A373403 -id:A373403 - cp's OEIS frontend

A053797 Lengths of successive gaps between squarefree numbers.

1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 3, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 3, 1, 2, 2, 2, 1

Offset: 1

N. J. A. Sloane, Apr 07 2000

From Gus Wiseman, Jun 11 2024: (Start)
Also the length of the n-th maximal run of nonsquarefree numbers. These runs begin:
       4
     8   9
      12
      16
      18
      20
    24  25
    27  28
      32
      36
      40
    44  45
  48  49  50
(End)

			The first gap is at 4 and has length 1; the next starts at 8 and has length 2 (since neither 8 nor 9 are squarefree).

Peter Kagey, Table of n, a(n) for n = 1..10000
M. Filaseta and O. Trifonov, On Gaps between Squarefree Numbers. In Analytic Number Theory, Vol 85, 1990, Birkhäuser, Basel, pp. 235-253.
E. Fogels, On the average values of arithmetic functions, Proc. Cambridge Philos. Soc. 1941, 37: 358-372.
L. Marmet, First occurrences of squarefree gaps...
L. Marmet, First occurrences of square-free gaps and an algorithm for their computation, arXiv preprint arXiv:1210.3829 [math.NT], 2012. - From _N. J. A. Sloane_, Jan 01 2013
K. F. Roth, On the gaps between squarefree numbers, J. London Math. Soc. 1951 (2) 26:263-268.
Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers

Gaps between terms of A005117.
For squarefree runs we have A120992, antiruns A373127 (firsts A373128).
For composite runs we have A176246 (rest of A046933), antiruns A373403.
For prime runs we have A251092 (rest of A175632), antiruns A027833.
Position of first appearance of n is A373199(n).
For antiruns instead of runs we have A373409.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A007674, A020754, A045882, A053806, A061398, A061399.

SF:= select(numtheory:-issqrfree,[$1..1000]):
map(`-`,select(`>`,SF[2..-1]-SF[1..-2],1),1); # Robert Israel, Sep 22 2015

ReplaceAll[Differences[Select[Range@384, SquareFreeQ]] - 1, 0 -> Nothing] (* Michael De Vlieger, Sep 22 2015 *)

Offset set to 1 by Peter Kagey, Sep 29 2015

A029707 Numbers n such that the n-th and the (n+1)-st primes are twin primes.

Original entry on oeis.org

2, 3, 5, 7, 10, 13, 17, 20, 26, 28, 33, 35, 41, 43, 45, 49, 52, 57, 60, 64, 69, 81, 83, 89, 98, 104, 109, 113, 116, 120, 140, 142, 144, 148, 152, 171, 173, 176, 178, 182, 190, 201, 206, 209, 212, 215, 225, 230, 234, 236, 253, 256, 262, 265, 268, 277

Offset: 1

N. J. A. Sloane, Dec 11 1999

nonn

Numbers m such that prime(m)^2 == 1 mod (prime(m) + prime(m + 1)). - Zak Seidov, Sep 18 2013

First differences are A027833. The complement is A049579. - Gus Wiseman, Dec 03 2024

Robert G. Wilson v, Table of n, a(n) for n = 1..86027
Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

Cf. A014574, A027833 (first differences), A007508. Equals PrimePi(A001359) (cf. A000720).
The complement is A049579, first differences A251092 except first term.
Lengths of runs of terms differing by 2 are A179067.
The first differences have run-lengths A373820 except first term.
A000040 lists the primes, differences A001223 (run-lengths A333254, A373821).
A038664 finds the first prime gap of 2n.
A046933 counts composite numbers between primes.
For prime runs: A005381, A006512, A025584, A067774.
Cf. A006560, A006562, A037201, A107770, A122535, A155752, A175632, A176246, A373819.

