A070940 -id:A070940 - cp's OEIS frontend

A372885 Prime numbers whose binary indices (positions of ones in reversed binary expansion) sum to another prime number.

2, 3, 11, 23, 29, 41, 43, 61, 71, 79, 89, 101, 103, 113, 131, 137, 149, 151, 163, 181, 191, 197, 211, 239, 269, 271, 281, 293, 307, 331, 349, 353, 373, 383, 401, 433, 457, 491, 503, 509, 523, 541, 547, 593, 641, 683, 701, 709, 743, 751, 761, 773, 827, 863, 887

Offset: 1

Gus Wiseman, May 19 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

The indices of these primes are A372886.

			The binary indices of 89 are {1,4,5,7}, with sum 17, which is prime, so 89 is in the sequence.
The terms together with their binary expansions and binary indices begin:
    2:         10 ~ {2}
    3:         11 ~ {1,2}
   11:       1011 ~ {1,2,4}
   23:      10111 ~ {1,2,3,5}
   29:      11101 ~ {1,3,4,5}
   41:     101001 ~ {1,4,6}
   43:     101011 ~ {1,2,4,6}
   61:     111101 ~ {1,3,4,5,6}
   71:    1000111 ~ {1,2,3,7}
   79:    1001111 ~ {1,2,3,4,7}
   89:    1011001 ~ {1,4,5,7}
  101:    1100101 ~ {1,3,6,7}
  103:    1100111 ~ {1,2,3,6,7}
  113:    1110001 ~ {1,5,6,7}
  131:   10000011 ~ {1,2,8}
  137:   10001001 ~ {1,4,8}
  149:   10010101 ~ {1,3,5,8}
  151:   10010111 ~ {1,2,3,5,8}
  163:   10100011 ~ {1,2,6,8}
  181:   10110101 ~ {1,3,5,6,8}
  191:   10111111 ~ {1,2,3,4,5,6,8}
  197:   11000101 ~ {1,3,7,8}

Robert Israel, Table of n, a(n) for n = 1..10000

For prime instead of binary indices we have A006450, prime case of A316091.
Prime numbers p such that A029931(p) is also prime.
Prime case of A372689.
The indices of these primes are A372886.
A000040 lists the prime numbers, A014499 their binary indices.
A019565 gives Heinz number of binary indices, adjoint A048675.
A058698 counts partitions of prime numbers, strict A064688.
A372687 counts strict partitions of prime binary rank, counted by A372851.
A372688 counts partitions of prime binary rank, with Heinz numbers A277319.
Binary indices:
- listed A048793, sum A029931
- reversed A272020
- opposite A371572, sum A230877
- length A000120, complement A023416
- min A001511, opposite A000012
- max A070939, opposite A070940
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
Cf. A000009, A029837, A035100, A038499, A096111, A372429, A372441, A372471, A372850, A372887.

filter:= proc(p)
  local L,i,t;
  L:= convert(p,base,2);
  isprime(add(i*L[i],i=1..nops(L)))
end proc:
select(filter, [seq(ithprime(i),i=1..200)]); # Robert Israel, Jun 19 2025

Select[Range[100],PrimeQ[#] && PrimeQ[Total[First/@Position[Reverse[IntegerDigits[#,2]],1]]]&]

A372886 Indices of prime numbers whose binary indices (positions of ones in reversed binary expansion) sum to another prime number.

Original entry on oeis.org

1, 2, 5, 9, 10, 13, 14, 18, 20, 22, 24, 26, 27, 30, 32, 33, 35, 36, 38, 42, 43, 45, 47, 52, 57, 58, 60, 62, 63, 67, 70, 71, 74, 76, 79, 84, 88, 94, 96, 97, 99, 100, 101, 108, 116, 124, 126, 127, 132, 133, 135, 137, 144, 150, 154, 156, 160, 161, 162, 164, 172

Offset: 1

Gus Wiseman, May 19 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

The prime numbers themselves are A372885(n).

