A093555 -id:A093555 - cp's OEIS frontend

A373576 Sums of maximal antiruns of prime-powers.

2, 3, 4, 12, 8, 49, 171, 2032, 5157, 3997521, 199713082, 561678378, 10122001905, 109934112352390774

Offset: 1

Gus Wiseman, Jun 17 2024

An antirun of a sequence (in this case A246655) is an interval of positions at which consecutive terms differ by more than one.

			The maximal antiruns of powers of primes begin:
   2
   3
   4
   5   7
   8
   9  11  13  16
  17  19  23  25  27  29  31

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

See link for composite, prime, nonsquarefree, and squarefree runs/antiruns.
Prime-power runs: A373675, min A373673, max A373674, length A174965.
Non-prime-power runs: A373678, min A373676, max A373677, length A110969.
Prime-power antiruns: A373576 (this sequence), min A120430, max A006549, length A373671.
Non-prime-power antiruns: A373679, min A373575, max A255346, length A373672.
A000040 lists the primes, differences A001223.
A000961 lists all powers of primes. A246655 lists just prime-powers.
A025528 counts prime-powers up to n.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A356068 counts non-prime-powers up to n.
A361102 lists all non-prime-powers (A024619 if not including 1).
Cf. A001359, A014963, A027833, A029707, A038664, A067774, A251092, A293697, A371201, A373401, A373669.

Total/@Split[Select[Range[1000],PrimePowerQ],#1+1!=#2&]//Most

a(14) from Giorgos Kalogeropoulos, Jun 18 2024

A373675 Sums of maximal runs of powers of primes A000961.

Original entry on oeis.org

15, 24, 11, 13, 33, 19, 23, 25, 27, 29, 63, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 255, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239

Offset: 1

Gus Wiseman, Jun 16 2024

nonn

A run of a sequence (in this case A000961) is an interval of positions at which consecutive terms differ by one.

			The maximal runs of powers of primes begin:
   1   2   3   4   5
   7   8   9
  11
  13
  16  17
  19
  23
  25
  27
  29
  31  32
  37
  41
  43
  47
  49

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

A000040 lists the primes, differences A001223.
A000961 lists all powers of primes (A246655 if not including 1).
A025528 counts prime-powers up to n.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A361102 lists all non-prime-powers (A024619 if not including 1).
See link for composite, prime, nonsquarefree, and squarefree runs.
Prime-power runs: A373675, min A373673, max A373674, length A174965.
Non-prime-power runs: A373678, min A373676, max A373677, length A110969.
Prime-power antiruns: A373576, min A120430, max A006549, length A373671.
Non-prime-power antiruns: A373679, min A373575, max A255346, length A373672.
Cf. A005117, A014963, A029707, A038664, A046933, A054265, A065855, A071148, A293697, A371201.

pripow[n_]:=n==1||PrimePowerQ[n];
Total/@Split[Select[Range[nn],pripow],#1+1==#2&]//Most

A373679 Sums of maximal antiruns of non-prime-powers.

Original entry on oeis.org

43, 53, 21, 163, 34, 35, 74, 39, 126, 45, 144, 51, 106, 55, 56, 57, 180, 128, 134, 69, 216, 75, 76, 77, 324, 85, 86, 87, 178, 91, 92, 93, 94, 95, 194, 99, 306, 105, 324, 111, 226, 115, 116, 117, 118, 119, 242, 123, 379, 262, 133, 134, 135, 414, 141, 142, 143

Offset: 1

Gus Wiseman, Jun 17 2024

nonn

An antirun of a sequence (in this case A361102) is an interval of positions at which consecutive terms differ by more than one.

