A103668 -id:A103668 - cp's OEIS frontend

A088700 Number of primes between successive semiprimes.

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

Offset: 1

Reinhard Zumkeller, Oct 08 2003

nonn

a(n) = 0 for almost all n. The average order of a(n) is 1/log log n. - Charles R Greathouse IV, Apr 29 2012

			a(34)=3, as there are three primes between A001358(34)=19*5=95 and A001358(34+1)=53*2=106: 97, 101 and 103.

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Prime Counting Function
Eric Weisstein's World of Mathematics, Semiprime

Cf. A001358, (semiprimes), A088701, A103668 (semiprimes between primes).

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

Select[Range[400], PrimeOmega[#] == 2&] // PrimePi // Differences (* Jean-François Alcover, Oct 12 2021 *)

a(n) = A000720(A001358(n+1)) - A000720(A001358(n));

a(A088701(n)) = n and a(k) <> n for 1 <= k < A088701(n).

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

A195416 Primes p such that there is only one semiprime number between prime(p) and prime(p+1).

Original entry on oeis.org

2, 3, 19, 29, 31, 59, 73, 149, 179, 199, 211, 271, 281, 311, 337, 347, 349, 359, 433, 449, 479, 523, 547, 659, 661, 683, 727, 733, 809, 821, 823, 829, 857, 877, 947, 977, 983, 1049, 1129, 1181, 1201, 1213, 1291, 1297, 1301, 1307, 1321, 1367, 1429, 1433, 1439, 1451, 1489, 1493, 1511, 1553, 1567

Offset: 1

Juri-Stepan Gerasimov, Sep 17 2011

nonn

			a(1)=2 because 2 is a prime and prime(2)=3<(only one semiprime 4)<5=prime(2+1), a(2)=3 because 3 is a prime and prime(3)=5<(only one semiprime 6)<7=prime(3+1), a(3)=19 because 19 is a prime and prime(19)=67<(only one semiprime 69)<71=prime(19+1).

Cf. A103668, A001358, A006450, A195190.

Corrected by R. J. Mathar, Sep 20 2011

cp's OEIS Frontend

A088700 Number of primes between successive semiprimes.

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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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A195416 Primes p such that there is only one semiprime number between prime(p) and prime(p+1).

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