			The binary indices of 89 = prime(24) are {1,4,5,7}, with sum 17, which is prime, so 24 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Numbers k such that A029931(prime(k)) is prime.
Indices of primes that belong to A372689.
The indexed prime numbers themselves are A372885.
A000040 lists the prime numbers, A014499 their binary indices
A006450 lists primes of prime index, prime case of A316091.
A019565 gives Heinz number of binary indices, adjoint A048675.
A038499 counts partitions of prime length, strict A085756.
Binary indices:
- listed A048793, sum A029931
- reversed A272020
- opposite A371572, sum A230877
- length A000120, complement A023416
- min A001511, opposite A000012
- max A070939, opposite A070940
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
A058698 counts partitions of prime numbers, strict A064688.
A372687 counts strict partitions of prime binary rank, counted by A372851.
A372688 counts partitions of prime binary rank, with Heinz numbers A277319.
Cf. A029837, A158704, A158705, A035100, A372429, A372471, A372850.

filter:= proc(p)
  local L,i,t;
  L:= convert(p,base,2);
  isprime(add(i*L[i],i=1..nops(L)))
end proc:
select(t -> filter(ithprime(t)), [$1..1000]); # Robert Israel, Jun 19 2025

Select[Range[100],PrimeQ[Total[First /@ Position[Reverse[IntegerDigits[Prime[#],2]],1]]]&]

A372890 Sum of binary ranks of all integer partitions of n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1).

Original entry on oeis.org

0, 1, 4, 10, 25, 52, 115, 228, 471, 931, 1871, 3687, 7373, 14572, 29049, 57694, 115058, 229101, 457392, 912469, 1822945, 3640998, 7277426, 14544436, 29079423, 58137188, 116254386, 232465342, 464889800, 929691662, 1859302291, 3718428513, 7436694889, 14873042016

Offset: 0

Gus Wiseman, May 23 2024

nonn

			The partitions of 4 are (4), (3,1), (2,2), (2,1,1), (1,1,1,1), with respective binary ranks 8, 5, 4, 4, 4 with sum 25, so a(4) = 25.

Alois P. Heinz, Table of n, a(n) for n = 0..3321

For Heinz number (not binary rank) we have A145519, row sums of A215366.
For Heinz number the strict version is A147655, row sums of A246867.
The strict version is A372888, row sums of A118462.
A005117 gives Heinz numbers of strict integer partitions.
A048675 gives binary rank of prime indices, distinct A087207.
A061395 gives greatest prime index, least A055396.
A118457 lists strict partitions in Mathematica order.
A277905 groups all positive integers by binary rank of prime indices.
Binary indices (A048793):
- length A000120, complement A023416
- min A001511, opposite A000012
- max A029837 or A070939, opposite A070940
- sum A029931, product A096111
- reverse A272020
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
- opposite A371572, sum A230877
Cf. A000041, A005940, A018819, A019565, A066633, A225620, A344086, A365410.

b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n],
      b(n, i-1)+(p->[0, p[1]*2^(i-1)]+p)(b(n-i, min(n-i, i))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..33);  # Alois P. Heinz, May 23 2024

Table[Total[Total[2^(#-1)]&/@IntegerPartitions[n]],{n,0,10}]

From Alois P. Heinz, May 23 2024: (Start)
a(n) = Sum_{k=1..n} 2^(k-1) * A066633(n,k).
a(n) mod 2 = A365410(n-1) for n>=1. (End)

A119387 a(n) is the number of binary digits (1's and nonleading 0's) which remain unchanged in their positions when n and (n+1) are written in binary.