			The maximal antiruns of non-prime-powers begin:
   1   6  10  12  14
  15  18  20
  21
  22  24  26  28  30  33
  34
  35
  36  38
  39
  40  42  44
  45
  46  48  50
  51
  52  54
  55
  56
  57
  58  60  62
  63  65

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

See link for composite, prime, nonsquarefree, and squarefree runs/antiruns.
Prime-power runs: A373675, min A373673, max A373674, length A174965.
Non-prime-power runs: A373678, min A373676, max A373677, length A110969.
Prime-power antiruns: A373576, min A120430, max A006549, length A373671.
Non-prime-power antiruns: A373679 (this sequence), min A373575, max A255346, length A373672.
A000040 lists the primes, differences A001223.
A000961 lists all powers of primes. A246655 lists just prime-powers.
A025528 counts prime-powers up to n.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A356068 counts non-prime-powers up to n.
A361102 lists all non-prime-powers (A024619 if not including 1).
Cf. A001359, A014963, A027833, A029707, A038664, A054265, A067774, A071148, A251092, A371201, A373401, A373669.

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

A373575 Numbers k such that k and k-1 both have at least two distinct prime factors. First element of the n-th maximal antirun of non-prime-powers.

Original entry on oeis.org

1, 15, 21, 22, 34, 35, 36, 39, 40, 45, 46, 51, 52, 55, 56, 57, 58, 63, 66, 69, 70, 75, 76, 77, 78, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 99, 100, 105, 106, 111, 112, 115, 116, 117, 118, 119, 120, 123, 124, 130, 133, 134, 135, 136, 141, 142, 143, 144, 145

Offset: 1

Gus Wiseman, Jun 18 2024

nonn

The last element of the same antirun is given by A255346.

An antirun of a sequence (in this case A361102) is an interval of positions at which consecutive terms differ by more than one.

			The maximal antiruns of non-prime-powers begin:
   1   6  10  12  14
  15  18  20
  21
  22  24  26  28  30  33
  34
  35
  36  38
  39
  40  42  44
  45
  46  48  50

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Runs of prime-powers:
- length A174965
- min A373673
- max A373674
- sum A373675
Runs of non-prime-powers:
- length A110969
- min A373676
- max A373677
- sum A373678
Antiruns of prime-powers:
- length A373671
- min A120430
- max A006549
- sum A373576
Antiruns of non-prime-powers:
- length A373672
- min A373575 (this sequence)
- max A255346
- sum A373679
A000961 lists all powers of primes. A246655 lists just prime-powers.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A356068 counts non-prime-powers up to n.
A361102 lists all non-prime-powers (A024619 if not including 1).
Various run-lengths: A053797, A120992, A175632, A176246.
Various antirun-lengths: A027833, A373127, A373403, A373409.
Cf. A001359, A008864, A014963, A067774, A251092, A373669, A373670.

Select[Range[100],!PrimePowerQ[#]&&!PrimePowerQ[#-1]&]
Join[{1},SequencePosition[Table[If[PrimeNu[n]>1,1,0],{n,150}],{1,1}][[;;,2]]] (* Harvey P. Dale, Feb 23 2025 *)

A373673 First element of each maximal run of powers of primes (including 1).

Original entry on oeis.org

1, 7, 11, 13, 16, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239

Offset: 1

Gus Wiseman, Jun 15 2024

nonn

A run of a sequence (in this case A000961) is an interval of positions at which consecutive terms differ by one.
The last element of the same run is A373674.
Consists of all powers of primes k such that k-1 is not a power of primes.

			The maximal runs of powers of primes begin:
   1   2   3   4   5
   7   8   9
  11
  13
  16  17
  19
  23
  25
  27
  29
  31  32
  37
  41
  43
  47
  49

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

For composite antiruns we have A005381, max A068780, length A373403.
For prime antiruns we have A006512, max A001359, length A027833.
For composite runs we have A008864, max A006093, length A176246.
For prime runs we have A025584, max A067774, length A251092 or A175632.
For runs of prime-powers:
- length A174965
- min A373673 (this sequence)
- max A373674
- sum A373675
For runs of non-prime-powers:
- length A110969 (firsts A373669, sorted A373670)
- min A373676
- max A373677
- sum A373678
For antiruns of prime-powers:
- length A373671
- min A120430
- max A006549
- sum A373576
For antiruns of non-prime-powers:
- length A373672
- min A373575
- max A255346
- sum A373679
A000961 lists all powers of primes (A246655 if not including 1).
A025528 counts prime-powers up to n.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A361102 lists all non-prime-powers (A024619 if not including 1).
Cf. A000040, A007674, A014963, A053806, A054265, A068781, A072284, A073051, A120992, A373400.

pripow[n_]:=n==1||PrimePowerQ[n];
Min/@Split[Select[Range[100],pripow],#1+1==#2&]//Most

A373676 First element of each maximal run of non-prime-powers.