A029707 := proc(n)
    numtheory[pi](A001359(n)) ;
end proc:
seq(A029707(n),n=1..30); # R. J. Mathar, Feb 19 2017

Select[ Range@300, PrimeQ[ Prime@# + 2] &] (* Robert G. Wilson v, Mar 11 2007 *)
Flatten[Position[Flatten[Differences/@Partition[Prime[Range[100]],2,1]], 2]](* Harvey P. Dale, Jun 05 2014 *)

def A029707(n) :
   a = [ ]
   for i in (1..n) :
      if (nth_prime(i+1)-nth_prime(i) == 2) :
         a.append(i)
   return(a)
A029707(277) # Jani Melik, May 15 2014

a(n) = A107770(n) - 1. - Juri-Stepan Gerasimov, Dec 16 2009

A027833 Distances between successive 2's in sequence A001223 of differences between consecutive primes.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 3, 6, 2, 5, 2, 6, 2, 2, 4, 3, 5, 3, 4, 5, 12, 2, 6, 9, 6, 5, 4, 3, 4, 20, 2, 2, 4, 4, 19, 2, 3, 2, 4, 8, 11, 5, 3, 3, 3, 10, 5, 4, 2, 17, 3, 6, 3, 3, 9, 9, 2, 6, 2, 6, 5, 6, 2, 3, 2, 3, 9, 4, 7, 3, 7, 20, 4, 7, 6, 5, 3, 7, 3, 20, 2, 14, 4, 10, 2, 3, 6, 4, 2, 2, 7, 2, 6, 3

Offset: 1

Jean-Marc MALASOMA (Malasoma(AT)entpe.fr)

nonn

a(n) = number of primes p such that A014574(n) < p < A014574(n+1). - Thomas Ordowski, Jul 20 2012
Conjecture: a(n) < log(A014574(n))^2. - Thomas Ordowski, Jul 21 2012
Conjecture: All positive integers are represented in this sequence. This is verified up to 184, by searching up to prime indexes of ~128000000. The rate of filling-in the smallest remaining gap among the integers, and the growth in the maximum value found, both slow down considerably relative to a fixed quantity of twin prime incidences examined in each pass. The maximum value found was 237. - Richard R. Forberg, Jul 28 2016
All positive integers below 312 are in this sequence. - Charles R Greathouse IV, Aug 01 2016
From Gus Wiseman, Jun 11 2024: (Start)
Also the length of the n-th maximal antirun of prime numbers > 3, where an antirun is an interval of positions at which consecutive terms differ by more than 2. These begin:
   5
   7  11
  13  17
  19  23  29
  31  37  41
  43  47  53  59
  61  67  71
  73  79  83  89  97 101
(End)

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

First differences of A029707 and A155752 = A029707 - 1. M. F. Hasler, Jul 24 2012
Positions of first appearances are A373401, sorted A373402.
Functional neighbors: A001359, A006512, A251092 or A175632, A373127 (firsts A373128, sorted A373200), A373403, A373405, A373409.
A000040 lists the primes, differences A001223.
A002808 lists the composite numbers, differences A073783.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
Cf. A025157, A038664, A071148,  A176246, A350842, A373406.

A027833 := proc(n)
    local plow,phigh ;
    phigh := A001359(n+1) ;
    plow := A001359(n) ;
    numtheory[pi](phigh)-numtheory[pi](plow) ;
end proc:
seq(A027833(n),n=1..100) ; # R. J. Mathar, Jan 20 2025

Differences[Flatten[Position[Differences[Prime[Range[500]]],2]]] (* Harvey P. Dale, Nov 17 2018 *)
Length/@Split[Select[Range[4,10000],PrimeQ[#]&],#1+2!=#2&]//Most (* Gus Wiseman, Jun 11 2024 *)

n=1; p=5; forprime(q=7,1e3, if(q-p==2, print1(n", "); n=1, n++); p=q) \\ Charles R Greathouse IV, Aug 01 2016

def A027833(n) :
   a = [ ]
   st = 2
   for i in (3..n) :
      if (nth_prime(i+1)-nth_prime(i) == 2) :
         a.append(i-st)
         st = i
   return(a)
A027833(496) # Jani Melik, May 15 2014

