cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-18 of 18 results.

A372687 Number of prime numbers whose binary indices sum to n. Number of strict integer partitions y of n such that Sum_i 2^(y_i-1) is prime.

Original entry on oeis.org

0, 0, 1, 1, 1, 0, 2, 1, 2, 0, 3, 3, 1, 4, 1, 6, 5, 8, 4, 12, 8, 12, 7, 20, 8, 16, 17, 27, 19, 38, 19, 46, 33, 38, 49, 65, 47, 67, 83, 92, 94, 113, 103, 130, 146, 127, 215, 224, 176, 234, 306, 270, 357, 383, 339, 393, 537, 540, 597, 683, 576, 798, 1026, 830, 1157
Offset: 0

Views

Author

Gus Wiseman, May 15 2024

Keywords

Comments

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).

Examples

			The a(2) = 1 through a(17) = 8 prime numbers:
  2  3  5  .  17  11  19  .  257  131  73  137  97  521  4099  1031
              7       13     67   41       71       263  2053  523
                             37   23       43       139  1033  269
                                           29       83   193   163
                                                    53   47    149
                                                    31         101
                                                               89
                                                               79
The a(2) = 1 through a(11) = 3 strict partitions:
  (2)  (2,1)  (3,1)  .  (5,1)    (4,2,1)  (4,3,1)  .  (9,1)    (6,4,1)
                        (3,2,1)           (5,2,1)     (6,3,1)  (8,2,1)
                                                      (7,2,1)  (5,3,2,1)

Crossrefs

For all positive integers (not just prime) we get A000009.
Number of prime numbers p with A029931(p) = n.
For odd instead of prime we have A096765, even A025147, non-strict A087787
Number of times n appears in A372429.
Number of rows of A372471 with sum n.
The non-strict version is A372688 (or A372887), ranks A277319 (or A372850).
These (strict) partitions have Heinz numbers A372851.
A014499 lists binary indices of prime numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A038499 counts partitions of prime length, strict A085756.
A048793 lists binary indices:
- length A000120
- min A001511
- sum A029931
- max A070939
- reverse A272020
A058698 counts partitions of prime numbers, strict A064688.
A096111 gives product of binary indices.
A372689 lists numbers whose binary indices sum to a prime.
A372885 lists primes whose binary indices sum to a prime, indices A372886.
Cf. A000040, A005940, A023506, A029837, A035100, A071814, A230877, A231204, A359359, A372436, A372441.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&PrimeQ[Total[2^#]/2]&]],{n,0,30}]

A372850 Numbers whose distinct prime indices are the binary indices of some prime number.

Original entry on oeis.org

3, 6, 9, 10, 12, 18, 20, 22, 24, 27, 30, 36, 40, 42, 44, 46, 48, 50, 54, 60, 66, 70, 72, 80, 81, 84, 88, 90, 92, 96, 100, 102, 108, 114, 118, 120, 126, 130, 132, 140, 144, 150, 160, 162, 168, 176, 180, 182, 184, 192, 198, 200, 204, 216, 228, 236, 238, 240, 242
Offset: 1

Views

Author

Gus Wiseman, May 16 2024

Keywords

Comments

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.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Note the function taking a set s to its rank Sum_i 2^(s_i-1) is the inverse of A048793 (binary indices).

Examples

			The distinct prime indices of 45 are {2,3}, which are the binary indices of 6, which is not prime, so 45 is not in the sequence.
The distinct prime indices of 60 are {1,2,3}, which are the binary indices of 7, which is prime, so 60 is in the sequence.
The terms together with their prime indices begin:
    3: {2}
    6: {1,2}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   18: {1,2,2}
   20: {1,1,3}
   22: {1,5}
   24: {1,1,1,2}
   27: {2,2,2}
   30: {1,2,3}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   46: {1,9}
   48: {1,1,1,1,2}
   50: {1,3,3}
   54: {1,2,2,2}
   60: {1,1,2,3}
   66: {1,2,5}
   70: {1,3,4}

Crossrefs

For even instead of prime we have A005408, with multiplicity A003159.
For odd instead of prime we have A005843, with multiplicity A036554.
For prime indices with multiplicity we have A277319, counted by A372688.
The squarefree case is A372851, counted by A372687.
Partitions of this type are counted by A372887.
A014499 lists binary indices of prime numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A038499 counts partitions of prime length, strict A085756.
A048793 and A272020 (reverse) list binary indices:
- length A000120
- min A001511
- sum A029931
- max A070939
A058698 counts partitions of prime numbers, strict A064688.
A372885 lists primes whose binary indices sum to a prime, indices A372886.
Cf. A000040, A005940, A023506, A029837, A035100, A071814, A096111, A372429, A372436, A372441, A372471, A372689.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],PrimeQ[Total[2^(Union[prix[#]]-1)]]&]

Formula

Numbers k such that Sum_{i:prime(i)|k} 2^(i-1) is prime, where the sum is over the distinct prime indices of k.

