A373126 -id:A373126 - cp's OEIS frontend

A372889 Greatest squarefree number <= 2^n.

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

A373415 Maximum of the n-th maximal run of squarefree numbers.

Original entry on oeis.org

3, 7, 11, 15, 17, 19, 23, 26, 31, 35, 39, 43, 47, 51, 53, 55, 59, 62, 67, 71, 74, 79, 83, 87, 89, 91, 95, 97, 103, 107, 111, 115, 119, 123, 127, 131, 134, 139, 143, 146, 149, 151, 155, 159, 161, 163, 167, 170, 174, 179, 183, 187, 191, 195, 197, 199, 203, 206

Offset: 1

Gus Wiseman, Jun 05 2024

nonn

The minimum is given by A072284.
A run of a sequence (in this case A005117) is an interval of positions at which consecutive terms differ by one.
Consists of all squarefree numbers k such that k + 1 is not squarefree.

			Row-maxima 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

Functional neighbors: A006093, A007674, A067774, A072284, A120992, A373413.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A049093, A049094, A061398, A069010, A077641, A077643, A112925, A112926, A143658, A372683, A373125, A373126.

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

a(n) = A070321(A072284(n+1) - 1).

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

Gus Wiseman, May 31 2024

nonn

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

			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

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.

Table[Total[PrimePi/@Select[Range[2^(n-1)+1,2^n],PrimeQ]],{n,10}]

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

cp's OEIS Frontend

A372889 Greatest squarefree number <= 2^n.

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A373415 Maximum of the n-th maximal run of squarefree numbers.

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A373124 Sum of indices of primes between powers of 2.

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