A109288 -id:A109288 - cp's OEIS frontend

A109287 4-almost primes equal to p*q + 1, where p and q are (not necessarily distinct) primes.

16, 36, 40, 56, 88, 135, 156, 184, 204, 210, 220, 248, 250, 260, 296, 306, 315, 328, 330, 340, 342, 372, 414, 459, 472, 490, 516, 536, 546, 580, 584, 636, 650, 686, 690, 708, 714, 732, 735, 738, 804, 808, 819, 836, 850, 852, 870, 872, 940, 950, 966, 975, 996

Offset: 1

Giovanni Teofilatto and Jonathan Vos Post, Aug 20 2005

4-almost primes of the form semiprime + 1.

			a(1) = 16 because (3*5+1)=(2^4) = 16.
a(2) = 36 because (5*7+1)=((2^2)*(3^2)) = 36.
a(3) = 40 because (3*13+1)=((2^3)*5) = 40.
a(4) = 56 because (5*11+1)=((2^3)*7) = 56.
a(5) = 88 because (3*29+1)=((2^3)*11) = 88.
a(6) = 135 because (2*67+1)=((3^3)*5) = 135.
a(7) = 156 because (5*31+1)=((2^2)*3*13) = 156.
a(8) = 184 because (3*61+1)=((2^3)*23) = 184.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Primes are in A000040. Semiprimes are in A001358. 4-almost primes are in A014613.
Primes of the form semiprime + 1 are in A005385 (safe primes).
Semiprimes of the form semiprime + 1 are in A109373.
3-almost primes of the form semiprime + 1 are in A109067.
4-almost primes of the form semiprime + 1 are in this sequence.
5-almost primes of the form semiprime + 1 are in A109383.
Least n-almost prime of the form semiprime + 1 are in A128665.
Similar to A076153; after A076153(0)=3 next difference is A076153(100)=1792 whereas A109287(100)=1794.
Cf. A077065, A079148, A092307, A109288, A109289, A109290.

bo[n_] := Plus @@ Last /@ FactorInteger[n]; Select[Range[1000], bo[ # ] == 4 && bo[ # - 1] == 2 &] (* Ray Chandler, Aug 27 2005 *)

is(n)=bigomega(n)==4 && bigomega(n-1)==2 \\ Charles R Greathouse IV, Sep 16 2015

a(n) is in this sequence iff a(n) is in A014613 and (a(n)-1) is in A001358.

Extended by Ray Chandler, Aug 27 2005

Edited by Ray Chandler, Mar 20 2007

A263990 Nonsquare numbers k such that k and k+1 are semiprimes.

Original entry on oeis.org

14, 21, 33, 34, 38, 57, 85, 86, 93, 94, 118, 122, 133, 141, 142, 145, 158, 177, 201, 202, 205, 213, 214, 217, 218, 253, 298, 301, 302, 326, 334, 381, 393, 394, 445, 446, 453, 481, 501, 514, 526, 537, 542, 553, 565, 622, 633, 634, 694, 697, 698, 706, 717, 745, 766, 778, 793, 802, 817, 842, 865, 878

Offset: 1

Zak Seidov, Oct 31 2015

nonn

If k and k+1 are semiprimes then k+1 is always nonsquare while k can be a square (see A263951). The sequence gives the nonsquare terms of A070552. Each of the numbers k and k+1 is a product of two distinct primes.
Numbers that are terms in A070552 but not in A263951.
The subsequence of triples of consecutive squarefree semiprimes is A039833. - R. J. Mathar, Aug 13 2019

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Subsequence of A070552, A086263.

Cf. A006881, A039833, A109288, A263951.

Select[Range[1000], ! IntegerQ[Sqrt[#]] && 2 == PrimeOmega[#] == PrimeOmega[# + 1] &]

is(n)=if(n%2, isprime((n+1)/2) && bigomega(n)==2 && !isprimepower(n), isprime(n/2) && bigomega(n+1)==2) \\ Charles R Greathouse IV, Apr 25 2016

a(n) = A109288(n) - 1. - Amiram Eldar, Aug 08 2025

A109290 Composite numbers which are not of the forms p*q -+ 1, where p and q are (not necessarily distinct) primes.

