A070552 -id:A070552 - cp's OEIS frontend

A108197 Number of composite numbers between two successive semiprimes.

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

A179502 Numbers k with the property that k^2, k^2+1 and k^2+2 are all semiprimes.

Original entry on oeis.org

11, 29, 79, 271, 379, 461, 521, 631, 739, 881, 929, 1459, 1531, 1709, 2161, 2239, 2341, 2729, 3049, 3491, 3709, 4021, 4349, 4561, 4691, 5021, 5281, 5851, 5879, 6301, 6329, 6829, 7559, 8009, 9151, 10069, 10099, 10151, 10529, 10891, 11719, 11959, 11969, 13799, 14051, 14159

Offset: 1

Zak Seidov, Jan 08 2011

nonn

From the first 10^6 primes, 6680 are terms of the sequence.
Also, all numbers k^2+1 are twice prime, and k^2+2 are thrice prime.
The number of terms less than 10^m beginning with m = 1: 0, 3, 11, 35, 160, 759, 4668, 30319, 204439, ..., .
The number of terms less than the (10^m)-th prime beginning with m = 1: 2, 7, 33, 165, 941, 6680, 48977, 373627, ..., .

Zak Seidov, Table of n, a(n) for n = 1..6680

n^2 are squares in A070552, which is a subsequence of A056809 (m and m+1 are semiprimes) and A001358 (semiprimes).

The sequence is a subsequence of A048161.

fQ[n_] := PrimeQ[(n^2 + 1)/2] && PrimeQ[(n^2 + 2)/3]; Select[ Prime@ Range@ 1667, fQ] (* Robert G. Wilson v, Feb 26 2011 *)
Select[Range[15000],PrimeOmega[#^2+{0,1,2}]=={2,2,2}&] (* Harvey P. Dale, May 12 2025 *)

{n=10;for(i=1,10^4,n=nextprime(n+1);n2=n^2;if(2==bigomega(n2+1)&&2==bigomega(n2+2),print1(n,",")))}

A188059 Numbers k with the property that k, k+1 and 2*k+1 are all semiprimes.

Original entry on oeis.org

25, 34, 38, 57, 93, 118, 133, 145, 177, 201, 205, 213, 218, 298, 334, 361, 381, 394, 446, 501, 633, 694, 698, 842, 865, 878, 898, 921, 1114, 1141, 1226, 1285, 1293, 1465, 1513, 1654, 1713, 1726, 1761, 1857, 1893, 1941, 1981, 2018, 2041, 2217, 2306, 2426, 2433, 2577, 2581, 2734, 2746, 2901, 2973, 3133, 3193, 3214, 3241, 3386, 3578, 3661, 3693, 3746, 3754, 3777, 3826, 3957

Offset: 1

Zak Seidov, Mar 20 2011

nonn

Numbers k such that 2k+1 is a semiprime and the sum of two consecutive semiprimes (k and k+1).

			25 is a term: k = 25 = 5*5, k+1 = 26 = 2*13, 2k+1 = 51 = 3*17.

Zak Seidov, Table of n, a(n) for n = 1..1309 (terms < 200000)

Cf. A001358 (semiprimes).

Cf. A176896 (safe semiprimes), A111153 (Sophie Germain semiprimes), A070552.

Select[Range[4000],Union[PrimeOmega[{#,#+1,2 #+1}]]=={2}&] (* Harvey P. Dale, May 11 2012 *)

Equals A111153 intersect A070552. - M. F. Hasler, Mar 20 2011

A217222 Initial terms of sets of 8 consecutive semiprimes with gap 2.