A372851 Squarefree numbers whose prime indices are the binary indices of some prime number.

Original entry on oeis.org

3, 6, 10, 22, 30, 42, 46, 66, 70, 102, 114, 118, 130, 182, 238, 246, 266, 318, 330, 354, 370, 402, 406, 434, 442, 510, 546, 646, 654, 690, 762, 770, 798, 930, 938, 946, 962, 986, 1066, 1102, 1122, 1178, 1218, 1222, 1246, 1258, 1334, 1378, 1430, 1482, 1578
Offset: 1

Views

Author

Gus Wiseman, May 16 2024

Keywords

Comments

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.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Note the function taking a set s to its rank Sum_i 2^(s_i-1) is the inverse of A048793 (binary indices).

Examples

			The prime indices of 70 are {1,3,4}, which are the binary indices of 13, which is prime, so 70 is in the sequence.
The prime indices of 15 are {2,3}, which are the binary indices of 6, which is not prime, so 15 is not in the sequence.
The terms together with their prime indices begin:
    3: {2}
    6: {1,2}
   10: {1,3}
   22: {1,5}
   30: {1,2,3}
   42: {1,2,4}
   46: {1,9}
   66: {1,2,5}
   70: {1,3,4}
  102: {1,2,7}
  114: {1,2,8}
  118: {1,17}
  130: {1,3,6}
  182: {1,4,6}
  238: {1,4,7}
  246: {1,2,13}
  266: {1,4,8}
  318: {1,2,16}
  330: {1,2,3,5}
  354: {1,2,17}
  370: {1,3,12}
  402: {1,2,19}

Crossrefs

[Warning: do not confuse A372887 with the strict case A372687.]
For odd instead of prime we have A039956.
For even instead of prime we have A056911.
Strict partitions of this type are counted by A372687.
Non-strict partitions of this type are counted by A372688, ranks A277319.
The nonsquarefree version is A372850, counted by A372887.
A014499 lists binary indices of prime numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A038499 counts partitions of prime length, strict A085756.
A048793 and A272020 (reverse) list binary indices:
- length A000120
- min A001511
- sum A029931
- max A070939
A058698 counts partitions of prime numbers, strict A064688.
A372885 lists primes whose binary indices sum to a prime, indices A372886.
Cf. A000040, A005940, A025147, A035100, A071814, A096111, A096765, A231204, A372429, A372471, A372689.

Programs

  • Mathematica
    Select[Range[100],SquareFreeQ[#] && PrimeQ[Total[2^(PrimePi/@First/@FactorInteger[#]-1)]]&]

Formula

Squarefree numbers k such that Sum_{i:prime(i)|k} 2^(i-1) is prime, where the sum is over the (distinct) prime indices of k.

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

Views

Author

Gus Wiseman, May 19 2024

Keywords

Comments

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).

Examples

			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)

Crossrefs

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.

Programs

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

A372686 Sorted list of positions of first appearances in A014499 (number of ones in binary expansion of each prime).

Original entry on oeis.org

1, 2, 4, 9, 11, 31, 64, 76, 167, 309, 502, 801, 1028, 6363, 7281, 12079, 12251, 43237, 43390, 146605, 291640, 951351, 1046198, 2063216, 3957778, 11134645, 14198321, 28186247, 54387475, 105097565, 249939829, 393248783, 751545789, 1391572698, 2182112798, 8242984130
Offset: 1

Views

Author

Gus Wiseman, May 14 2024

Keywords

Comments

The unsorted version is A372517.

Examples

			The sequence contains 9 because the first 9 terms of A014499 are 1, 2, 2, 3, 3, 3, 2, 3, 4, and the last of these is the first position of 4.

Links

Crossrefs

Positions of first appearances in A014499.
The unsorted version is A372517.
For binary length we have A372684, primes A104080, firsts of A035100.
Taking primes gives A372685, unsorted version A061712.
A000120 counts ones in binary expansion (binary weight), zeros A080791.
A029837 gives greatest binary index, least A001511.
A030190 gives binary expansion, reversed A030308.
A035103 counts zeros in binary expansion of each prime, firsts A372474.
A048793 lists binary indices, reverse A272020, sum A029931.
A070939 gives length of binary expansion (number of bits).
A372471 lists binary indices of primes.
Cf. A000040, A005940, A059015, A066195, A069010, A071814, A211997, A372429, A372433, A372473, A372516.

