A373414 -id:A373414 - cp's OEIS frontend

A375707 First differences minus 1 of nonsquarefree numbers.

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

Offset: 1

Gus Wiseman, Sep 16 2024

Also the number of squarefree numbers between the nonsquarefree numbers A013929(n) and A013929(n+1).
Delete all 0's to get A120992.
The image is {0,1,2,3}.
Add 1 to all terms for A078147.

			The runs of squarefree numbers begin:
  (5,6,7)
  ()
  (10,11)
  (13,14,15)
  (17)
  (19)
  (21,22,23)
  ()
  (26)
  ()
  (29,30,31)
  (33,34,35)

Amiram Eldar, Table of n, a(n) for n = 1..10000

Positions of 0, 1, 2, 3 are A375709, A375710, A375711, A375712. This is a set partition of the positive integers into four blocks.
For runs of squarefree numbers:
- length: A120992, anti A373127
- min: A072284, anti A373408
- max: A373415, anti A007674
- sum: A373413, anti A373411
For runs of nonsquarefree numbers:
- length: A053797, anti A373409
- min: A053806, anti A373410
- max: A376164, anti A068781
- sum: A373414, anti A373412
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A046933 counts composite numbers between consecutive primes.
A073784 counts primes between consecutive composite numbers.
A093555 counts non-prime-powers between consecutive prime-powers.
Cf. A061398, A061399, A077641, A077643, A143658, A173143, A251092, A294242, A373125, A373126, A373573.

Differences[Select[Range[100],!SquareFreeQ[#]&]]-1

lista(nmax) = {my(prev = 4); for (n = 5, nmax, if(!issquarefree(n), print1(n - prev - 1, ", "); prev = n));} \\ Amiram Eldar, Sep 17 2024

Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = 6/(Pi^2-6) = 1.550546... . - Amiram Eldar, Sep 17 2024

A373406 Sum of the n-th maximal run of odd primes differing by two.

Original entry on oeis.org

15, 24, 36, 23, 60, 37, 84, 47, 53, 120, 67, 144, 79, 83, 89, 97, 204, 216, 113, 127, 131, 276, 300, 157, 163, 167, 173, 360, 384, 396, 211, 223, 456, 233, 480, 251, 257, 263, 540, 277, 564, 293, 307, 624, 317, 331, 337, 696, 353, 359, 367, 373, 379, 383

Offset: 1

Gus Wiseman, Jun 05 2024

nonn

The length of this run is given by A251092.
For this sequence we define a run to be an interval of positions at which consecutive terms differ by two. Normally, a run has consecutive terms differing by one, but odd prime numbers already differ by at least two.
Contains A054735 (sums of twin prime pairs) without its first two terms and A007510 (non-twin primes) as subsequences. - R. J. Mathar, Jun 07 2024

			Row-sums of:
   3   5   7
  11  13
  17  19
  23
  29  31
  37
  41  43
  47
  53
  59  61
  67
  71  73
  79
  83
  89
  97

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

The partial sums are a subset of A071148 (partial sums of odd primes).
Functional neighbors: A025584, A054265, A067774, A251092 (or A175632), A373405, A373413, A373414.
A000040 lists the primes, differences A001223.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
Cf. A005117, A007053, A029707, A038664, A293697, A350842, A371201.

Total/@Split[Select[Range[3,100],PrimeQ],#1+2==#2&]//Most

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

A375705 Sum of the n-th maximal run of adjacent (increasing by one at a time) non-perfect-powers.

Original entry on oeis.org

5, 18, 75, 164, 26, 118, 102, 510, 791, 1160, 1629, 2210, 369, 253, 2040, 3756, 4745, 3914, 1764, 3978, 2994, 8720, 10421, 6003, 5984, 14459, 16820, 19425, 13446, 8328, 25415, 28824, 32525, 36530, 40851, 45500, 50489, 55830, 37259, 23276, 67616, 74085, 80954

Offset: 1

Gus Wiseman, Aug 29 2024

nonn

Non-perfect-powers (A007916) are numbers without a proper integer root.

			The list of all non-perfect-powers, split into runs, begins:
   2   3
   5   6   7
  10  11  12  13  14  15
  17  18  19  20  21  22  23  24
  26
  28  29  30  31
  33  34  35
  37  38  39  40  41  42  43  44  45  46  47  48
Row n begins with A375703(n), ends with A375704(n), adds up to a(n), and has length A375702(n).

