A070552 -id:A070552 - cp's OEIS frontend

A092128 Numbers n such that n, n+2, n+4, n+6, n+8, n+10 are semiprimes.

1383, 3091, 3093, 5609, 8129, 8131, 8133, 9753, 9983, 9985, 9987, 10401, 11013, 12053, 13637, 16499, 22457, 30991, 43339, 45803, 49083, 53761, 55559, 55561, 58277, 63047, 63951, 64829, 69603, 71727, 76803, 80633, 92603, 92605, 98493

Offset: 1

Zak Seidov, Feb 22 2004

Semiprimes in arithmetic progression. All terms are odd, see also A056809.

Cf. A056809, A070552, A092125, A092126, A092127, A092129.

PrimeFactorExponentsAdded[n_] := Plus @@ Flatten[Table[ #[[2]], {1}] & /@ FactorInteger[n]]; Select[ Range[ 99210], PrimeFactorExponentsAdded[ # ] == PrimeFactorExponentsAdded[ # + 2] == PrimeFactorExponentsAdded[ # + 4] == PrimeFactorExponentsAdded[ # + 6] == PrimeFactorExponentsAdded[ # + 8] == PrimeFactorExponentsAdded[ # + 10] == 2 &] (* Robert G. Wilson v, Feb 24 2004 *)
spQ[n_]:=PrimeOmega[n]==2; Select[Range[100000],AllTrue[#+{0,2,4,6,8,10},spQ]&] (* Harvey P. Dale, Dec 19 2021 *)

More terms from Don Reble, Feb 23 2004

More terms from Robert G. Wilson v, Feb 24 2004

A092127 Numbers n such that n, n+2, n+4, n+6, n+8 are semiprimes.

Original entry on oeis.org

213, 1133, 1343, 1383, 1385, 1835, 1937, 2315, 2807, 3091, 3093, 3095, 3147, 3977, 4313, 4529, 4835, 5089, 5609, 5611, 6185, 6533, 7141, 8129, 8131, 8133, 8135, 9753, 9755, 9983, 9985, 9987, 9989, 10401, 10403, 11013, 11015, 11099, 11663, 12053

Offset: 1

Zak Seidov, Feb 22 2004

Semiprimes in arithmetic progression. All terms are odd, see also A056809.

Cf. A056809, A070552, A092125, A092126, A092128, A092129.

PrimeFactorExponentsAdded[n_] := Plus @@ Flatten[Table[ #[[2]], {1}] & /@ FactorInteger[n]]; Select[ Range[ 12054], PrimeFactorExponentsAdded[ # ] == PrimeFactorExponentsAdded[ # + 2] == PrimeFactorExponentsAdded[ # + 4] == PrimeFactorExponentsAdded[ # + 6] == PrimeFactorExponentsAdded[ # + 8] == 2 &] (* Robert G. Wilson v, Feb 24 2004 *)

A109373 Semiprimes of the form semiprime + 1.

Original entry on oeis.org

10, 15, 22, 26, 34, 35, 39, 58, 86, 87, 94, 95, 119, 122, 123, 134, 142, 143, 146, 159, 178, 202, 203, 206, 214, 215, 218, 219, 254, 299, 302, 303, 327, 335, 362, 382, 394, 395, 446, 447, 454, 482, 502, 515, 527, 538, 543, 554, 566, 623, 634, 635, 695, 698

Offset: 1

Jonathan Vos Post, Aug 24 2005

			a(1) = 10 because (3*3+1)=(2*5) = 10.
a(2) = 15 because (2*7+1)=(3*5) = 15.
a(3) = 22 because (3*7+1)=(2*11) = 22.
a(4) = 26 because (5*5+1)=(2*13) = 26.
a(5) = 34 because (3*11+1)=(2*17) = 34.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Primes are in A000040. Semiprimes are in A001358.
Primes of the form semiprime + 1 are in A005385 (safe primes).
Semiprimes of the form semiprime + 1 are in this sequence.
3-almost primes of the form semiprime + 1 are in A109067.
4-almost primes of the form semiprime + 1 are in A109287.
5-almost primes of the form semiprime + 1 are in A109383.
Least n-almost prime of the form semiprime + 1 are in A128665.
Cf. A056809, A077065, A079148, A086005, A086006, A092307.
Subsequence of A088707; A064911.

