A143658 -id:A143658 - cp's OEIS frontend

A372684 Least k such that prime(k) >= 2^n.

1, 3, 5, 7, 12, 19, 32, 55, 98, 173, 310, 565, 1029, 1901, 3513, 6543, 12252, 23001, 43391, 82026, 155612, 295948, 564164, 1077872, 2063690, 3957810, 7603554, 14630844, 28192751, 54400029, 105097566, 203280222, 393615807, 762939112, 1480206280, 2874398516, 5586502349

Offset: 1

Gus Wiseman, May 30 2024

nonn

			The numbers prime(a(n)) together with their binary expansions and binary indices begin:
        2:                       10 ~ {2}
        5:                      101 ~ {1,3}
       11:                     1011 ~ {1,2,4}
       17:                    10001 ~ {1,5}
       37:                   100101 ~ {1,3,6}
       67:                  1000011 ~ {1,2,7}
      131:                 10000011 ~ {1,2,8}
      257:                100000001 ~ {1,9}
      521:               1000001001 ~ {1,4,10}
     1031:              10000000111 ~ {1,2,3,11}
     2053:             100000000101 ~ {1,3,12}
     4099:            1000000000011 ~ {1,2,13}
     8209:           10000000010001 ~ {1,5,14}
    16411:          100000000011011 ~ {1,2,4,5,15}
    32771:         1000000000000011 ~ {1,2,16}
    65537:        10000000000000001 ~ {1,17}
   131101:       100000000000011101 ~ {1,3,4,5,18}
   262147:      1000000000000000011 ~ {1,2,19}
   524309:     10000000000000010101 ~ {1,3,5,20}
  1048583:    100000000000000000111 ~ {1,2,3,21}
  2097169:   1000000000000000010001 ~ {1,5,22}
  4194319:  10000000000000000001111 ~ {1,2,3,4,23}
  8388617: 100000000000000000001001 ~ {1,4,24}

The opposite (greatest k such that prime(k) <= 2^n) is A007053.
Positions of first appearances in A035100.
The distance from prime(a(n)) to 2^n is A092131.
Counting zeros instead of all bits gives A372474, firsts of A035103.
Counting ones instead of all bits gives A372517, firsts of A014499.
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
For squarefree numbers between primes:
- sum A373197
- length A373198 = A061398 - 1
- min A000040
- max A112925, opposite A112926
Cf. A000120, A029837, A029931, A030190, A049095, A069010, A070939, A077641, A080791, A211997.

Table[PrimePi[If[n==1,2,NextPrime[2^n]]],{n,30}]

a(n) = primepi(nextprime(2^n)); \\ Michel Marcus, May 31 2024

a(n>1) = A007053(n) + 1.
a(n) = A000720(A104080(n)).
prime(a(n)) = A104080(n).
prime(a(n)) - 2^n = A092131(n).

More terms from Michel Marcus, May 31 2024

A372540 Least k such that the k-th squarefree number has binary expansion of length n. Index of the smallest squarefree number >= 2^n.

Original entry on oeis.org

1, 2, 4, 7, 12, 21, 40, 79, 158, 315, 625, 1246, 2492, 4983, 9963, 19921, 39845, 79689, 159361, 318726, 637462, 1274919, 2549835, 5099651, 10199302, 20398665, 40797328, 81594627, 163189198, 326378285, 652756723, 1305513584, 2611027095, 5222054082, 10444108052

Offset: 0

Gus Wiseman, May 10 2024

			The squarefree numbers A005117(a(n)) together with their binary expansions and binary indices begin:
       1:                  1 ~ {1}
       2:                 10 ~ {2}
       5:                101 ~ {1,3}
      10:               1010 ~ {2,4}
      17:              10001 ~ {1,5}
      33:             100001 ~ {1,6}
      65:            1000001 ~ {1,7}
     129:           10000001 ~ {1,8}
     257:          100000001 ~ {1,9}
     514:         1000000010 ~ {2,10}
    1027:        10000000011 ~ {1,2,11}
    2049:       100000000001 ~ {1,12}
    4097:      1000000000001 ~ {1,13}
    8193:     10000000000001 ~ {1,14}
   16385:    100000000000001 ~ {1,15}
   32770:   1000000000000010 ~ {2,16}
   65537:  10000000000000001 ~ {1,17}
  131073: 100000000000000001 ~ {1,18}