Original entry on oeis.org

8129, 237449, 401429, 452639, 604487, 858179, 1471727, 1999937, 2376893, 2714987, 3111977, 3302039, 3869237, 4622087, 7813559, 9795449, 10587899, 10630739, 11389349, 14186387, 14924153, 15142547, 15757337, 18017687, 18271829, 19732979, 22715057, 25402907

Offset: 1

Zak Seidov, Sep 28 2012

nonn

All terms == 11 (mod 18).
Also all terms of sets of 8 consecutive semiprimes are odd, e.g., {8129, 8131, 8133, 8135, 8137, 8139, 8141, 8143} is the smallest set of 8 consecutive semiprimes.
Note that in all cases "9th term" (in this case 8143+2=8145) is divisible by 9 and hence is not semiprime.
Also note that all seven "intermediate" even integers (in this case {8130, 8132, 8134, 8136, 8138, 8140, 8142}) have at least three prime factors counting with multiplicity. Up to n = 40*10^9 there are 5570 terms of this sequence.

Zak Seidov, Table of n, a(n) for n = 1..5570

Subsequence of A082919.

Cf. A056809, A070552, A092125-A092129, A092207-A092209.

Transpose[Select[Partition[Select[Range[26*10^6],PrimeOmega[#] == 2&],8,1], Union[ Differences[#]]=={2}&]][[1]] (* Harvey P. Dale, Sep 02 2015 *)

A131457 a(n+1) is the next semiprime such that a(n+1)-1 divides (a(1)...a(n))^2.

Original entry on oeis.org

4, 9, 10, 21, 22, 25, 26, 33, 34, 35, 46, 49, 51, 55, 57, 58, 65, 69, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 161, 166, 169, 177, 178, 183, 185, 187, 201, 202, 203, 205, 206, 209, 213

Offset: 1

Jonathan Vos Post, Oct 21 2007

This is to semiprimes A001358 as A007459 is to primes A000040.

			a(1) = 4 because 4 = 2^2 is the first semiprime.
a(2) = 9 because 9 = 3^2 is the next semiprime after 4, where 9-1=8 divides 4^2 = 16.
a(3) = 10 because 10 = 2*5 is the next semiprime after 9 where 10-9=9 divides (4*9)^2.
a(4) = 21 because 21 = 3*7 is the next semiprime after 10, where 10-1=9 divides (4*9*10)^2.
a(5) = 22 because 22 = 2*11 is the next semiprime after 21, where 21-1=20 divides (4*9*10*21)^2.

Cf. A000040, A001358, A007459, A070552, A109373.

isA001358 := proc(n) if numtheory[bigomega](n) = 2 then true ; else false; fi ; end: A131457 := proc(n) option remember ; local a,prevpr; if n =1 then 4; else prevpr := (mul(A131457(i),i=1..n-1))^2 ; a := A131457(n-1)+1 ; while not isA001358(a) or prevpr mod (a-1) <> 0 do a := a+1 ; od; RETURN(a) ; fi ; end: seq(A131457(n),n=1..80) ; # R. J. Mathar, Oct 30 2007

semiprimeQ[n_] := PrimeOmega[n] == 2;
a[n_] := a[n] = Module[{k, prevpr}, If[n == 1, 4, prevpr = Product[a[i], {i, 1, n-1}]^2; k = a[n-1]+1; While[!semiprimeQ[k] || Mod[prevpr, k-1] != 0, k++]; Return[k]]];
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jan 28 2024, after R. J. Mathar *)

Corrected and extended by R. J. Mathar, Oct 30 2007

A178034 a(n) = binomial(n*Omega(n),Omega(n)) / n.

Original entry on oeis.org

1, 1, 1, 7, 1, 11, 1, 253, 17, 19, 1, 595, 1, 27, 29, 39711, 1, 1378, 1, 1711, 41, 43, 1, 138415, 49, 51, 3160, 3403, 1, 3916, 1, 25637001, 65, 67, 69, 477191, 1, 75, 77, 657359, 1, 7750, 1, 8515, 8911, 91, 1, 132563501, 97, 11026, 101, 11935, 1, 1633355

