A088700 -id:A088700 - cp's OEIS frontend

A103668 Number of semiprimes between prime(n) and prime(n+1).

0, 1, 1, 2, 0, 2, 0, 2, 2, 0, 3, 2, 0, 1, 2, 3, 0, 2, 1, 0, 2, 1, 3, 4, 0, 0, 1, 0, 1, 6, 1, 2, 0, 5, 0, 1, 3, 1, 1, 2, 0, 3, 0, 1, 0, 6, 7, 1, 0, 0, 2, 0, 2, 2, 2, 2, 0, 1, 1, 0, 3, 7, 1, 0, 1, 6, 2, 3, 0, 0, 2, 3, 1, 1, 2, 1, 4, 1, 2, 4, 0, 2, 0, 1, 0, 3, 3, 1, 0, 1, 4, 3, 1, 2, 2, 1, 5, 0, 7, 3, 3, 2, 2, 0, 1

Offset: 1

Zak Seidov, Feb 12 2005

			a(4)=2 because between prime(4)=7 and prime(5)=11 there are two semiprimes: 3*3 and 2*5.
a(11)=3 because between p(11)=31 and p(12)=37 there are three semiprimes: 33=3*11, 34=2*17 and 35=5*7.

T. D. Noe, Table of n, a(n) for n=1..1000

The first occurrence of k = 0, 1, 2, ... is at position 1, 2, 4, 11, 24, 34, 30, 47, ... (A103669).
Primes: A000040, semiprimes: A001358, number of primes between two successive semiprimes: A088700.
Cf. A103654, A103655, A103669.

fQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; f[n_] := Count[fQ /@ Range[Prime[n] + 1, Prime[n + 1] - 1], True]; Table[ f[n], {n, 105}] (* Robert G. Wilson v, May 07 2005 *)
Table[Count[Range[Prime[n],Prime[n+1]],?(PrimeOmega[#]==2&)],{n,110}] (* _Harvey P. Dale, Sep 29 2019 *)

A103654 Primes which are the average of two successive semiprimes.

Original entry on oeis.org

5, 53, 67, 89, 113, 131, 173, 211, 251, 293, 307, 337, 379, 409, 449, 487, 491, 499, 631, 683, 701, 727, 751, 769, 787, 919, 941, 953, 991, 1009, 1039, 1051, 1063, 1117, 1193, 1259, 1399, 1459, 1471, 1499, 1511, 1567, 1627, 1697, 1709, 1733, 1759, 1787, 1801

Offset: 1

Zak Seidov, Feb 12 2005

nonn

			a(3)=67 because 65 and 69 are two successive semiprimes closest to 67 and 67=(65+69)/2;a(333)=22679 because 22677 and 22691 are two successive semiprimes closest to 22679 and 22679=(22677+22681)/2.

Zak Seidov, Table of n, a(n) for n = 1..10000

Indices of these primes: A103655. Primes: A000040, semiprimes: A001358, number of primes between successive semiprimes: A088700, number of semiprimes between two successive primes: A103668.

Cf. A000040, A088700, A103655, A001358, A103668, A103669.

list(lim)=my(v=List(),u=v,t,lim2=lim+log(lim)^2);forprime(p=2,sqrt(lim2),t=p;forprime(q=p,lim2\t,listput(v,t*q)));v=vecsort(Vec(v));for(i=2,#v,t=(v[i]+v[i-1])/2;if(denominator(t)==1&&isprime(t),if(t>lim,break,listput(u,t))));Vec(u) \\ Charles R Greathouse IV, Oct 08 2012

p=(q+r)/2, where q

p are two successive semiprimes closest to p.

A103655 Indices of primes which are the average of two successive semiprimes.

Original entry on oeis.org

3, 16, 19, 24, 30, 32, 40, 47, 54, 62, 63, 68, 75, 80, 87, 93, 94, 95, 115, 124, 126, 129, 133, 136, 138, 157, 160, 162, 167, 169, 175, 177, 179, 187, 196, 205, 222, 232, 233, 239, 240, 247, 258, 265, 267, 270, 274, 277, 279, 298, 299, 318, 327, 335, 336, 339

Offset: 1

Zak Seidov, Feb 12 2005

nonn

			19 is a member because p(19)=67; 65 and 69 are two successive semiprimes closest to 67 and 67=(65+69)/2.

Zak Seidov, Table of n, a(n) for n = 1..10000

Corresponding primes: A103654. Primes: A000040, semiprimes: A001358, number of primes between two successive semiprimes: A088700, number of semiprimes between two successive primes: A103668.

Cf. A000040, A088700, A103654, A001358, A103668, A103669.

PrimePi[#]&/@Select[Mean/@Partition[Select[Range[2500],PrimeOmega[#]==2&],2,1],PrimeQ] (* Harvey P. Dale, Sep 08 2024 *)

a(n)=pi(A103654(n)).