For nonprime numbers we have A054265, anti-runs A373404.
For nonsquarefree numbers we have A373414, anti-runs A373412.
For squarefree numbers we have A373413, anti-runs A373411.
For prime-powers we have A373675, anti-runs A373576.
For non-prime-powers we have A373678, anti-runs A373679.
The anti-run version is A375737, sums of A375736.
A001597 lists perfect-powers, differences A053289.
A007916 lists non-perfect-powers, differences A375706.
A046933 counts composite numbers between primes.
For runs of non-perfect-powers:
- length: A375702 = A053289(n+1) - 1
- first: A375703 (same as A216765 with 2 exceptions)
- last: A375704 (same as A045542 with 8 removed)
- sum: A375705 (this)
Cf. A006093, A006549, A251092, A255346, A371201, A373197, A375708.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Total/@Split[Select[Range[100],radQ],#1+1==#2&]//Most

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

A375737 Sum of the n-th maximal anti-run of adjacent (increasing by more than one at a time) non-perfect-powers.

Original entry on oeis.org

2, 8, 6, 17, 11, 12, 13, 14, 32, 18, 19, 20, 21, 22, 23, 78, 29, 30, 64, 34, 72, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 98, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 128, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 162, 83, 84, 85, 86, 87

Offset: 1

Gus Wiseman, Sep 10 2024

nonn

Non-perfect-powers (A007916) are numbers with no proper integer roots.

An anti-run of a sequence is an interval of positions at which consecutive terms differ by more than one.

			The initial anti-runs are the following, whose sums are a(n):
  (2)
  (3,5)
  (6)
  (7,10)
  (11)
  (12)
  (13)
  (14)
  (15,17)
  (18)
  (19)
  (20)
  (21)
  (22)
  (23)
  (24,26,28)

For nonprime numbers we have A373404, runs A054265.
For squarefree numbers we have A373411, runs A373413.
For nonsquarefree numbers we have A373412, runs A373414.
For prime-powers we have A373576, runs A373675.
For non-prime-powers we have A373679, runs A373678.
For anti-runs of non-perfect-powers:
- length: A375736
- first: A375738
- last: A375739
- sum: A375737 (this)
For runs of non-perfect-powers:
- length: A375702
- first: A375703
- last: A375704
- sum: A375705
A001597 lists perfect-powers, differences A053289.
A007916 lists non-perfect-powers, differences A375706.
Cf. A007674, A045542, A046933, A053797, A216765, A251092, A373403, A375714.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Total/@Split[Select[Range[100],radQ],#1+1!=#2&]//Most

A376164 Maximum of the n-th maximal run of nonsquarefree numbers (increasing by 1 at a time).

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 25, 28, 32, 36, 40, 45, 50, 52, 54, 56, 60, 64, 68, 72, 76, 81, 84, 88, 90, 92, 96, 100, 104, 108, 112, 117, 121, 126, 128, 132, 136, 140, 144, 148, 150, 153, 156, 160, 162, 164, 169, 172, 176, 180, 184, 189, 192, 196, 198, 200, 204, 208

Offset: 1

Gus Wiseman, Sep 15 2024

nonn

			The maximal runs of nonsquarefree numbers begin:
       4
     8   9
      12
      16
      18
      20
    24  25
    27  28
      32
      36
      40
    44  45
  48  49  50

For length instead of maximum we have A053797 (firsts A373199).
For lengths of anti-runs we have A373409 (firsts A373573).
For sum instead of maximum we have A373414, anti A373412.
For minimum instead of maximum we have A053806, anti A373410.
For anti-runs instead of runs we have A068781.
For squarefree instead of nonsquarefree we have A373415, anti A007674.
For nonprime instead of nonsquarefree we have A006093 with 2 removed.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147, sums A329472.
A061398 counts squarefree numbers between primes, nonsquarefree A061399.
A120992 gives squarefree run-lengths, anti A373127 (firsts A373128).
A373413 adds up each maximal run of squarefree numbers, min A072284.
A375707 counts squarefree numbers between consecutive nonsquarefree numbers.
Cf. A000040, A020754, A045882, A049094, A054265, A077641, A101836, A143658, A294242, A375709.

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

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A375707 First differences minus 1 of nonsquarefree numbers.

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A373406 Sum of the n-th maximal run of odd primes differing by two.

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

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A375705 Sum of the n-th maximal run of adjacent (increasing by one at a time) non-perfect-powers.

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

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A375737 Sum of the n-th maximal anti-run of adjacent (increasing by more than one at a time) non-perfect-powers.

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A376164 Maximum of the n-th maximal run of nonsquarefree numbers (increasing by 1 at a time).

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