A110969 Length of the runs of ones in A014963.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 4, 3, 1, 3, 1, 3, 5, 1, 2, 2, 3, 1, 5, 1, 1, 5, 7, 3, 1, 3, 1, 3, 7, 3, 1, 2, 5, 1, 9, 1, 5, 5, 3, 1, 3, 5, 1, 9, 1, 3, 1, 11, 11, 3, 1, 3, 5, 1, 1, 7, 4, 5, 5, 1, 5, 3, 1, 5, 3, 13, 3, 1, 3, 13, 5, 5, 3, 1, 3, 5, 1, 5, 5, 5, 3, 5, 7, 3, 7

Offset: 1

Franz Vrabec, Sep 27 2005

nonn

Unbounded sequence.
From A373669 we see that 10 first appears at a(28195574) = 10.
Also run-lengths of non-prime-powers (assuming 1 is not a prime-power), where a run of a sequence (in this case A361102) is an interval of positions at which consecutive terms differ by one. Also nonzero differences of consecutive prime-powers minus one. - Gus Wiseman, Jun 18 2024

			a(5)=2 because the fifth run of ones in A014963 is of length 2.

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A014963.
Positions of first appearances are A373670, sorted A373669.
For runs of prime-powers:
- length A174965, antiruns A373671
- min A373673, antiruns A120430
- max A373674, antiruns A006549
- sum A373675, antiruns A373576
For runs of non-prime-powers:
- length A110969 (this sequence), antiruns A373672
- min A373676, antiruns A373575
- max A373677, antiruns A255346
- sum A373678, antiruns 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, A067774, A251092.

Length /@ SplitBy[Table[Exp[MangoldtLambda[n]], {n, 400}], # != 1 &][[ ;; -1 ;; 2]] (* Michael De Vlieger, Mar 21 2024 *)
DeleteCases[Differences[Select[Range[100],PrimePowerQ]]-1,0] (* Gus Wiseman, Jun 18 2024 *)

\\ b(n) returns boolean of A014963(n) == 1.
b(n)={my(t); !isprime(if(ispower(n, ,&t), t, n))}
seq(n)={my(k=1, i=0, L=List()); while(#Lk, listput(L, i-k)); k = i+1)); Vec(L)} \\ Andrew Howroyd, Jan 02 2020

Terms a(41) and beyond from Andrew Howroyd, Jan 02 2020

A065310 Number of occurrences of n-th prime in A065308, where A065308(j) = prime(j - pi(j)).

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Oct 29 2001

nonn

Seems identical to A054546. Each odd prime arises once or twice!?

First differences of A018252 (positive nonprime numbers). Including 0 gives A054546. Removing 1 gives A073783. - Gus Wiseman, Sep 15 2024

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000040, A000720, A054576, A065308-A065309.
For twin 2's see A169643.
Positions of 1's are A375926, complement A014689 (except first term).
Other families of numbers and their first-differences:
For prime numbers (A000040) we have A001223.
For composite numbers (A002808) we have A073783.
For nonprime numbers (A018252) we have A065310 (this).
For perfect powers (A001597) we have A053289.
For non-perfect-powers (A007916) we have A375706.
For squarefree numbers (A005117) we have A076259.
For nonsquarefree numbers (A013929) we have A078147.
For prime-powers inclusive (A000961) we have A057820.
For prime-powers exclusive (A246655) we have A057820(>1).
For non-prime-powers inclusive (A024619) we have A375735.
For non-prime-powers exclusive (A361102) we have A375708.
Cf. A046933, A053707, A054546, A110969, A176246, A251092, A373403, A375929.

t=Table[Prime[w-PrimePi[w]], {w, a, b}] Table[Count[t, Prime[n]], {n, c, d}]
Differences[Select[Range[100],!PrimeQ[#]&]] (* Gus Wiseman, Sep 15 2024 *)

{ p=1; f=2; m=1; for (n=1, 1000, a=0; p=nextprime(p + 1); while (p==f, a++; m++; f=prime(m - primepi(m))); write("b065310.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 16 2009

A373127 Length of the n-th maximal antirun of squarefree numbers differing by more than one.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 4, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 5, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 4, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 2, 3, 1, 2, 2, 1, 2, 1, 2, 4, 2, 1, 4, 1, 3, 2, 1, 2, 1, 2, 1, 2, 2, 1, 4, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 4, 1, 3, 4, 1, 2, 1

Offset: 1

Gus Wiseman, Jun 05 2024

nonn

The sum of this antirun is given by A373411.