A103669 First occurrence of just n semiprimes occurs between the a(n)-th prime and the next prime.

Original entry on oeis.org

1, 2, 4, 11, 24, 34, 30, 47, 221, 259, 189, 375, 429, 217, 1831, 1879, 1229, 3795, 3644, 4522, 2225, 10229, 14862, 4612, 34202, 38590, 66762, 14357, 44227, 40933, 33608, 161441, 31545, 111924, 415069, 278832, 126172, 1576499, 104071, 271743, 786922, 3183065, 4875380, 3166684, 2219883, 6080675, 6443469, 1319945

Offset: 1

Zak Seidov, Feb 12 2005

k(31)>78496, k(32)=31545.

The first occurrence of k in A103668. - Robert G. Wilson v, May 07 2005

			n=3, k=11, p(11)=31, p(12)=37, three semiprimes between 31 and 37 are 33=3*11, 34=2*17,35=5*7.

Primes: A000040, semiprimes: A001358, number of primes between two successive semiprimes: A088700, number of semiprimes between two successive primes: A103668.

Cf. A000040, A088700, A001358, A103654, A103655, A103668.

fQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; f[n_] := Count[fQ /@ Range[Prime[n] + 1, Prime[n + 1] - 1], True];
t = Table[ f[n], {n, 1600000}]; Position[t, #, 1, 1] & /@ Range[0, 41] // Flatten (* Or *) t = Table[0, {50}]; Do[ a = f[n]; If[ t[[a + 1]] == 0, t[[a + 1]] = n], {n, 1600000}]

More terms from Robert G. Wilson v, May 07 2005

A088701 Smallest semiprime such that n primes will follow until the next semiprime.

Original entry on oeis.org

9, 4, 10, 95, 818, 2681, 16053, 35981, 797542, 1069541, 340894, 6974465, 3586843, 2070050198, 5238280946, 14499777101, 2601693504238, 7472089131123

Offset: 0

Reinhard Zumkeller, Oct 08 2003

Is this sequence infinite? - David A. Corneth, Aug 17 2018

Cf. A214520 (primes that are the only prime between consecutive semiprimes).

om = Array[PrimeOmega, 1100000]; sp = Flatten@ Position[om, 2]; Table[ sp[[ SelectFirst[ Range[Length[sp] - 1], Count[Take[om, {sp[[#]], sp[[# + 1]]}], x_ /; x == 1] == j &, 0]]], {j, 0, 10}] (* Giovanni Resta, Aug 16 2018 *)

use ntheory ":all";
my($l,$nextn,@C)=(4,0);
forcomposites {
  if (is_semiprime($_)) {
    my $c = prime_count($l+1,$_-1);
    if (!defined $C[$c]) {
      $C[$c] = $l;
      while (defined $C[$nextn]) { print "$nextn $C[$nextn]\n"; $nextn++; }
    }
    $l = $_;
  }
} 5,1e7;  # Dana Jacobsen, Aug 16 2018

A088700(a(n)) = n and A088700(k) <> n for 1 <= k < a(n).

a(11)-a(15) from Donovan Johnson, Mar 14 2010
a(16) from Giovanni Resta, Aug 17 2018
a(17) from Giovanni Resta, Aug 18 2018

A108197 Number of composite numbers between two successive semiprimes.

Original entry on oeis.org

0, 1, 0, 1, 0, 3, 0, 1, 0, 4, 0, 0, 1, 0, 4, 1, 1, 2, 1, 0, 1, 2, 2, 2, 2, 3, 1, 0, 0, 2, 1, 0, 0, 7, 2, 2, 2, 0, 1, 0, 0, 4, 2, 0, 4, 0, 0, 1, 0, 6, 1, 0, 1, 3, 1, 6, 0, 2, 1, 1, 4, 4, 0, 0, 1, 0, 2, 2, 0, 0, 1, 0, 0, 1, 3, 5, 1, 7, 1, 2, 0, 3, 2, 1, 1, 4, 2, 6, 1, 1, 2, 2, 0, 1, 0, 0, 1, 2, 2, 3, 1, 1, 2, 0, 1

Offset: 1

Giovanni Teofilatto, Jun 15 2005

This is to A046933 as semiprimes A001358 are to primes A000040. This is to composites A002808 as A088700 is to primes. a(A070552(i)) = 0. - Jonathan Vos Post, Oct 10 2007

a(n) = 0 if A001358(n) is in A070552. - Jonathan Vos Post, Mar 11 2007

			a(1) = 0 because between 2*2 and 2*3 there is 5 and it is not composite.
a(2) = 1 because between 2*3 and 3*3 there is 8 = 2*2*2;
a(6) = 3 because between 3*5 and 3*7 there are three composite numbers: {16, 18, 20}.
a(10) = 4 because between 2*13 and 3*11 there are four composite numbers: {27, 28, 30, 32}.
a(15) = 4 because there are four composites {40,42,44,45} between semiprime(15)=39 and semiprime(16)=46.

