A103655 -id:A103655 - 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.

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

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

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