a109373 n = a109373_list !! (n-1)
a109373_list = filter ((== 1) . a064911) a088707_list
-- Reinhard Zumkeller, Feb 20 2012

fQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; Select[ Range[ 700], fQ[ # - 1] && fQ[ # ] &] (* Robert G. Wilson v *)
With[{sps=Select[Range[700],PrimeOmega[#]==2&]},Transpose[Select[ Partition[ sps,2,1],#[[2]]-#[[1]]==1&]][[2]]] (* Harvey P. Dale, Sep 05 2012 *)

is(n)=bigomega(n)==2 && bigomega(n-1)==2 \\ Charles R Greathouse IV, Jan 31 2017

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

a(n) = A070552(n) + 1.

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

A123017 Semiprimes k such that k+3 is also a semiprime.

Original entry on oeis.org

6, 22, 35, 46, 55, 62, 74, 82, 91, 115, 118, 119, 142, 143, 155, 158, 166, 202, 203, 206, 214, 215, 218, 259, 262, 295, 298, 299, 302, 323, 326, 355, 358, 362, 391, 395, 451, 466, 478, 482, 502, 511, 514, 526, 535, 542, 551, 559, 562, 583, 586, 611, 623, 626

Offset: 1

Jonathan Vos Post, Nov 04 2006

When a(n+1) = a(n) + 3 we have that a(n) is a semiprime such that a(n) and a(n)+3 and a(n) + 3 + 3 are all semiprimes, hence at least 3 semiprimes in arithmetic progression with common difference 3. This subsequence begins 115, 155. There cannot be 4 semiprimes in arithmetic progression with common difference 3, starting with k, because modulo 4 we have {k, k+3, k+6, k+9} == {k+0, k+3, k+2, k+1} and one of these must be divisible by 4, hence a nonsemiprime (eliminating k = 4 by inspection).

			a(1) = 6 because 6 = 2 * 3 is semiprime and 6 + 3 = 9 = 3^2 is semiprime.
a(2) = 22 because 22 = 2 * 11 and 22 + 3 = 25 = 5^2.
a(3) = 35 because 35 = 5 * 7  and 35 + 3 = 38 = 2 * 19.
a(4) = 46 because 46 = 2 * 23 and 46 + 3 = 49 = 7^2.
a(5) = 55 because 55 = 5 * 11 and 55 + 3 = 58 = 2 * 29.

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

Cf. A001358, A056809, A070552, A092125, A092126, A092127, A092128, A092129.

semiprimeQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; Select[ Range@ 670, semiprimeQ[ # ] && semiprimeQ[ # + 3] &] (* Robert G. Wilson v, Aug 31 2007 *)
SequencePosition[Table[If[PrimeOmega[n]==2,1,0],{n,700}],{1,,,1}][[All, 1]] (* Requires Mathematica version 10 or later *)  (* Harvey P. Dale, Mar 03 2017 *)

{a(n)} = {k such that k is in A001358 and k+3 is in A001358}.

More terms from Robert G. Wilson v, Aug 31 2007

A124570 Array read by antidiagonals: T(d,k) (k >= 1, d = 1,2,3,4,5,6,...) = smallest semiprime s of k (not necessarily consecutive) semiprimes in arithmetic progression with common difference d, or 0 if there is no such arithmetic progression.

Original entry on oeis.org

4, 4, 4, 4, 9, 4, 4, 4, 33, 4, 4, 6, 91, 0, 4, 4, 6, 115, 213, 0, 4, 4, 4, 6, 0, 213, 0, 4, 4, 4, 4, 111, 0, 1383, 0, 4, 4, 14, 9, 0, 201, 0, 3091, 0, 4, 4, 6, 51, 203, 0, 201, 0, 8129, 0, 4, 4, 6, 6, 0, 1333, 0, 481, 0, 0, 0, 4, 4, 4, 77, 69, 0, 1333, 0, 5989, 0, 0, 0, 4

