A349833.
A230624 is a subsequence.
A row of A350601.

A349833 Even numbers that are "generated" (in Kaprekar's sense) in all four bases 2, 4, 6, and 8.

Original entry on oeis.org

0, 2, 10, 14, 22, 24, 28, 36, 38, 44, 50, 58, 60, 62, 66, 68, 74, 76, 82, 84, 92, 94, 96, 98, 106, 110, 118, 120, 122, 132, 134, 136, 140, 154, 156, 158, 162, 170, 176, 178, 186, 196, 198, 206, 210, 214, 216, 222, 228, 234, 244, 246, 252, 258, 260, 262, 264, 268, 274, 284, 286

Offset: 1

N. J. A. Sloane, Jan 07 2022

Using Max Alekseyev's PARI "Gen" program (see A010061), we run
vector(500,k,length(Gen(k,2))),
vector(500,k,length(Gen(k,4))),
vector(500,k,length(Gen(k,6))),
vector(500,k,length(Gen(k,8))),
to find the numbers that are generated in bases 2, 4, 6, and 8, and then take the even numbers that are common to all four lists.

N. J. A. Sloane, Table of n, a(n) for n = 1..2275

Cf. A003052, A010061, A228082, A349829, A349830, A349831, A349832.
A230624 is a subsequence.
A row of A350601.

A374101 Numbers k such that k and k+2 are both self numbers (A003052).

Original entry on oeis.org

1, 3, 5, 7, 108, 209, 310, 411, 512, 613, 714, 815, 916, 1109, 1210, 1311, 1412, 1513, 1614, 1715, 1816, 1917, 2110, 2211, 2312, 2413, 2514, 2615, 2716, 2817, 2918, 3111, 3212, 3313, 3414, 3515, 3616, 3717, 3818, 3919, 4112, 4213, 4314, 4415, 4516, 4617, 4718

Offset: 1

Amiram Eldar, Jun 28 2024

The least difference between consecutive self numbers is 2 (see Griffin N. Macris's proof at A010061 that may be adapted to other bases).

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

Subsequence of A003052.

Cf. A010061, A339216 (binary analog).

seq[max_] := Module[{c = Complement[Range[max], Table[n + DigitSum[n], {n, 1, max}]], d, ind}, d = Differences[c]; ind = Position[d, 2] // Flatten; c[[ind]]]; seq[5000]

A232228 a(1)=1; thereafter a(n) = 2^(number of bits in binary expansion of a(n-1)) + 1 + a(n-1).

Original entry on oeis.org

1, 4, 13, 30, 63, 128, 385, 898, 1923, 3972, 8069, 16262, 32647, 65416, 130953, 262026, 524171, 1048460, 2097037, 4194190, 8388495, 16777104, 33554321, 67108754, 134217619, 268435348, 536870805, 1073741718, 2147483543, 4294967192, 8589934489, 17179869082, 34359738267, 68719476636

Offset: 1

N. J. A. Sloane, Nov 22 2013

nonn

An infinite subsequence of A010061.

Harvey P. Dale, Table of n, a(n) for n = 1..1001
D. W. Bange, Solution to Problem E 2408, Amer. Math. Monthly, 81 (1974), 407.
Index entries for Colombian or self numbers and related sequences

Cf. A010061.

f:=proc(n) option remember; if n=1 then 1 else 2^(nops(convert(f(n-1),base,2)))+1+f(n-1); fi; end;
[seq(f(n),n=1..40)];

NestList[2^IntegerLength[#,2]+1+#&,1,40] (* Harvey P. Dale, May 05 2020 *)

A374102 Prime binary self (or Colombian) numbers: primes not expressible as the sum of an integer and its binary weight.

Original entry on oeis.org

13, 23, 37, 71, 83, 97, 113, 233, 311, 313, 337, 359, 373, 401, 409, 449, 457, 499, 569, 593, 599, 601, 617, 641, 643, 739, 761, 809, 821, 853, 881, 883, 929, 947, 953, 977, 1103, 1129, 1153, 1193, 1217, 1237, 1249, 1297, 1303, 1319, 1321, 1429, 1459, 1489, 1499

Offset: 1

Amiram Eldar, Jun 28 2024

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

Intersection of A000040 and A010061.
A374103 is a subsequence.
Cf. A000120, A006378 (decimal analog).

With[{m = 240}, Complement[Prime[Range[m]], Table[n + DigitCount[n, 2, 1], {n, 0, Prime[m]}]]]

A374103 Numbers k such that k and k+2 are both prime binary self (or Colombian) numbers (A374102).

Original entry on oeis.org

311, 599, 641, 881, 1319, 1697, 1721, 2657, 2969, 3257, 4019, 4127, 4337, 4721, 5009, 6449, 6569, 6689, 6761, 7547, 9041, 9239, 10457, 10529, 11171, 11699, 11939, 13691, 16229, 19379, 20147, 20357, 20477, 22697, 23057, 24977, 25169, 26249, 26681, 26729, 27059

Offset: 1

Amiram Eldar, Jun 28 2024

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

Subsequence of A001359, A339216 and A374102.

Cf. A010061, A006378.

seq[max_] := Module[{c = Complement[Prime[Range[max]], Table[n + DigitCount[n, 2, 1], {n, 0, Prime[max]}]], d, ind}, d = Differences[c]; ind = Position[d, 2] // Flatten; c[[ind]]]; seq[3000]

cp's OEIS Frontend

A350601 Array read by antidiagonals: row n lists even numbers that are "generated" (in Kaprekar's sense) in all bases 2, 4, 6, ..., 2n.

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A227359 Natural numbers that are not of the form (k +- sum of binary digits of k) for any k.

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Blitz3D

A227361 If n is even, then a(n) = n + bitsum(n), else a(n) = n - bitsum(n), where bitsum(n) is the count of binary 1's in n, A000120.

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Blitz3D

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PARI

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A227408 Set of all n, where n = r(s(n)) = s(r(n)), given that r(n) = n+bitcount(n), s(n) = n-bitcount(n), and bitcount(n) is the count of binary 1's in n.

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PARI

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Extensions

A349832 Even numbers that are "generated" (in Kaprekar's sense) in all three bases 2, 4, and 6.

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A349833 Even numbers that are "generated" (in Kaprekar's sense) in all four bases 2, 4, 6, and 8.

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A374101 Numbers k such that k and k+2 are both self numbers (A003052).

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Mathematica

A232228 a(1)=1; thereafter a(n) = 2^(number of bits in binary expansion of a(n-1)) + 1 + a(n-1).

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Maple

Mathematica

A374102 Prime binary self (or Colombian) numbers: primes not expressible as the sum of an integer and its binary weight.

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