Original entry on oeis.org

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

Offset: 0

Leroy Quet, Jul 26 2006

The largest k for which A220645(n,k) > 0 is k = a(n). That is, a(n) is the largest power of 2 that divides binomial(n,i) for 0 <= i <= n. - T. D. Noe, Dec 18 2012

a(n) is the distance between the first and last 1's in the binary expansion of n+1; see examples and formulae. - David James Sycamore, Feb 21 2023

			9 in binary is 1001. 10 (decimal) is 1010 in binary. 2 binary digits remain unchanged (the leftmost two digits) between 1001 and 1010. So a(9) = 2.
From _David James Sycamore_, Feb 26 2023: (Start)
Number of bits surviving transition from n to n+1 = distance between first and last 1's in binary expansion of n+1 (no need to compare n and n+1). Examples:
n = 2^k - 1: distance between 1's in n+1 = 2^k is 0; a(n) = 0 (all bits change).
82 in binary is 1010010, and 83 is 1010011 distance between 1's in 83 = 6 = a(82).
Show visually for a(327) = 5:
 n   = 327 = 101000111
             ^^^^^      5 unchanged bits.
 n+1 = 328 = 101001000
             ^    ^     distance between 1's = 5. (End)

Paolo Xausa, Table of n, a(n) for n = 0..10000 (terms 0..1023 from T. D. Noe)
Lukas Spiegelhofer and Michael Wallner, Divisibility of binomial coefficients by powers of primes, arXiv:1604.07089 [math.NT], 2016. Mentions this sequence.
Index entries for sequences related to binary expansion of n

Cf. A048881, A086784.
Cf. A070940.
Cf. A000265.
Cf. A001511, A070939.
Cf. A373709 (partial sums).

#include 
#define NMAX 200
int sameD(int a, int b) { int resul=0 ; while(a>0 && b >0) { if( (a &1) == (b & 1)) resul++ ; a >>= 1 ; b >>= 1 ; } return resul ; }
int main(int argc, char*argv[])
{ for(int n=0;nR. J. Mathar, Jul 29 2006 */

int A119387(int n)
{
    int m=n+1;
    while (!(m&1)) m>>=1;
    int m_bits = 0;
    while (m>>=1) m_bits++;
    return m_bits;
}
/* Laura Monroe, Oct 18 2020 */

a119387 n = length $ takeWhile (< a070940 n) [1..n]
-- Reinhard Zumkeller, Apr 22 2013

a:= n-> ilog2(n+1)-padic[ordp](n+1, 2):
seq(a(n), n=0..128);  # Alois P. Heinz, Jun 28 2021

a = {0}; Table[b = IntegerDigits[n, 2]; If[Length[a] == Length[b], c = 1; While[a[[c]] == b[[c]], c++]; c--, c = 0]; a = b; c, {n, 101}] (* T. D. Noe, Dec 18 2012 *)
(* Second program, faster *)
Array[Last[#] - First[#] &@ Position[IntegerDigits[#, 2], 1][[All, 1]] &, 2^14] (* Michael De Vlieger, Feb 22 2023 *)
Table[BitLength[k] - 1 - IntegerExponent[k, 2], {k, 100}] (* Paolo Xausa, Oct 01 2024 *)

a(n) = n++; local(c); c=0; while(2^(c+1)Ralf Stephan, Oct 16 2013; corrected by Michel Marcus, Jun 28 2021 */

a(n) = my(x=Vecrev(binary(n)), y=Vecrev(binary(n+1))); sum(k=1, min(#x, #y), x[k] == y[k]); \\ Michel Marcus, Jun 27 2021

a(n) = exponent(n+1) - valuation(n+1, 2); \\ Antoine Mathys, Nov 20 2024

def A119387(n): return (n+1).bit_length()-(n+1&-n-1).bit_length() # Chai Wah Wu, Jul 07 2022

a(n) = A048881(n) + A086784(n+1). (A048881(n) is the number of 1's which remain unchanged between binary n and (n+1). A086784(n+1) is the number of nonleading 0's which remain unchanged between binary n and (n+1).)
a(A000225(n))=0. - R. J. Mathar, Jul 29 2006
a(n) = -valuation(H(n)*n,2) where H(n) is the n-th harmonic number. - Benoit Cloitre, Oct 13 2013
a(n) = A000523(n+1) - A007814(n+1) = floor(log(n+1)/log(2)) - valuation(n+1,2). - Benoit Cloitre, Oct 13 2013 [corrected by David James Sycamore, Feb 28 2023]
Recurrence: a(2n) = floor(log_2(n)) except a(0) = 0, a(2n+1) = a(n). - Ralf Stephan, Oct 16 2013, corrected by Peter J. Taylor, Mar 01 2020
a(n) = floor(log_2(A000265(n+1))). - Laura Monroe, Oct 18 2020
a(n) = A070939(n+1) - A001511(n+1). - David James Sycamore, Feb 24 2023