Chai Wah Wu, Table of n, a(n) for n = 0..73

Counting zeros instead of length gives A372473, firsts of A372472.
For prime instead of squarefree we have:
- zeros A372474, firsts of A035103
- ones A372517, firsts of A014499
- bits A372684, firsts of A035100
Positions of first appearances in A372475, run-lengths A077643.
For weight instead of length we have A372541, firsts of A372433.
Indices of the squarefree numbers listed by A372683.
A000120 counts ones in binary expansion (binary weight), zeros A080791.
A005117 lists squarefree numbers.
A030190 gives binary expansion, reversed A030308.
A070939 counts bits, binary length, or length of binary expansion.
Cf. A029931, A048793, A049093, A049094, A059015, A069010, A143658, A211997, A230877.

nn=1000;
ssnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[Max@@#]&];
dcs=IntegerLength[Select[Range[nn],SquareFreeQ],2];
Table[Position[dcs,i][[1,1]],{i,ssnm[dcs]}]

from itertools import count
from math import isqrt
from sympy import mobius, factorint
def A372540(n): return next(sum(mobius(a)*(k//a**2) for a in range(1, isqrt(k)+1)) for k in count(1<Chai Wah Wu, May 12 2024

A005117(a(n)) = A372683(n).

a(n) = A143658(n)+1 for n > 1. - Chai Wah Wu, Aug 26 2024

a(24)-a(34) from Chai Wah Wu, May 12 2024

A373125 Difference between 2^n and the least squarefree number >= 2^n.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, May 28 2024

nonn

For prime instead of squarefree we have A092131, opposite A013603.
For primes instead of powers of 2: A240474, A240473, A112926, A112925.
Difference between 2^n and A372683(n).
The opposite is A373126, delta of A372889.
A005117 lists squarefree numbers, first differences A076259.
A053797 gives lengths of gaps between squarefree numbers.
A061398 counts squarefree numbers between primes (exclusive).
A070939 or (preferably) A029837 gives length of binary expansion.
A077643 counts squarefree terms between powers of 2, run-lengths of A372475.
A143658 counts squarefree numbers up to 2^n.
Cf. A372473 (firsts of A372472), A372541 (firsts of A372433).
For primes between powers of 2:
- sum A293697 (except initial terms)
- length A036378
- min A104080 or A014210, indices A372684 (firsts of A035100)
- max A014234, delta A013603
Cf. A010036, A029931, A049093-A049096, A077641, A372540, A373197, A373198.

Table[NestWhile[#+1&,2^n,!SquareFreeQ[#]&]-2^n,{n,0,100}]

a(n) = A372683(n)-2^n. - R. J. Mathar, May 31 2024

A373412 Sum of the n-th maximal antirun of nonsquarefree numbers differing by more than one.

Original entry on oeis.org

12, 99, 52, 180, 93, 49, 335, 279, 156, 629, 99, 540, 237, 245, 125, 521, 567, 450, 963, 340, 347, 728, 1386, 1080, 1637, 243, 244, 1511, 1610, 555, 852, 1171, 2142, 960, 985, 1689, 343, 1042, 351, 1068, 724, 732, 1116, 1905, 1980, 2898, 424, 2161, 3150, 2339

Offset: 1

Gus Wiseman, Jun 06 2024

nonn

The length of this antirun is given by A373409.

An antirun of a sequence (in this case A013929) is an interval of positions at which consecutive terms differ by more than one.

			Row-sums of:
   4   8
   9  12  16  18  20  24
  25  27
  28  32  36  40  44
  45  48
  49
  50  52  54  56  60  63
  64  68  72  75
  76  80
  81  84  88  90  92  96  98
  99

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers

The partial sums are a subset of A329472.
Functional neighbors: A068781, A373404, A373405, A373409, A373410, A373411, A373414.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A025157, A049093, A049094, A061399, A101836, A112925, A143658, A294242, A350842, A372683, A373197.

Total/@Split[Select[Range[100],!SquareFreeQ[#]&],#1+1!=#2&]//Most

A373126 Difference between 2^n and the greatest squarefree number <= 2^n.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, May 29 2024

nonn

			The greatest squarefree number <= 2^21 is 2097149, and 2^21 = 2097152, so a(21) = 3.

For prime instead of squarefree we have A013603, opposite A092131.
For primes instead of powers of 2: A240474, A240473, A112926, A112925.
Difference between 2^n and A372889.
The opposite is A373125, delta of A372683.
A005117 lists squarefree numbers, first differences A076259.
A053797 gives lengths of gaps between squarefree numbers.
A061398 counts squarefree numbers between primes (exclusive).
A070939 or (preferably) A029837 gives length of binary expansion.
A077643 counts squarefree terms between powers of 2, run-lengths of A372475.
A143658 counts squarefree numbers up to 2^n.
Cf. A372473 (firsts of A372472), A372541 (firsts of A372433).
For primes between powers of 2:
- sum A293697 (except initial terms)
- length A036378
- min A104080 or A014210, indices A372684 (firsts of A035100)
- max A014234
Cf. A010036, A029931, A046933, A049093-A049096, A077641, A372540, A373197.