Offset: 1

Jonathan Vos Post, Nov 04 2006

Comment from Hugo van der Sanden Aug 14 2021: (Start)
Row d=12 starts 4 9 9 10 10 469 3937 7343 7343 44719 78937 78937 78937 78937 55952333 233761133 597191343199.
Row d=18 starts 4 4 15 15 15 695 695 1727 7711 13951 13951 46159 400847 400847 400847 65737811 13388955301 934046384293.
Row d=24 starts 4 9 9 10 10 793 4819 6415 7271 14069 14069 14069 31589 67344271 616851797 48299373047 48299373047 20302675273219.
Row d=30 starts 4 4 9 25 25 2779 2779 6347 6347 6347 10811 10811 87109 87109 87109 1513723 15009191 15009191 316612697 316612697 1275591688621.
Row d=36 starts 4 10 10 10 15 1333 3161 4997 6865 34885 142171 834863 1327447 35528747 720945097 63389173477 63389173477 16074207679897 41728758250241.
Row d=42 starts 4 4 9 35 35 2701 2987 2987 7729 26995 26995 185795 307553 708385 708385 708385 1090198367 1819546069 20263042201 5672249016001.
Later terms in these rows are always >10^14. (End)
If p is the least prime that does not divide d, then T(d,k) <= p^2 if k >= p^2 (i.e. any a.p. of length >= p^2 with difference d contains a term divisible by p^2, and the only semiprime divisible by p^2 is p^2). Thus every row is eventually 0. - Robert Israel, Aug 11 2024

			Array begins:
d.\...k=1.k=2.k=3.k=4.k=5..k=6..k=7..k=8....k=9..k=10.k=11..k=12.
0..|..4...4...4...4...4....4....4....4......4....4.....4.....4...
1..|..4...9...33..0...0....0....0....0......0....0.....0.....0....
2..|..4...4...91..213.213..1383.3091.8129...0....0.....0.....0.....
3..|..4...6...115.0...0....0....0....0......0....0.....0.....0.....
4..|..4...6...6...111.201..201..481..5989...0....0.....0.....0....
5..|..4...4...4...0...0....0....0....0......0....0.....0.....0.....
6..|..4...4...9...203.1333.1333.1333.2159...8309.18799.60499.60499
7..|..4...14..51..0...0....0....0....0......0....0.....0.....0.....
8..|..4...6...6...69..473..511..511..112697.0....0.....0.....0.....
9..|..4...6...77..0...0....0....0....0......0....0.....0.....0.....
10.|..4...4...15..289.289..289..1631.13501..0....0.....0.....0.....
11.|..4...4...4...0...0....0....0....0......0....0.....0.....0.....
Example for row 3: 115 = 5 * 23 is semiprime, 115+3 = 118 = 2 * 59 is semiprime and 115+3+3 = 121 = 11^2 is semiprime, so T(3,3) = 115.

R. J. Mathar, Table for d <= 999 (PDF)

Semiprime analog of A124064.

Cf. A125025 (row lengths), A001358, A056809, A070552, A092125, A092126, A092127, A092128, A092129, A124064, A092209 (row d=2), A091016 (row d=6).

T(1,2)=A070552(1). T(1,3)=A056809(1). T(2,4)=A092126(1). T(2,5)=A092127(1). T(2,6)=A092128(1). T(2,7)=A092129(1). T(2,8)=A082919(1). T(3,2)=A123017(1). T(d,1)=A001358(1). - R. J. Mathar, Aug 05 2021

Corrected and extended by R. J. Mathar, Nov 06 2006

Definition clarified by Robert Israel, Aug 11 2024

A123255 Numbers k such that 4k+1, 4k+2, and 4k+3 are all semiprimes.

Original entry on oeis.org

8, 21, 23, 30, 35, 50, 53, 54, 75, 98, 111, 158, 174, 210, 230, 260, 284, 315, 336, 350, 410, 440, 459, 473, 485, 495, 525, 545, 554, 576, 590, 608, 615, 629, 660, 680, 683, 774, 846, 900, 923, 966, 975, 989, 1071, 1103, 1133, 1148, 1220, 1400, 1430, 1463, 1499

Offset: 1

Jonathan Vos Post, Oct 09 2006

4k+4 = 4*(k+1) = 2*2*(k+1) cannot be semiprime as well, as it has at least 3 prime factors with multiplicity. Thus there are no four consecutive semiprimes.

			a(1) = 8 because 4*8+1 = 33 = 3*11 is semiprime and 4*8+2 = 34 = 2*17 is semiprime and 4*8+3 = 35 = 3*5 is semiprime.