More terms from R. J. Mathar, Jul 29 2006

Edited by Charles R Greathouse IV, Aug 04 2010

A372888 Sum of binary ranks of all strict integer partitions of n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1).

Original entry on oeis.org

0, 1, 2, 7, 13, 31, 66, 138, 279, 581, 1173, 2375, 4783, 9630, 19316, 38802, 77689, 155673, 311639, 623845, 1248179, 2497719, 4996387, 9995304, 19992908, 39990902, 79986136, 159983241, 319975073, 639971495, 1279962115, 2559966847, 5119970499, 10240030209

Offset: 0

Gus Wiseman, May 23 2024

nonn

			The strict partitions of 6 are (6), (5,1), (4,2), (3,2,1), with respective binary ranks 32, 17, 10, 7 with sum 66, so a(6) = 66.

Alois P. Heinz, Table of n, a(n) for n = 0..3321

Row sums of A118462 (binary ranks of strict partitions).
For Heinz number the non-strict version is A145519, row sums of A215366.
For Heinz number (not binary rank) we have A147655, row sums of A246867.
The non-strict version is A372890.
A000009 counts strict partitions, ranks A005117.
A048675 gives binary rank of prime indices, distinct A087207.
A277905 groups all positive integers by binary rank of prime indices.
Binary indices (A048793):
- length A000120, complement A023416
- min A001511, opposite A000012
- max A029837 or A070939, opposite A070940
- sum A029931, product A096111
- reverse A272020
- complement A368494, sum A359400
- opposite A371572, sum A230877
- opposite complement A371571, sum A359359
Cf. A000041, A005940, A015716, A018819, A019565, A118457, A225620, A344086.

b:= proc(n, i) option remember; `if`(i*(i+1)/2 [0, p[1]*2^(i-1)]
          +p)(b(n-i, min(n-i, i-1)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..33);  # Alois P. Heinz, May 23 2024

Table[Total[Total[2^(#-1)]& /@ Select[IntegerPartitions[n],UnsameQ@@#&]],{n,0,10}]

a(n) = Sum_{k=1..n} 2^(k-1) * A015716(n,k). - Alois P. Heinz, May 24 2024

A372887 Number of integer partitions of n whose distinct parts are the binary indices of some prime number.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 6, 8, 12, 14, 21, 29, 36, 48, 56, 74, 94, 123, 144, 195, 235, 301, 356, 456, 538, 679, 803, 997, 1189, 1467, 1716, 2103, 2488, 2968, 3517, 4185, 4907, 5834, 6850, 8032, 9459, 11073, 12933, 15130, 17652, 20480, 24011, 27851, 32344, 37520

Offset: 0

Gus Wiseman, May 19 2024

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

Note the inverse of A048793 (binary indices) takes a set s to Sum_i 2^(s_i-1).

			The partition y = (4,3,1,1) has distinct parts {1,3,4}, which are the binary indices of 13, which is prime, so y is counted under a(9).
The a(2) = 1 through a(9) = 14 partitions:
  (2)  (21)  (22)   (221)   (51)     (331)     (431)      (3321)
             (31)   (311)   (222)    (421)     (521)      (4221)
             (211)  (2111)  (321)    (511)     (2222)     (4311)
                            (2211)   (2221)    (3221)     (5211)
                            (3111)   (3211)    (3311)     (22221)
                            (21111)  (22111)   (4211)     (32211)
                                     (31111)   (5111)     (33111)
                                     (211111)  (22211)    (42111)
                                               (32111)    (51111)
                                               (221111)   (222111)
                                               (311111)   (321111)
                                               (2111111)  (2211111)
                                                          (3111111)
                                                          (21111111)