Table[2^n-NestWhile[#-1&,2^n,!SquareFreeQ[#]&],{n,0,100}]

a(n) = 2^n-A372889(n). - R. J. Mathar, May 31 2024

A373410 Minimum of the n-th maximal antirun of nonsquarefree numbers differing by more than one.

Original entry on oeis.org

4, 9, 25, 28, 45, 49, 50, 64, 76, 81, 99, 100, 117, 121, 125, 126, 136, 148, 153, 169, 172, 176, 189, 208, 225, 243, 244, 245, 261, 276, 280, 289, 297, 316, 325, 333, 343, 344, 351, 352, 361, 364, 369, 376, 388, 405, 424, 425, 441, 460, 476, 477, 496, 508, 513

Offset: 1

Gus Wiseman, Jun 06 2024

nonn

The maximum is given by A068781.
An antirun of a sequence (in this case A013929) is an interval of positions at which consecutive terms differ by more than one.
Consists of 4 and all nonsquarefree numbers n such that n - 1 is also nonsquarefree.

			Row-minima of:
   4   8
   9  12  16  18  20  24
  25  27
  28  32  36  40  44
  45  48
  49
  50  52  54  56  60  63
  64  68  72  75
  76  80
  81  84  88  90  92  96  98
  99

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers

Functional neighbors: A005381, A006512, A053806, A068781, A373408, A373409, A373412.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A049094, A061399, A077641, A077643, A112926, A143658, A294242, A350842, A372683, A372684, A373125.

First/@Split[Select[Range[100],!SquareFreeQ[#]&],#1+1!=#2&]

a(1) = 4; a(n>1) = A068781(n-1) + 1.

A373411 Sum of the n-th maximal antirun of squarefree numbers differing by more than one.

Original entry on oeis.org

1, 2, 8, 6, 17, 24, 14, 72, 22, 78, 30, 64, 34, 72, 38, 80, 42, 89, 263, 58, 120, 127, 66, 136, 70, 144, 151, 78, 161, 168, 86, 360, 94, 293, 102, 208, 106, 216, 110, 224, 114, 233, 241, 379, 130, 264, 271, 138, 280, 142, 288, 600, 312, 158, 648, 166, 510, 351

Offset: 1

Gus Wiseman, Jun 05 2024

nonn

The length of this antirun is given by A373127.

An antirun of a sequence (in this case A005117) is an interval of positions at which consecutive terms differ by more than one.

			Row-sums of:
   1
   2
   3  5
   6
   7 10
  11 13
  14
  15 17 19 21
  22
  23 26 29
  30
  31 33
  34
  35 37
  38
  39 41
  42
  43 46
  47 51 53 55 57

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers

The partial sums are a subset of A173143.
Functional neighbors: A007674, A373127 (firsts A373128, sorted firsts A373200), A373404, A373405, A373408, A373412, A373413.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A025157, A049093, A061398, A077641, A077643, A112925, A112926, A143658, A372683, A373123, A373197, A373198.

Total/@Split[Select[Range[100],SquareFreeQ],#1+1!=#2&]//Most

A373123 Sum of all squarefree numbers from 2^(n-1) to 2^n - 1.

Original entry on oeis.org

1, 5, 18, 63, 218, 891, 3676, 15137, 60580, 238672, 953501, 3826167, 15308186, 61204878, 244709252, 979285522, 3917052950, 15664274802, 62663847447, 250662444349, 1002632090376, 4010544455838, 16042042419476, 64168305037147, 256675237863576

Offset: 1

Gus Wiseman, May 27 2024

nonn

			This is the sequence of row sums of A005117 treated as a triangle with row-lengths A077643:
   1
   2   3
   5   6   7
  10  11  13  14  15
  17  19  21  22  23  26  29  30  31
  33  34  35  37  38  39  41  42  43  46  47  51  53  55  57  58  59  61  62

Counting all numbers (not just squarefree) gives A010036.
For the sectioning of A005117:
Row-lengths are A077643, partial sums A143658.
First column is A372683, delta A373125, indices A372540, firsts of A372475.
Last column is A372889, delta A373126, indices A143658, diffs A077643.
For primes instead of powers of two:
- sum A373197
- length A373198 = A061398 - 1
- maxima A112925, opposite A112926
For prime instead of squarefree:
- sum A293697 (except initial terms)
- length A036378
- min A104080 or A014210, indices A372684 (firsts of A035100)
- max A014234, delta A013603
A000120 counts ones in binary expansion (binary weight), zeros A080791.
A005117 lists squarefree numbers, first differences A076259.
A030190 gives binary expansion, reversed A030308.
A070939 or (preferably) A029837 gives length of binary expansion.
Cf. A372473 (firsts of A372472), A372541 (firsts of A372433).
Cf. A029931, A048793, A049093, A049094, A059015, A069010, A077641.