Offset: 1

Michel Lagneau, May 17 2010

nonn

Omega(.) = A001222(.) is the number of prime divisors of n (counted with multiplicity).
binomial(nk,k)= n*binomial(nk-1,k-1) ensures that all entries are integers.
Subcases for this sequence:
If n is prime, Omega(n) = 1, and a(n) = binomial (n,1) / n = 1.
If n and n+1 are products of two primes (A070552), then Omega(n) = Omega(n+1) = 2, and binomial(n*Omega(n), Omega(n)) / n = binomial(2*n, 2) / n = 2*n-1 and binomial(2*(n+1), 2) / (n+1) = 2*n+1, and we obtain two consecutive numbers of the form (x, x+2), for example (17,19), (27,29), (41,43),... at n =9, 14...
Chaining this property: If n, n+1, and n+2 are semiprimes (A056809) , we find three consecutive numbers of the form (x, x+2,x+4), for example (65, 67, 69), (169, 171, 173), at n=33, 85.
At places where Omega(n)=3, we find the subsequence A060544, for example a(8) = A060544(8).
At places where Omega(n)=4, we find the subsequence A015219.

			a(8) = binomial(8*Omega(8),Omega(8))/8 = binomial(8*3,3)/8 = 2024/8 = 253.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001358, A038456

A178034 := proc(n)
        local o ;
        o := numtheory[bigomega](n) ;
        binomial(n*o,o)/n ;
end proc: # R. J. Mathar, Jul 08 2012

bon[n_]:=Module[{o=PrimeOmega[n]},Binomial[n*o,o]/n]; Array[bon,60] (* Harvey P. Dale, Jul 22 2014 *)

a(n)=my(b=bigomega(n));binomial(n*b,b)/n \\ Charles R Greathouse IV, Oct 25 2012

A368670 Numbers k such that k, k + 1, k + 2, and k + 4 are all semiprimes.

Original entry on oeis.org

141, 201, 213, 217, 301, 1137, 1345, 1401, 1761, 1837, 1893, 1941, 1981, 2101, 3097, 3865, 3957, 4413, 4533, 4881, 5997, 6157, 6241, 7113, 7141, 7165, 7401, 7977, 8185, 8257, 8913, 9753, 9985, 10117, 11013, 11181, 11377, 11757, 12057, 13953, 14037, 14253, 14917, 14977, 14997, 16177, 16293, 16437, 16593

Offset: 1

Robert Israel, Jan 02 2024

nonn

k, k + 1, k + 2 and k + 3 can't all be semiprimes, as one of them is divisible by 4.

All terms == 1 (mod 4).

			a(3) = 213 is a term because 213 = 3 * 71, 214 = 2 * 107, 215 = 5 * 43 and 217 = 7 * 31 are all semiprimes.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001358, A056809, A070552.

select(t -> numtheory:-bigomega(t)=2 and numtheory:-bigomega(t+1)=2 and numtheory:-bigomega(t+2)=2 and numtheory:-bigomega(t+4)=2, 4 * [$1..10000] +~ 1);

Select[Range[17000], PrimeOmega[#] == PrimeOmega[#+1] == PrimeOmega[#+2] == PrimeOmega[#+4] == 2 &] (* Stefano Spezia, Jan 02 2024 *)

A128686 Least number of the form semiprime - 1 which is the product of exactly n primes.

Original entry on oeis.org

3, 9, 8, 24, 32, 64, 128, 864, 1792, 1536, 2048, 4096, 8192, 24576, 73728, 98304, 131072, 393216, 524288, 1048576, 3145728, 6291456, 8388608, 37748736, 50331648, 150994944, 301989888, 268435456, 1207959552, 1610612736, 2147483648, 4294967296, 19327352832, 25769803776, 115964116992, 171798691840, 343597383680, 927712935936, 2886218022912, 1099511627776, 20890720927744, 24189255811072

Offset: 1

Ray Chandler, Mar 20 2007

nonn

Cf. A005383, A070552, A128665.

a(27)-a(42) from Donovan Johnson, Nov 24 2010

A131208 Greatest prime divisor of all composite numbers between n-th semiprime and next semiprime, or 1 if there are no such composite numbers.