Original entry on oeis.org

1, 6, 10, 12, 14, 18, 20, 24, 26, 28, 30, 33, 38, 42, 44, 48, 50, 54, 60, 62, 65, 68, 72, 74, 80, 82, 84, 90, 98, 102, 104, 108, 110, 114, 122, 126, 129, 132, 138, 140, 150, 152, 158, 164, 168, 170, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234

Offset: 1

Gus Wiseman, Jun 16 2024

nonn

We consider 1 to be a power of a prime and a non-prime-power, but not a prime-power.
A run of a sequence (in this case A000961) is an interval of positions at which consecutive terms differ by one.
The last element of the same run is A373677.
Consists of 1 and all non-prime-powers k such that k-1 is a power of a prime.

			The maximal runs of non-prime-powers begin:
   1
   6
  10
  12
  14  15
  18
  20  21  22
  24
  26
  28
  30
  33  34  35  36
  38  39  40
  42
  44  45  46
  48
  50  51  52
  54  55  56  57  58
  60

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

See link for prime, composite, squarefree, and nonsquarefree runs/antiruns.
For runs of powers of primes:
- length A174965
- min A373673
- max A373674
- sum A373675
For runs of non-prime-powers:
- length A110969 (firsts A373669, sorted A373670)
- min A373676 (this sequence)
- max A373677
- sum A373678
For antiruns of prime-powers:
- length A373671
- min A120430
- max A006549
- sum A373576
For antiruns of non-prime-powers:
- length A373672
- min A373575
- max A255346
- sum A373679
A000961 lists all powers of primes. A246655 is just prime-powers so lacks 1.
A025528 counts prime-powers up to n.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A361102 lists all non-prime-powers (A024619 if not including 1).
Cf. A000040, A007674, A014963, A038664, A053806, A054265, A068781, A072284, A073051, A373400.

Select[Range[100],#==1||!PrimePowerQ[#]&&PrimePowerQ[#-1]&]

A373677 Last element of each maximal run of non-prime-powers.

Original entry on oeis.org

1, 6, 10, 12, 15, 18, 22, 24, 26, 28, 30, 36, 40, 42, 46, 48, 52, 58, 60, 63, 66, 70, 72, 78, 80, 82, 88, 96, 100, 102, 106, 108, 112, 120, 124, 126, 130, 136, 138, 148, 150, 156, 162, 166, 168, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238

Offset: 1

Gus Wiseman, Jun 16 2024

nonn

We consider 1 to be a power of a prime and a non-prime-power, but not a prime-power.
A run of a sequence (in this case A000961) is an interval of positions at which consecutive terms differ by one.
The first element of the same run is A373676.
Consists of all non-prime-powers k such that k+1 is a prime-power.

			The maximal runs of non-prime-powers begin:
   1
   6
  10
  12
  14  15
  18
  20  21  22
  24
  26
  28
  30
  33  34  35  36
  38  39  40
  42
  44  45  46
  48
  50  51  52
  54  55  56  57  58
  60

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

See link for prime, composite, squarefree, and nonsquarefree runs/antiruns.
For runs of powers of primes:
- length A174965
- min A373673
- max A373674
- sum A373675
For runs of non-prime-powers:
- length A110969 (firsts A373669, sorted A373670)
- min A373676
- max A373677 (this sequence)
- sum A373678
For antiruns of prime-powers:
- length A373671
- min A120430
- max A006549
- sum A373576
For antiruns of non-prime-powers:
- length A373672
- min A373575
- max A255346
- sum A373679
A000961 lists all powers of primes. A246655 is just prime-powers so lacks 1.
A025528 counts prime-powers up to n.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A361102 lists all non-prime-powers (A024619 if not including 1).
Cf. A000040, A005381, A007674, A014963, A038664, A068780, A068781, A073051, A373400, A373401.