Programs

  • Mathematica
    First/@GatherBy[Range[1000],DigitCount[Prime[#],2,1]&]

Formula

prime(a(n)) = A372685(n).

Extensions

a(26)-a(36) from Pontus von Brömssen, May 15 2024

A372538 Numbers k such that the number of ones minus the number of zeros in the binary expansion of the k-th prime number is 1.

Original entry on oeis.org

3, 8, 20, 23, 24, 26, 30, 58, 61, 63, 65, 67, 78, 80, 81, 82, 84, 88, 185, 187, 194, 200, 201, 203, 213, 214, 215, 221, 225, 226, 227, 234, 237, 246, 249, 253, 255, 256, 257, 259, 266, 270, 280, 284, 287, 290, 573, 578, 586, 588, 591, 593, 611, 614, 615, 626
Offset: 1

Views

Author

Gus Wiseman, May 13 2024

Keywords

Examples

			The binary expansion of 83 is (1,0,1,0,0,1,1) with ones minus zeros 4 - 3 = 1, and 83 is the 23rd prime, so 23 is in the sequence.
The primes A000040(a(n)) together with their binary expansions and binary indices begin:
     5:           101 ~ {1,3}
    19:         10011 ~ {1,2,5}
    71:       1000111 ~ {1,2,3,7}
    83:       1010011 ~ {1,2,5,7}
    89:       1011001 ~ {1,4,5,7}
   101:       1100101 ~ {1,3,6,7}
   113:       1110001 ~ {1,5,6,7}
   271:     100001111 ~ {1,2,3,4,9}
   283:     100011011 ~ {1,2,4,5,9}
   307:     100110011 ~ {1,2,5,6,9}
   313:     100111001 ~ {1,4,5,6,9}
   331:     101001011 ~ {1,2,4,7,9}
   397:     110001101 ~ {1,3,4,8,9}
   409:     110011001 ~ {1,4,5,8,9}
   419:     110100011 ~ {1,2,6,8,9}
   421:     110100101 ~ {1,3,6,8,9}
   433:     110110001 ~ {1,5,6,8,9}
   457:     111001001 ~ {1,4,7,8,9}
  1103:   10001001111 ~ {1,2,3,4,7,11}
  1117:   10001011101 ~ {1,3,4,5,7,11}
  1181:   10010011101 ~ {1,3,4,5,8,11}
  1223:   10011000111 ~ {1,2,3,7,8,11}

Crossrefs

Restriction of A031448 to the primes, positions of ones in A145037.
Taking primes gives A095073, negative A095072.
Positions of ones in A372516, absolute value A177718.
Cf. A066196, A095070-A095075, A177796, A372539.
A000120 counts ones in binary expansion (binary weight), zeros A080791.
A030190 gives binary expansion, reversed A030308.
A035103 counts zeros in binary expansion of primes, firsts A372474.
A048793 lists binary indices, reverse A272020, sum A029931.
A070939 gives the length of an integer's binary expansion.
A101211 lists run-lengths in binary expansion, row-lengths A069010.
A372471 lists binary indices of primes.
Cf. A000040, A003714, A035100, A037861, A053738, A061712, A066195, A104080, A211997, A372429, A372517, A372686.

Programs

  • Mathematica
    Select[Range[1000],DigitCount[Prime[#],2,1]-DigitCount[Prime[#],2,0]==1&]

A372539 Numbers k such that the number of ones minus the number of zeros in the binary expansion of the k-th prime number is -1.

Original entry on oeis.org

7, 19, 21, 25, 56, 57, 59, 60, 62, 68, 71, 77, 79, 87, 175, 177, 179, 180, 186, 188, 189, 192, 193, 195, 196, 197, 204, 210, 212, 216, 218, 243, 244, 248, 254, 262, 263, 265, 279, 567, 572, 576, 577, 583, 592, 598, 599, 600, 602, 603, 605, 606, 610, 613, 616
Offset: 1

