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.

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A372517 Least k such that the k-th prime number has exactly n ones in its binary expansion.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, May 12 2024

Keywords

Comments

In other words, the a(n)-th prime is the least with binary weight n. The sorted version is A372686.

Examples

			The primes A000040(a(n)) 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}
      311:              100110111 ~ {1,2,3,5,6,9}
      127:                1111111 ~ {1,2,3,4,5,6,7}
      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}
     8191:          1111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13}
    73727:      10001111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13,17}
    63487:       1111011111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,13,14,15,16}

Crossrefs

Positions firsts of first appearances in A014499.
Taking primes gives A061712.
Counting zeros (weight) gives A372474, firsts of A035103.
For binary length we have A372684 (take primes A104080), firsts of A035100.
The sorted version is A372686, taking primes A372685.
A000120 counts ones in binary expansion (binary weight), zeros A080791.
A029837 gives greatest binary index, least A001511.
A030190 gives binary expansion, reversed A030308.
A048793 lists binary indices, reverse A272020, sum A029931.
A372471 lists binary indices of primes.
Cf. A000040, A066195, A069010, A070939, A071814, A211997, A372429, A372516.

Programs

  • Mathematica
    spsm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[Max@@#]&];
j=DigitCount[#,2,1]&/@Select[Range[1000],PrimeQ];
Table[Position[j,k][[1,1]],{k,spsm[j]}]
  • PARI
    a(n) = my(k=1, p=2); while(hammingweight(p) !=n, p = nextprime(p+1); k++); k; \\ Michel Marcus, May 13 2024
  • Python
    from itertools import count
from sympy import isprime, primepi
from sympy.utilities.iterables import multiset_permutations
def A372517(n):
    for l in count(n-1):
        m = 1<Chai Wah Wu, May 13 2024

Formula

A000040(a(n)) = A061712(n).

Extensions

a(32)-a(36) from Pontus von Brömssen, May 13 2024

A372541 Least k such that the k-th squarefree number has exactly n ones in its binary expansion.

Original entry on oeis.org

1, 3, 6, 11, 20, 60, 78, 157, 314, 624, 1245, 3736, 4982, 9962, 19920, 39844, 79688, 239046, 318725, 956194, 1912371, 2549834, 5099650, 15298984, 20398664, 40797327, 81594626, 163189197, 326378284, 979135127, 1305513583, 2611027094, 5222054081, 10444108051
Offset: 0

Views

Author

Gus Wiseman, May 09 2024

Keywords

Examples

			The squarefree numbers A005117(a(n)) together with their binary expansions and binary indices begin:
       1:                   1 ~ {1}
       3:                  11 ~ {1,2}
       7:                 111 ~ {1,2,3}
      15:                1111 ~ {1,2,3,4}
      31:               11111 ~ {1,2,3,4,5}
      95:             1011111 ~ {1,2,3,4,5,7}
     127:             1111111 ~ {1,2,3,4,5,6,7}
     255:            11111111 ~ {1,2,3,4,5,6,7,8}
     511:           111111111 ~ {1,2,3,4,5,6,7,8,9}
    1023:          1111111111 ~ {1,2,3,4,5,6,7,8,9,10}
    2047:         11111111111 ~ {1,2,3,4,5,6,7,8,9,10,11}
    6143:       1011111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,13}
    8191:       1111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13}
   16383:      11111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13,14}
   32767:     111111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
   65535:    1111111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16}
  131071:   11111111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17}

Links

Crossrefs

Positions of firsts appearances in A372433.
Counting zeros instead of ones gives A372473, firsts in A372472.
For prime instead of squarefree we have A372517, firsts of A014499.
Counting bits (length) gives A372540, firsts of A372475, runs A077643.
A000120 counts ones in binary expansion (binary weight), zeros A080791.
A005117 lists squarefree numbers.
A030190 gives binary expansion, reversed A030308.
A048793 lists positions of ones in reversed binary expansion, sum A029931.
A145037, A097110 count ones minus zeros, for primes A372516, A177796.
A371571 lists positions of zeros in binary expansion, sum A359359.
A371572 lists positions of ones in binary expansion, sum A230877.
A372515 lists positions of zeros in reversed binary expansion, sum A359400.
Cf. A023416, A049093, A049094, A069010, A070939, A071403, A211997, A280296, A372474.

Programs

  • Mathematica
    nn=10000;
spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
dcs=DigitCount[Select[Range[nn],SquareFreeQ],2,1];
Table[Position[dcs,i][[1,1]],{i,spnm[dcs-1]}]
  • Python
    from math import isqrt
from itertools import count
from sympy import factorint, mobius
from sympy.utilities.iterables import multiset_permutations
def A372541(n):
    if n==0: return 1
    for l in count(n):
        m = 1<Chai Wah Wu, May 10 2024

Extensions

a(23)-a(33) from Chai Wah Wu, May 10 2024

A372516 Number of ones minus number of zeros in the binary expansion of the n-th prime number.

Original entry on oeis.org

0, 2, 1, 3, 2, 2, -1, 1, 3, 3, 5, 0, 0, 2, 4, 2, 4, 4, -1, 1, -1, 3, 1, 1, -1, 1, 3, 3, 3, 1, 7, -2, -2, 0, 0, 2, 2, 0, 2, 2, 2, 2, 6, -2, 0, 2, 2, 6, 2, 2, 2, 6, 2, 6, -5, -1, -1, 1, -1, -1, 1, -1, 1, 3, 1, 3, 1, -1, 3, 3, -1, 3, 5, 3, 5, 7, -1, 1, -1, 1, 1
Offset: 1

Views

Author

Gus Wiseman, May 13 2024

Keywords

Comments

Absolute value is A177718.