Table[Total[Select[Range[2^(n-1),2^n-1],SquareFreeQ]],{n,10}]

a(n) = my(s=0); forsquarefree(i=2^(n-1), 2^n-1, s+=i[1]); s; \\ Michel Marcus, May 29 2024

A373413 Sum of the n-th maximal run of squarefree numbers.

Original entry on oeis.org

6, 18, 21, 42, 17, 19, 66, 26, 90, 102, 114, 126, 93, 51, 53, 55, 174, 123, 198, 210, 147, 234, 165, 258, 89, 91, 282, 97, 306, 318, 330, 342, 237, 245, 127, 390, 267, 414, 426, 291, 149, 151, 309, 474, 161, 163, 498, 170, 347, 534, 546, 558, 381, 582, 197

Offset: 1

Gus Wiseman, Jun 05 2024

nonn

The length of this run is given by A120992.

A run of a sequence (in this case A005117) is an interval of positions at which consecutive terms differ by one.

			Row-sums of:
   1   2   3
   5   6   7
  10  11
  13  14  15
  17
  19
  21  22  23
  26
  29  30  31
  33  34  35
  37  38  39
  41  42  43
  46  47
  51
  53
  55
  57  58  59

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers

The partial sums are a subset of A173143.
Functional neighbors: A054265, A072284, A120992, A373406, A373411, A373414, A373415.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A049093, A049095, A061398, A077641, A077643, A143658, A371201, A373123, A373125, A373126, A373197.

Total/@Split[Select[Range[100],SquareFreeQ],#1+1==#2&]//Most

A372889 Greatest squarefree number <= 2^n.

Original entry on oeis.org

1, 2, 3, 7, 15, 31, 62, 127, 255, 511, 1023, 2047, 4094, 8191, 16383, 32767, 65535, 131071, 262142, 524287, 1048574, 2097149, 4194303, 8388607, 16777214, 33554431, 67108863, 134217727, 268435455, 536870911, 1073741822, 2147483647, 4294967295, 8589934591

Offset: 0

Gus Wiseman, May 27 2024

nonn

			The terms together with their binary expansions and binary indices begin:
      1:               1 ~ {1}
      2:              10 ~ {2}
      3:              11 ~ {1,2}
      7:             111 ~ {1,2,3}
     15:            1111 ~ {1,2,3,4}
     31:           11111 ~ {1,2,3,4,5}
     62:          111110 ~ {2,3,4,5,6}
    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}
   4094:    111111111110 ~ {2,3,4,5,6,7,8,9,10,11,12}
   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}

Positions of these terms in A005117 are A143658.
For prime instead of squarefree we have A014234, delta A013603.
For primes instead of powers of two we have A112925, opposite A112926.
Least squarefree number >= 2^n is A372683, delta A373125, indices A372540.
The opposite for prime instead of squarefree is A372684, firsts of A035100.
The delta (difference from 2^n) is A373126.
A000120 counts ones in binary expansion (binary weight), zeros A080791.
A005117 lists squarefree numbers, first differences A076259.
A030190 gives binary expansion, reversed A030308, length A070939 or A029837.
A061398 counts squarefree numbers between primes, exclusive.
A077643 counts squarefree terms between powers of 2, run-lengths of A372475.
Cf. A029931, A048793, A049093, A049094, A077641, A372433, A372472, A372473, A372541, A373197, A373198.

Table[NestWhile[#-1&,2^n,!SquareFreeQ[#]&],{n,0,15}]

a(n) = my(k=2^n); while (!issquarefree(k), k--); k; \\ Michel Marcus, May 29 2024

a(n) = A005117(A143658(n)).

a(n) = A070321(2^n). - R. J. Mathar, May 31 2024

cp's OEIS Frontend

A372684 Least k such that prime(k) >= 2^n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A372540 Least k such that the k-th squarefree number has binary expansion of length n. Index of the smallest squarefree number >= 2^n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A373125 Difference between 2^n and the least squarefree number >= 2^n.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A373412 Sum of the n-th maximal antirun of nonsquarefree numbers differing by more than one.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A373126 Difference between 2^n and the greatest squarefree number <= 2^n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A373410 Minimum of the n-th maximal antirun of nonsquarefree numbers differing by more than one.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A373411 Sum of the n-th maximal antirun of squarefree numbers differing by more than one.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A373123 Sum of all squarefree numbers from 2^(n-1) to 2^n - 1.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica