A371571 -id:A371571 - cp's OEIS frontend

A372689 Positive integers whose binary indices (positions of ones in reversed binary expansion) sum to a prime number.

2, 3, 4, 6, 9, 11, 12, 16, 18, 23, 26, 29, 33, 38, 41, 43, 44, 48, 50, 55, 58, 61, 64, 69, 71, 72, 74, 79, 81, 86, 89, 91, 92, 96, 101, 103, 104, 106, 111, 113, 118, 121, 131, 132, 134, 137, 142, 144, 149, 151, 152, 154, 159, 163, 164, 166, 169, 174, 176, 181

Offset: 1

Gus Wiseman, May 18 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.

Note the function taking a set s to its binary rank Sum_i 2^(s_i-1) is the inverse of A048793 (binary indices).

			The terms together with their binary expansions and binary indices begin:
   2:      10 ~ {2}
   3:      11 ~ {1,2}
   4:     100 ~ {3}
   6:     110 ~ {2,3}
   9:    1001 ~ {1,4}
  11:    1011 ~ {1,2,4}
  12:    1100 ~ {3,4}
  16:   10000 ~ {5}
  18:   10010 ~ {2,5}
  23:   10111 ~ {1,2,3,5}
  26:   11010 ~ {2,4,5}
  29:   11101 ~ {1,3,4,5}
  33:  100001 ~ {1,6}
  38:  100110 ~ {2,3,6}
  41:  101001 ~ {1,4,6}
  43:  101011 ~ {1,2,4,6}
  44:  101100 ~ {3,4,6}
  48:  110000 ~ {5,6}
  50:  110010 ~ {2,5,6}
  55:  110111 ~ {1,2,3,5,6}
  58:  111010 ~ {2,4,5,6}
  61:  111101 ~ {1,3,4,5,6}

Numbers k such that A029931(k) is prime.
Union of prime-indexed rows of A118462.
For even instead of prime we have A158704, odd A158705.
For prime indices instead of binary indices we have A316091.
The prime case is A372885, indices 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.
A372471 lists binary indices of primes, row-sums A372429.
A372687 counts strict partitions of prime binary rank, counted by A372851.
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. A035100, A071814, A096111, A023506, A277319, A372688, A372850, A372887.

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

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

Original entry on oeis.org

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)

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}]

A373120 Number of distinct possible binary ranks of 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

1, 1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 26, 33, 43, 55, 70, 89, 109, 136, 167, 206, 251, 306, 371, 445, 535, 639, 759, 904, 1069, 1262, 1489, 1747, 2047, 2390, 2784, 3237, 3754, 4350, 5027, 5798, 6680, 7671, 8808, 10091, 11543, 13190, 15040, 17128, 19477, 22118

Offset: 0

Gus Wiseman, May 26 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, so a(4) = 3.

The strict case is A000009.
A048675 gives binary rank of prime indices, distinct A087207.
A118462 lists binary ranks of strict integer partitions, row sums A372888.
A277905 groups all positive integers by binary rank of prime indices.
A372890 adds up binary ranks of integer partitions.
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, A018819, A158704, A158705, A225620, A231204, A372688.

Table[Length[Union[Total[2^(#-1)]&/@IntegerPartitions[n]]],{n,0,15}]

cp's OEIS Frontend

A372689 Positive integers whose binary indices (positions of ones in reversed binary expansion) sum to a prime number.

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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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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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Mathematica

A373120 Number of distinct possible binary ranks of 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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Mathematica