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 11-15 of 15 results.

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

Views

Author

Gus Wiseman, May 27 2024

Keywords

Examples

			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

Crossrefs

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.

Programs

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

Views

Author

Gus Wiseman, Jun 05 2024

Keywords

Comments

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.

Examples

			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

Links

Crossrefs

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.

Programs

  • Mathematica
    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

Views

Author

Gus Wiseman, May 27 2024

Keywords

Examples

			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}

Crossrefs

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.

Programs

  • Mathematica
    Table[NestWhile[#-1&,2^n,!SquareFreeQ[#]&],{n,0,15}]
  • PARI
    a(n) = my(k=2^n); while (!issquarefree(k), k--); k; \\ Michel Marcus, May 29 2024

Formula

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

Views

Author

Gus Wiseman, Jun 05 2024

Keywords

Comments

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.

Examples

			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

Links

Crossrefs

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.

Programs

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

Formula

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

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
Previous Showing 11-15 of 15 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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