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

			Row-lengths 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

Positions of first appearances are A373128, sorted A373200.
Functional neighbors: A007674, A027833 (partial sums A029707), A120992, A373403, A373408, A373409, A373411.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A077643 counts squarefree numbers with n bits, sum A373123.
Cf. A049093, A049094, A061398, A077641, A112925, A112926, A174965, A350842, A372683, A373125, A373126, A373197, A373198.

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

A373671 Length of the n-th maximal antirun of prime-powers.

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 7, 26, 27, 1007, 5558, 5734, 31209

Offset: 1

Gus Wiseman, Jun 14 2024

An antirun of a sequence (in this case A000961 without 1) is an interval of positions at which consecutive terms differ by more than one.

			The maximal antiruns of prime-powers 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.

For prime antiruns we have A027833.
For nonsquarefree runs we have A053797, firsts A373199.
For non-prime-powers runs we have A110969, firsts A373669, sorted A373670.
For squarefree runs we have A120992.
For prime-power runs we have A174965.
For prime runs we have A175632.
For composite runs we have A176246, firsts A073051, sorted A373400.
For squarefree antiruns we have A373127, firsts A373128.
For composite antiruns we have A373403.
For antiruns of prime-powers:
- length A373671 (this sequence)
- min A120430
- max A006549
For antiruns of non-prime-powers:
- length A373672
- min A373575
- max A255346
A000961 lists the powers of primes (including 1).
A025528 counts prime-powers up to n.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A361102 lists the non-prime-powers (not including 1 A024619).
Cf. A000040, A001359, A008864, A014963, A038664, A054265, A067774, A251092, A373401.

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

Partial sums are A025528(A006549(n)).

A373672 Length of the n-th maximal antirun of non-prime-powers.

Original entry on oeis.org

5, 3, 1, 6, 1, 1, 2, 1, 3, 1, 3, 1, 2, 1, 1, 1, 3, 2, 2, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 1, 3, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1

Offset: 1

Gus Wiseman, Jun 14 2024

nonn

An antirun of a sequence (in this case A361102 or A024619 with 1) 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

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

For prime antiruns we have A027833.
For nonsquarefree runs we have A053797, firsts A373199.
For non-prime-powers runs we have A110969, firsts A373669, sorted A373670.
For squarefree runs we have A120992.
For prime-power runs we have A174965.
For prime runs we have A175632.
For composite runs we have A176246, firsts A073051, sorted A373400.
For squarefree antiruns we have A373127, firsts A373128.
For composite antiruns we have A373403.
For antiruns of prime-powers:
- length A373671
- min A120430
- max A006549
For antiruns of non-prime-powers:
- length A373672 (this sequence), firsts (3,7,2,25,1,4)
- min A373575
- max A255346
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).
Cf. A001359, A008864, A014963, A038664, A054265, A067774, A251092, A373401.

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

Partial sums are A356068(A255346(n)).

A373576 Sums of maximal antiruns of prime-powers.

Original entry on oeis.org

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

cp's OEIS Frontend

A053797 Lengths of successive gaps between squarefree numbers.

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A029707 Numbers n such that the n-th and the (n+1)-st primes are twin primes.

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A027833 Distances between successive 2's in sequence A001223 of differences between consecutive primes.

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A110969 Length of the runs of ones in A014963.

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A065310 Number of occurrences of n-th prime in A065308, where A065308(j) = prime(j - pi(j)).

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PARI

A373127 Length of the n-th maximal antirun of squarefree numbers differing by more than one.

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A373671 Length of the n-th maximal antirun of prime-powers.

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