Semiprime analog of A046933.

Cf. A001358, A002808, A046933, A065855, A070552, A088700.

with(numtheory): sp:=proc(n) if bigomega(n)=2 then n else fi end: SP:=[seq(sp(n),n=1..450)]: for j from 1 to nops(SP)-1 do ct:=0: for i from SP[j]+1 to SP[j+1]-1 do if isprime(i)=false then ct:=ct+1 else ct:=ct fi: od: a[j]:=ct: od:seq(a[j],j=1..nops(SP)-1); # Emeric Deutsch, Mar 30 2007
A001358 := proc(nmin) local a,n ; a :=[] ; n := 1 ; while nops(a) < nmin do if numtheory[bigomega](n) = 2 then a := [op(a),n] ; fi ; n := n+1 ; od: RETURN(a) ; end: A000720 := proc(n) numtheory[pi](n) ; end: A065855 := proc(n) n-A000720(n)-1 ; end: A108197 := proc(nmin) local a,n,a001358 ; a001358 := A001358(nmin+1) ; a := [] ; for n from 1 to nops(a001358)-1 do a := [op(a), A065855(op(n+1,a001358))-A065855(op(n,a001358))-1 ] ; od; RETURN(a) ; end: A108197(100) ; # R. J. Mathar, Oct 23 2007

terms = 105;
cc = Select[Range[4 terms], CompositeQ] /. c_ /; PrimeOmega[c] == 2 -> 0;
SequenceReplace[cc, {0, c___ /; FreeQ[{c}, 0]} :> Length[{c}]][[;; terms]] (* Jean-François Alcover, Mar 31 2020 *)

a(n) = A065855(A001358(n+1)) - A065855(A001358(n)) - 1. - R. J. Mathar, Oct 23 2007

a(n)=A065516(n)-1-A088700(n). - R. J. Mathar, Jul 31 2008

Corrected and extended by Ray Chandler, Jul 07 2005
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jul 13 2007
Further edited by N. J. A. Sloane at the suggestion of R. J. Mathar, Jul 01 2008

A176451 Number of primes between two consecutive nonprimes in A037143.

Original entry on oeis.org

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

Offset: 1

Juri-Stepan Gerasimov, Apr 18 2010

2 together with the number of primes between successive semiprimes.

			a(1) = 2 because A037143(1) = 1 (nonprime) < A037143(2) = 2 (prime) < A037143(3) = 3 (prime) < A037143(4) = 4 (nonprime).

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

Cf. A001358, A037143, A088700.

Join[{2}, -1 + Differences[Position[Select[PrimeOmega[Range[400]], # < 3 &], 2] // Flatten]] (* Amiram Eldar, Sep 07 2024 *)

a(n) = A088700(n-1) for n >= 2.

A130973 Number of primes between successive pairs of twin primes, for a(n) > 0.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 4, 2, 1, 3, 1, 2, 3, 10, 4, 7, 4, 3, 2, 1, 2, 18, 2, 2, 17, 1, 2, 6, 9, 3, 1, 1, 1, 8, 3, 2, 15, 1, 4, 1, 1, 7, 7, 4, 4, 3, 4, 1, 1, 7, 2, 5, 1, 5, 18, 2, 5, 4, 3, 1, 5, 1, 18, 12, 2, 8, 1, 4, 2, 5, 4, 1, 1, 1, 9, 10

Offset: 1

Omar E. Pol, Aug 23 2007

a(k) corresponds to the k-th term in the isolated prime sequence A007510 or A134797. a(1) corresponds to 23. a(2) corresponds to 37. a(3) corresponds to 47 and 53. - Enrique Navarrete, Jan 28 2017

Lengths of the runs of consecutive integers in A176656. - R. J. Mathar, Feb 19 2017

C. K. Caldwell, The Prime Pages.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A001223, A007510 (isolated primes), A027883, A048614, A048198, A052011, A052012, A061273, A076777, A073784, A082602, A088700, A179067 (clusters of twin primes).

cp's OEIS Frontend

A103668 Number of semiprimes between prime(n) and prime(n+1).

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A103654 Primes which are the average of two successive semiprimes.

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A103655 Indices of primes which are the average of two successive semiprimes.

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A103669 First occurrence of just n semiprimes occurs between the a(n)-th prime and the next prime.

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A088701 Smallest semiprime such that n primes will follow until the next semiprime.

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Perl

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A108197 Number of composite numbers between two successive semiprimes.

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A176451 Number of primes between two consecutive nonprimes in A037143.

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A130973 Number of primes between successive pairs of twin primes, for a(n) > 0.

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