Select[Range[100],!PrimePowerQ[#]&&PrimePowerQ[#+1]&]

A373678 Sums of maximal runs of non-prime-powers.

Original entry on oeis.org

1, 6, 10, 12, 29, 18, 63, 24, 26, 28, 30, 138, 117, 42, 135, 48, 153, 280, 60, 125, 131, 207, 72, 380, 80, 82, 430, 651, 297, 102, 315, 108, 333, 819, 369, 126, 259, 670, 138, 1296, 150, 770, 800, 495, 168, 513, 880, 180, 1674, 192, 585, 198, 2255, 2387, 675

Offset: 1

Gus Wiseman, Jun 16 2024

nonn

We consider 1 to be a power of a prime and a non-prime-power, but not a prime-power.

A run of a sequence (in this case A361102) is an interval of positions at which consecutive terms differ by one.

			The maximal runs of non-powers of primes begin:
   1
   6
  10
  12
  14  15
  18
  20  21  22
  24
  26
  28
  30
  33  34  35  36
  38  39  40
  42
  44  45  46
  48
  50  51  52
  54  55  56  57  58
  60

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

A000040 lists the primes, differences A001223.
A000961 lists all powers of primes (A246655 if not including 1).
A025528 counts prime-powers up to n.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A361102 lists all non-prime-powers (A024619 if not including 1).
See link for composite, prime, nonsquarefree, and squarefree runs.
Prime-power runs: A373675, min A373673, max A373674, length A174965.
Non-prime-power runs: A373678, min A373676, max A373677, length A110969.
Prime-power antiruns: A373576, min A120430, max A006549, length A373671.
Non-prime-power antiruns: A373679, min A373575, max A255346, length A373672.
Cf. A005117, A014963, A038664, A046933, A054265, A065855, A071148, A293697, A371201, A373400.

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

A375707 First differences minus 1 of nonsquarefree numbers.

Original entry on oeis.org

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

A376597 Inflection and undulation points in the sequence of prime-powers inclusive (A000961).

Original entry on oeis.org

1, 2, 3, 6, 8, 14, 15, 16, 27, 32, 50, 61, 67, 72, 85, 92, 93, 124, 129, 132, 136, 141, 185, 190, 211, 214, 221, 226, 268, 292, 301, 302, 322, 374, 394, 423, 456, 463, 502, 503, 547, 559, 560, 593, 604, 640, 646, 663, 671, 675, 710, 726, 727, 746, 754, 755

Offset: 1

Gus Wiseman, Oct 05 2024

nonn

These are points at which the second differences (A376596) are zero.

Inclusive means 1 is a prime-power. For the exclusive version, subtract 1 and shift left.

			The prime-powers inclusive (A000961) are:
  1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, ...
with first differences (A057820):
  1, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, ...
with first differences (A376596):
  0, 0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, ...
with zeros (A376597) at:
  1, 2, 3, 6, 8, 14, 15, 16, 27, 32, 50, 61, 67, 72, 85, 92, 93, 124, 129, 132, ...

Gus Wiseman, Inflection and undulation points in the prime-powers inclusive.

The first differences were A057820, see also A053707, A376340.
These are the zeros of A376596 (sorted firsts A376653, exclusive A376654).
The complement is A376598.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
A064113 lists positions of adjacent equal prime gaps.
For prime-powers inclusive: A057820 (first differences), A376596 (second differences), A376598 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376599 (non-prime-power).
Cf. A025475, A053707, A069623, A093555, A174965, A361102, A376653, A376654.

Join@@Position[Differences[Select[Range[1000],#==1||PrimePowerQ[#]&],2],0]

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A373576 Sums of maximal antiruns of prime-powers.

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A373675 Sums of maximal runs of powers of primes A000961.

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A373679 Sums of maximal antiruns of non-prime-powers.

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A373575 Numbers k such that k and k-1 both have at least two distinct prime factors. First element of the n-th maximal antirun of non-prime-powers.

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A373673 First element of each maximal run of powers of primes (including 1).

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A373676 First element of each maximal run of non-prime-powers.

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A373677 Last element of each maximal run of non-prime-powers.

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A373678 Sums of maximal runs of non-prime-powers.

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