For odd instead of prime we have A000041, even A002865.
The strict case is A372687, ranks A372851.
Counting not just distinct parts gives A372688, ranks A277319.
These partitions have Heinz numbers A372850.
A014499 lists binary indices of prime numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A058698 counts partitions of prime numbers, strict A064688.
A372689 lists numbers whose binary indices sum to a prime.
A372885 lists primes whose binary indices sum to a prime, indices A372886.
Binary indices:
- listed A048793, sum A029931
- reversed A272020
- opposite A371572, sum A230877
- length A000120, complement A023416
- min A001511, opposite A000012
- max A070939, opposite A070940
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
Cf. A000040, A000041, A029837, A035100, A038499, A372429, A372471.

Table[Length[Select[IntegerPartitions[n], PrimeQ[Total[2^(Union[#]-1)]]&]],{n,0,30}]

A080080 T(n,k) = length of longest carry sequence when adding k to n in binary representation, 1 <= k <= n (triangular array).

Original entry on oeis.org

1, 0, 1, 2, 1, 1, 0, 0, 0, 1, 1, 0, 3, 1, 1, 0, 2, 2, 1, 1, 1, 3, 2, 2, 1, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 1, 0, 4, 1, 1, 0, 1, 1, 0, 0, 3, 3, 1, 1, 1, 2, 1, 1, 0, 4, 3, 3, 1, 2, 1, 1, 0, 0, 0, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 4, 2, 2, 2, 2, 1, 1, 1, 3, 1, 1, 0, 3, 3, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1

Offset: 1

Reinhard Zumkeller, Jan 26 2003

T(n,1) = A007814(n+1), T(n,n) = 1; for n>1: T(n,n-1) = A043545(n+1); T(n,k) <= A070940(n) = T(n, A080079(n)).

T(n,k) = A050600(n+k,k) - 1. - Reinhard Zumkeller, Aug 03 2014

			Triangle begins:
              1
            0   1
          2   1   1
        0   0   0   1
      1   0   3   1   1
    0   2   2   1   1   1
  3   2   2   1   2   1   1

Reinhard Zumkeller, Rows n=1..150 of triangle, flattened
Index entries for sequences related to binary expansion of n

Cf. A050600.

import Data.Bits (xor, (.&.), shiftL)
a080080 :: Int -> Int -> Int
a080080 n k = addc n k 0 where
   addc x y z | y == 0    = z - 1
              | otherwise = addc (x `xor` y) (shiftL (x .&. y) 1) (z + 1)
a080080_row n = map (a080080 n) [1..n]
a080080_tabl = map a080080_row [1..]
-- Reinhard Zumkeller, Apr 22 2013

A082909 a(n) = Sum_{d|n} (d mod 3).

Original entry on oeis.org

1, 3, 1, 4, 3, 3, 2, 6, 1, 6, 3, 4, 2, 6, 3, 7, 3, 3, 2, 9, 2, 6, 3, 6, 4, 6, 1, 8, 3, 6, 2, 9, 3, 6, 6, 4, 2, 6, 2, 12, 3, 6, 2, 9, 3, 6, 3, 7, 3, 9, 3, 8, 3, 3, 6, 12, 2, 6, 3, 9, 2, 6, 2, 10, 6, 6, 2, 9, 3, 12, 3, 6, 2, 6, 4, 8, 6, 6, 2, 15, 1, 6, 3, 8, 6, 6, 3, 12, 3, 6, 4, 9, 2, 6, 6, 9, 2, 9, 3, 13, 3, 6

Offset: 1

N. J. A. Sloane, Nov 02 2008

nonn

The old entry with this sequence number was a duplicate of A070940.

Cf. A004016, A093061.