Views

Author

Gus Wiseman, May 14 2024

Keywords

Examples

			The binary expansion of 17 is (1,0,0,0,1) with ones minus zeros 2 - 3 = -1, and 17 is the 7th prime, 7 is in the sequence.
The primes A000040(a(n)) together with their binary expansions and binary indices begin:
    17:         10001 ~ {1,5}
    67:       1000011 ~ {1,2,7}
    73:       1001001 ~ {1,4,7}
    97:       1100001 ~ {1,6,7}
   263:     100000111 ~ {1,2,3,9}
   269:     100001101 ~ {1,3,4,9}
   277:     100010101 ~ {1,3,5,9}
   281:     100011001 ~ {1,4,5,9}
   293:     100100101 ~ {1,3,6,9}
   337:     101010001 ~ {1,5,7,9}
   353:     101100001 ~ {1,6,7,9}
   389:     110000101 ~ {1,3,8,9}
   401:     110010001 ~ {1,5,8,9}
   449:     111000001 ~ {1,7,8,9}
  1039:   10000001111 ~ {1,2,3,4,11}
  1051:   10000011011 ~ {1,2,4,5,11}
  1063:   10000100111 ~ {1,2,3,6,11}
  1069:   10000101101 ~ {1,3,4,6,11}
  1109:   10001010101 ~ {1,3,5,7,11}
  1123:   10001100011 ~ {1,2,6,7,11}
  1129:   10001101001 ~ {1,4,6,7,11}
  1163:   10010001011 ~ {1,2,4,8,11}

Crossrefs

Restriction of A031444 (positions of '-1's in A145037) to A000040.
Taking primes gives A095072.
Positions of negative ones in A372516, absolute value A177718.
The negative version is A372538, taking primes A095073.
A000120 counts ones in binary expansion (binary weight), zeros A080791.
A030190 gives binary expansion, reversed A030308.
A035103 counts zeros in binary expansion of primes, firsts A372474.
A048793 lists binary indices, reverse A272020, sum A029931.
A070939 gives the length of an integer's binary expansion.
A101211 lists run-lengths in binary expansion, row-lengths A069010.
A372471 lists binary indices of primes.
Cf. A003714, A031448, A035100, A037861, A053738, A061712, A066195, A104080, A211997, A372429, A372517, A372686.

Programs

  • Mathematica
    Select[Range[1000],DigitCount[Prime[#],2,1]-DigitCount[Prime[#],2,0]==-1&]

A372685 Prime numbers such that no lesser prime has the same binary weight (number of ones in binary expansion).

Original entry on oeis.org

2, 3, 7, 23, 31, 127, 311, 383, 991, 2039, 3583, 6143, 8191, 63487, 73727, 129023, 131071, 522239, 524287, 1966079, 4128767, 14680063, 16250879, 33546239, 67108351, 201064447, 260046847, 536739839, 1073479679, 2147483647, 5335154687, 8581545983, 16911433727
Offset: 1

Views

Author

Gus Wiseman, May 10 2024

Keywords

Comments

The unsorted version is A061712.

Examples

			The terms together with their binary expansions and binary indices begin:
     2:            10 ~ {2}
     3:            11 ~ {1,2}
     7:           111 ~ {1,2,3}
    23:         10111 ~ {1,2,3,5}
    31:         11111 ~ {1,2,3,4,5}
   127:       1111111 ~ {1,2,3,4,5,6,7}
   311:     100110111 ~ {1,2,3,5,6,9}
   383:     101111111 ~ {1,2,3,4,5,6,7,9}
   991:    1111011111 ~ {1,2,3,4,5,7,8,9,10}
  2039:   11111110111 ~ {1,2,3,5,6,7,8,9,10,11}
  3583:  110111111111 ~ {1,2,3,4,5,6,7,8,9,11,12}
  6143: 1011111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,13}

Links

Crossrefs

This statistic (binary weight of primes) is A014499.
Sorted version of A061712.
For binary length instead of weight we have A104080, firsts of A035100.
These primes have indices A372686, sorted version of A372517.
A000120 counts ones in binary expansion (binary weight), zeros A080791.
A029837 gives greatest binary index, least A001511.
A030190 gives binary expansion, reversed A030308.
A035103 counts zeros in binary expansion of primes, firsts A372474.
A048793 lists binary indices, reverse A272020, sum A029931.
A372471 lists binary indices of primes.
Cf. A000040, A005940, A066195, A069010, A070939, A071814, A211997, A372429, A372516, A372684.

Programs

  • Mathematica
    First/@GatherBy[Select[Range[1000],PrimeQ],DigitCount[#,2,1]&]
  • Python
    from itertools import islice
from sympy import nextprime
def A372685_gen(): # generator of terms
    p, a = 1, {}
    while (p:=nextprime(p)):
        if (c:=p.bit_count()) not in a:
            yield p
        a[c] = p
A372685_list = list(islice(A372685_gen(),20)) # Chai Wah Wu, May 12 2024

Formula

a(n) = prime(A372686(n)).

Extensions

a(22)-a(33) from Chai Wah Wu, May 12 2024
Previous Showing 11-18 of 18 results.

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