Examples

			The binary expansion of 83 is (1,0,1,0,0,1,1), and 83 is the 23rd prime, so a(23) = 4 - 3 = 1.

Crossrefs

The sum instead of difference is A035100, firsts A372684 (primes A104080).
The negative version is A037861(A000040(n)).
Restriction of A145037 to the primes.
The unsigned version is A177718.
- Positions of zeros are A177796, indices of the primes A066196.
- Positions of positive terms are indices of the primes A095070.
- Positions of negative terms are indices of the primes A095071.
- Positions of negative ones are A372539, indices of the primes A095072.
- Positions of ones are A372538, indices of the primes A095073.
- Positions of nonnegative terms are indices of the primes A095074.
- Positions of nonpositive terms are indices of the primes A095075.
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 length of binary expansion.
A101211 lists run-lengths in binary expansion, row-lengths A069010.
A372471 lists the binary indices of each prime.
Cf. A000043, A003714, A005940, A059305, A061712, A066195, A071814, A211997, A372429, A372517, A372686.

Programs

  • Mathematica
    Table[DigitCount[Prime[n],2,1]-DigitCount[Prime[n],2,0],{n,100}]
DigitCount[#,2,1]-DigitCount[#,2,0]&/@Prime[Range[100]] (* Harvey P. Dale, May 09 2025 *)

Formula

a(n) = A000120(A000040(n)) - A080791(A000040(n)).
a(n) = A014499(n) - A035103(n).
a(n) = A145037(A000040(n))

A211998 Positions where the monotonicity of A061712 is broken.

Original entry on oeis.org

6, 14, 22, 30, 38, 62, 78, 94, 126, 174, 206, 254, 510, 542, 606, 766, 1022, 1278, 2046
Offset: 1

Views

Author

Vladimir Shevelev and Peter J. C. Moses, Oct 25 2012

Keywords

Crossrefs

Cf. A061712, A211997.

Programs

  • Mathematica
    (* This script is not convenient for more than 10 terms *) A061712[n_] := A061712[ n] = Module[{m, s, k, p}, For[m=0, True, m++, s = {1, Sequence @@ #, 1} & /@ Permutations[Join[Table[1, {n-2}], Table[0, {m}]]] // Sort; For[k=1, k <= Length[ s], k++, p = FromDigits[s[[k]], 2]; If[PrimeQ[p], Return[p]]]]]; A061712[1] = 2; Reap[Do[If[A061712[n+1] < A061712[n], Print[n]; Sow[n]], {n, 1, 250}]][[2, 1]] (* Jean-François Alcover, Mar 16 2015 *)

Formula

{n: A061712(n+1) < A061712(n)}.

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

A373124 Sum of indices of primes between powers of 2.

Original entry on oeis.org

1, 2, 7, 11, 45, 105, 325, 989, 3268, 10125, 33017, 111435, 369576, 1277044, 4362878, 15233325, 53647473, 189461874, 676856245, 2422723580, 8743378141, 31684991912, 115347765988, 421763257890, 1548503690949, 5702720842940, 21074884894536, 78123777847065
Offset: 0

Views

Author

Gus Wiseman, May 31 2024

Keywords

Comments

Sum of k such that 2^n+1 <= prime(k) <= 2^(n+1).

Examples

			Row-sums of the sequence of all positive integers as a triangle with row-lengths A036378:
   1
   2
   3  4
   5  6
   7  8  9 10 11
  12 13 14 15 16 17 18
  19 20 21 22 23 24 25 26 27 28 29 30 31
  32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

Crossrefs

For indices of primes between powers of 2:
- sum A373124 (this sequence)
- length A036378
- min A372684 (except initial terms), delta A092131
- max A007053
For primes between powers of 2:
- sum A293697
- length A036378
- min A104080 or A014210
- max A014234, delta A013603
For squarefree numbers between powers of 2:
- sum A373123
- length A077643, run-lengths of A372475
- min A372683, delta A373125, indices A372540
- max A372889, delta A373126, indices A143658
Cf. A000040, A000120, A014499, A029837, A029931, A035100, A069010, A070939, A112925, A112926, A211997.

Programs

  • Mathematica
    Table[Total[PrimePi/@Select[Range[2^(n-1)+1,2^n],PrimeQ]],{n,10}]
  • PARI
    ip(n) = primepi(1<A007053
t(n) = n*(n+1)/2; \\ A000217
a(n) = t(ip(n+1)) - t(ip(n)); \\ Michel Marcus, May 31 2024

A278477 Primes that set a new record for the Hamming weight.

Original entry on oeis.org

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

Views

Author

Joerg Arndt, Nov 23 2016

Keywords

Comments

The Mersenne primes (A000668) are a subsequence.

Links

Crossrefs

Cf. A000668, A061712, A211997.

Programs

  • Maple
    M:= 40: # to use A061712(1..M)
A061712:= proc(n) local d,c,cands;
  for d from 0 do
    cands:= map(t -> 2^(n+d)-1 - add(2^(n-1+d-j), j=t),
        combinat:-choose([$1..n-2+d], d));
    for c in cands do if  isprime(c) then return c fi od
  od
end proc:
A061712(1):= 2:
R:= map(A061712, [$1..M]):
R[select(t -> R[t] < `if`(isprime(2^(M+1)-1), 2^(M+1)-1, 2^(M+2)+2^M-1) and R[t] = min(R[t..-1]), [$1..nops(R)])]; # Robert Israel, Nov 23 2016
  • PARI
    {my(h=0);forprime(p=2,10^11,my(t=hammingweight(p));if(t>h,print1(p,", ");h=t));}
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