Table[Total[Mod[Divisors[n],3]],{n,110}] (* Harvey P. Dale, Jan 01 2019 *)
Table[DivisorSum[n,Mod[#,3]&],{n,110}] (* Harvey P. Dale, Jan 09 2022 *)

A199570 Table, each row contains the previous sequence in odd columns and the row number in even columns.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 4, 1, 4, 2, 4, 1, 4, 3, 4, 1, 4, 3, 4, 2, 4, 3, 4, 1, 5, 1, 5, 2, 5, 1, 5, 3, 5, 1, 5, 3, 5, 2, 5, 3, 5, 1, 5, 4, 5, 1, 5, 4, 5, 2, 5, 4, 5, 1, 5, 4, 5, 3, 5, 4, 5, 1, 5, 4, 5, 3, 5, 4, 5, 2, 5, 4, 5, 3, 5, 4, 5

Offset: 1

Franklin T. Adams-Watters, Nov 08 2011

			The table starts:
  1
  1 2
  1 3 1 3 2 3
  1 4 1 4 2 4 1 4 3 4 1 4 3 4 2 4 3 4
  ...

John Tyler Rascoe, Rows n = 1..9 of table, flattened

Cf. A025192 (row lengths), A070940.

n=4;v=vector(3^n);v[1]=1;for(k=1,n,for(i=(s=3^(k-1))+1,3^k,v[i]=if((i-s)%2,v[(i-s+1)\2],k+1)));v

def A199570_list(row):
    A = [1]
    for i in range(2,row+1):
        z = 2*(3**(i-2))
        for j in range(1,z+1):
            if j%2 != 0: A.append(A[int((j-1)/2)])
            else: A.append(i)
    return(A) # John Tyler Rascoe, Feb 19 2023

A328567 a(n) is the smallest positive integer divisible by n such that it is possible to strike out a digit from its binary expansion (apart from trailing zeros) so that the resulting number is nonzero and divisible by n.

Original entry on oeis.org

3, 6, 21, 12, 75, 42, 105, 24, 279, 150, 341, 84, 403, 210, 465, 48, 1071, 558, 1197, 300, 1323, 682, 1449, 168, 1575, 806, 1701, 420, 1827, 930, 1953, 96, 4191, 2142, 4445, 1116, 4699, 2394, 4953, 600, 5207, 2646, 5461, 1364, 5715, 2898, 5969, 336, 6223, 3150

Offset: 1

Rémy Sigrist, Oct 20 2019

This sequence is a binary variant of A309631.

This kind of sequence is well defined for any fixed base b > 1: for any n > 0: consider the concatenation in base b, say m, of n, "0", and n; m is a multiple of n, and removing the central "0" (which is not a trailing zero), gives another multiple of n.

			For n = 3:
- the first multiples of 3 are (in decimal and in binary), alongside the possible values resulting from striking out a non-trailing zero:
    3*k  bin(3*k)  striked (binary)
    ---  --------  ----------------
      3        11  1
      6       110  10
      9      1001  1, 100, 101
     12      1100  100
     15      1111  111
     18     10010  10, 1000, 1010
     21     10101  101, 1001, 1010, 1011, 1101
- 21 is the least appropriate multiple,
- so a(3) = 21.

Rémy Sigrist, Table of n, a(n) for n = 1..8192
Rémy Sigrist, PARI program for A328567

Cf. A070940, A309631.

PARI
```
See Links section.
```

Apparently, a(n)/n = 2^(1+A070940(n)) - 1.

cp's OEIS Frontend

A372885 Prime numbers whose binary indices (positions of ones in reversed binary expansion) sum to another prime number.

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A372886 Indices of prime numbers whose binary indices (positions of ones in reversed binary expansion) sum to another prime number.

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A372890 Sum of binary ranks of all integer partitions of n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1).

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A119387 a(n) is the number of binary digits (1's and nonleading 0's) which remain unchanged in their positions when n and (n+1) are written in binary.

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PARI

PARI

PARI

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A372888 Sum of binary ranks of all strict integer partitions of n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1).

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Maple

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A372887 Number of integer partitions of n whose distinct parts are the binary indices of some prime number.

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A080080 T(n,k) = length of longest carry sequence when adding k to n in binary representation, 1 <= k <= n (triangular array).

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