Original entry on oeis.org

1, 2, 1, 3, 1, 5, 1, 3, 1, 7, 1, 1, 3, 1, 11, 3, 5, 13, 7, 1, 5, 7, 17, 7, 19, 13, 7, 1, 1, 11, 23, 1, 1, 17, 11, 19, 29, 1, 5, 1, 1, 31, 13, 1, 23, 1, 1, 3, 1, 37, 13, 1, 5, 41, 7, 43, 1, 13, 23, 31, 47, 13, 1, 1, 17, 1, 23, 53, 1, 1, 3, 1, 1, 11, 37, 29, 59, 61, 31, 7

Offset: 1

Jonathan Vos Post, Oct 24 2007

Largest of all prime factors of the numbers between semiprime(n) and semiprime(n+1). Semiprime analog of A052248. a(A070552(n)) = 1. This sequence defines a mapping of semiprimes to primes.

Cf. A001358, A052248, A070552.

a(n) = MAX{(A001358(n) < k < A001358(n+1), A006530(k))}.

Corrected and extended by R. J. Mathar, Jan 15 2008

A133597 Array of semiprimes, read by antidiagonals, where row k is the first of pairs of consecutive semiprimes j and j+k.

Original entry on oeis.org

9, 4, 14, 6, 49, 21, 10, 22, 55, 25, 69, 51, 35, 91, 33, 15, 77, 58, 46, 119, 34, 26, 123, 106, 65, 62, 143, 38, 169, 39, 365, 161, 87, 74, 159, 57, 146, 437, 134, 371, 178, 111, 82, 183, 85, 237, 226, 458, 187, 505, 221, 129, 115, 185, 86

Offset: 1

Jonathan Vos Post, Dec 27 2007

Every semiprime occurs in this table exactly once. Note that similar tables exist for k-almost primes (integers with exactly k prime factors, with multiplicity), this being the k=2 slice of a 3-dimensional array.

			The array begins:
==================================================================
n=......1....2.....3....4....5....6....7....8....9...10
==================================================================
k=1.|...9...14....21...25...33...34...38...57...85...86....A070552
k=2.|...4...49....55...91..119..143..159..183..185..203....A136196
k=3.|...6...22....35...46...62...74...82..115..155..166....A264043
k=4.|. 10...51....58...65...87..111..129..209..249..274....A264044
k=5.|..69...77...106..161..178..221..254..309..314..329....A264045
k=6.|..15..123...365..371..505..545..573..591..649..707....A264046
k=7.|..26...39...134..187..194..267..519..566..655..771....
k=8.|.169..437...458..614..723..737..905..965.1047.1059....
k=9.|.146..226...278..346.1018.1177.1273.1546.1594.1865....
k=10|.237..427..1027.1101.1661.2723.2747.3173.3295.3669....A217030
==================================================================

Cf. A001358, A070552, A136196.

v = Select[Range[5000], PrimeOmega[#]==2 &]; L[k_] := L[k] = v[[Select[Range[Length[v]-1], v[[#+1]] - v[[#]] == k &]]]; Flatten@ Table[ Table[L[k-j+1][[j]], {j, k}], {k, 10}] (* Giovanni Resta, Jun 20 2016 *)

Corrected and edited by Giovanni Resta, Jun 20 2016

cp's OEIS Frontend

A108197 Number of composite numbers between two successive semiprimes.

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A179502 Numbers k with the property that k^2, k^2+1 and k^2+2 are all semiprimes.

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A188059 Numbers k with the property that k, k+1 and 2*k+1 are all semiprimes.

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A217222 Initial terms of sets of 8 consecutive semiprimes with gap 2.

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A131457 a(n+1) is the next semiprime such that a(n+1)-1 divides (a(1)...a(n))^2.

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A178034 a(n) = binomial(n*Omega(n),Omega(n)) / n.

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A368670 Numbers k such that k, k + 1, k + 2, and k + 4 are all semiprimes.

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A128686 Least number of the form semiprime - 1 which is the product of exactly n primes.

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