A103654 -id:A103654 - 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 *)

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

A212820 Balanced primes which are the average of two successive semiprimes.

Original entry on oeis.org

5, 53, 173, 211, 1511, 3307, 3637, 4457, 4993, 6863, 11411, 11731, 11903, 12653, 15907, 18223, 20107, 20201, 20347, 20731, 22051, 23801, 26041, 35911, 39113, 40493, 46889, 47303, 51551, 52529, 60083, 63559, 69623, 71011, 75787, 77081, 78803, 85049, 91297

Offset: 1

Gerasimov Sergey, May 28 2012

nonn

Prime p which is the average of the previous prime and the following prime and is also the average of two successive semiprimes.

			53 is in the sequence because it is the average of 47 and 59 (the two neighboring primes) and 51 and 55 (the two neighboring semiprimes).

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A006562, A103654.

with(numtheory):
prevsp:= proc(n) local k; for k from n-1 by -1
           while isprime(k) or bigomega(k)<>2 do od; k end:
nextsp:= proc(n) local k; for k from n+1
           while isprime(k) or bigomega(k)<>2 do od; k end:
a:= proc(n) option remember; local p;
      p:= `if`(n=1, 2, a(n-1));
      do p:= nextprime(p);
         if p=(prevprime(p)+nextprime(p))/2 and
            p=(prevsp(p)+nextsp(p))/2 then break fi
      od; p
    end:
seq (a(n), n=1..40);  # Alois P. Heinz, Jun 03 2012

prevsp[n_] := Module[{k}, For[k = n-1, PrimeQ[k] || PrimeOmega[k] != 2, k--]; k];
nextsp[n_] := Module[{k}, For[k = n+1, PrimeQ[k] || PrimeOmega[k] != 2 , k++]; k];
a[n_] := a[n] = Module[{p}, p = If[n==1, 2, a[n-1]]; While[True, p = NextPrime[p]; If[p == (NextPrime[p, -1] + NextPrime[p])/2 && p == (prevsp[p] + nextsp[p])/2, Break[]]]; p];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Mar 24 2017, after Alois P. Heinz *)

{ A212820 } = { A006562 } intersection { A103654 }.

cp's OEIS Frontend

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

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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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A212820 Balanced primes which are the average of two successive semiprimes.

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