Original entry on oeis.org

4, 6, 12, 18, 28, 30, 42, 44, 46, 49, 51, 55, 60, 62, 65, 69, 72, 74, 77, 80, 82, 91, 98, 99, 100, 102, 104, 106, 108, 111, 115, 125, 126, 129, 136, 138, 148, 150, 152, 153, 155, 161, 164, 166, 169, 171, 172, 174, 175, 180, 183, 185, 187, 189, 190, 192, 194, 196

Offset: 1

Giovanni Teofilatto, Aug 20 2005

nonn

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

Cf. A109287, A109288, A109289.

filter:= proc(n)
   if n::even then numtheory:-bigomega(n+1) <> 2 and numtheory:-bigomega(n-1) <> 2
   elif n mod 4 = 1 then not isprime(n) and not isprime((n+1)/2)
   else not isprime(n) and not isprime((n-1)/2)
   fi
end proc:
select(filter, [$4..200]); # Robert Israel, Apr 20 2021

bo[n_] := Plus @@ Last /@ FactorInteger[n]; Select[Range[2, 200], ! (PrimeQ[ # ] || bo[ # - 1] == 2 || bo[ # + 1] == 2) &] (* Ray Chandler, Aug 27 2005 *)

Corrected and extended by Ray Chandler, Aug 27 2005

A109289 Composite numbers which are not of the form p*q + 1, where p and q are distinct primes.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 18, 20, 21, 24, 25, 26, 28, 30, 32, 33, 38, 42, 44, 45, 46, 48, 49, 50, 51, 54, 55, 57, 60, 62, 64, 65, 68, 69, 72, 74, 76, 77, 80, 81, 82, 84, 85, 90, 91, 93, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 114, 115, 117, 118, 121, 122, 125, 126, 128

Offset: 1

Giovanni Teofilatto, Aug 20 2005

Cf. A109287, A109288, A109290.

Select[Range[2, 130], ! (PrimeQ[ # ] || (Plus @@ Last /@ FactorInteger[ # - 1] == 2 && Length[FactorInteger[ # - 1]] == 2)) &] (* Ray Chandler, Aug 25 2005 *)
fQ[n_] := Last /@ FactorInteger[n] != {1, 1}; Select[ Range[2, 128], !PrimeQ[ # ] && fQ[ # - 1] &] (* Robert G. Wilson v *)

Extended by Robert G. Wilson v and Ray Chandler, Aug 25 2005

A173967 Sums of two consecutive numbers that are nonsquare semiprimes.

Original entry on oeis.org

29, 43, 67, 69, 77, 115, 171, 173, 187, 189, 237, 245, 267, 283, 285, 291, 317, 355, 403, 405, 411, 427, 429, 435, 437, 507, 597, 603, 605, 653, 669, 763, 787, 789, 891, 893, 907, 963, 1003, 1029, 1053, 1075, 1085, 1107, 1131, 1245, 1267, 1269, 1389, 1395

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 03 2010

nonn

			14=2*7; 15=3*5; 14+15=29,
21=3*7; 22=2*11; 21+22=43,..

Cf. A173966, A173969, A006881, A263990, A109288.

f[n_]:=Last/@FactorInteger[n]=={1,1};lst={};Do[If[f[n],If[f[n+1],AppendTo[lst,2*n+1]]],{n,7!}];lst

a(n) = A263990(n) + A109288(n). - Andrey Zabolotskiy, Apr 07 2025

New name from Andrey Zabolotskiy, Apr 07 2025

cp's OEIS Frontend

A109287 4-almost primes equal to p*q + 1, where p and q are (not necessarily distinct) primes.

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A263990 Nonsquare numbers k such that k and k+1 are semiprimes.

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A109290 Composite numbers which are not of the forms p*q -+ 1, where p and q are (not necessarily distinct) primes.

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Maple

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A109289 Composite numbers which are not of the form p*q + 1, where p and q are distinct primes.

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Mathematica

Extensions

A173967 Sums of two consecutive numbers that are nonsquare semiprimes.

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Examples

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Mathematica

Formula

Extensions