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

Cf. A001358, A056809, A070552.

IsSemiprime:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ n: n in [2..1500] | IsSemiprime(4*n+1) and IsSemiprime(4*n+2) and IsSemiprime(4*n+3) ]; // Vincenzo Librandi, Dec 22 2010

Select[Range[1100],Union[PrimeOmega[4#+{1,2,3}]]=={2}&] (* Harvey P. Dale, Feb 02 2015 *)

from sympy import factorint, isprime
def issemiprime(n):
    return sum(factorint(n).values()) == 2 if n&1 else isprime(n//2)
def ok(n): return all(issemiprime(4*n+i) for i in (2, 1, 3))
print([k for k in range(1500) if ok(k)]) # Michael S. Branicky, Nov 26 2022

{k: 4k+1 is in A001358 AND 4k+2 is in A001358 AND 4k+3 is in A001358}.
{k: 4k+1 is in A070552 AND 4k+2 is in A070552}.
{(A056809(i)-1)/4}.

336 and 680 added by Vincenzo Librandi, Dec 22 2010

A126436 Number of composites between successive values of A014612.

Original entry on oeis.org

2, 3, 0, 5, 0, 0, 8, 0, 0, 3, 1, 7, 2, 0, 1, 2, 0, 1, 10, 4, 0, 1, 1, 2, 2, 1, 0, 6, 0, 3, 5, 7, 0, 2, 0, 7, 0, 3, 0, 0, 0, 0, 4, 3, 1, 1, 2, 9, 3, 9, 4, 0, 3, 1, 1, 1, 0, 0, 7, 1, 2, 3, 1, 2, 1, 2, 1, 0, 0, 0, 3, 1

Offset: 1

Jonathan Vos Post, Mar 12 2007

			a(1) = 2 because there are two composites {9,10} between A014612(1)=8 and A014612(2)=12.
a(2) = 3 because there are two composites {14, 15, 16} between A014612(2)=12 and A014612(3)=18.
a(3) = 0 because there are no composites between A014612(3)=18 and A014612(4)=20, only the prime 19.
a(7) = 8 because {32,33,34,35,36,38,39,40} between A014612(7)=30 and A014612(8)=42.

3-almost prime analog of A046933 = number of composites between successive primes.

Cf. A002808, A014612, A046933, A070552, A114403.

isA014612 :=proc(n) if numtheory[bigomega](n) = 3 then true ; else false ; fi ; end: isA002808 := proc(n) RETURN(not isprime(n) and n <> 1 ); end: A126436 := proc(nmax) local a ; a := -1 ; for n from 1 to nmax do if isA014612(n) then if a >= 0 then printf("%d,",a) ; fi ; a := 0 ; elif isA002808(n) and a>= 0 then a := a+1 ; fi ; od : end: A126436(300) : # R. J. Mathar, Apr 03 2007

nmax = 72;
S = Select[Range[300](* increase range if a(n) unevaluated *), PrimeOmega[#] == 3&];
a[n_ /; n+1 <= Length[S]] := Count[Range[S[[n]]+1, S[[n+1]]-1], _?CompositeQ];
Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Oct 26 2023 *)

a(n) <= A114403(n) - 1.

More terms from R. J. Mathar, Apr 03 2007

A340482 Numbers that are the product of two not necessarily distinct odd primes pq with the property that (pq+1)/2 and (p+q)/2 are primes.

Original entry on oeis.org

9, 21, 25, 33, 57, 85, 93, 121, 133, 145, 177, 205, 213, 217, 253, 361, 393, 445, 553, 565, 633, 697, 793, 817, 841, 865, 913, 933, 973, 1137, 1285, 1345, 1417, 1437, 1465, 1477, 1513, 1537, 1717, 1765, 1837, 1857, 1893, 2101, 2173, 2245, 2305, 2517, 2577, 2581, 2605, 2641, 2653

Offset: 1

Hartmut F. W. Hoft, Jan 09 2021

nonn

For the squares p^2 in this sequence the area of the central region of the three regions in the symmetric representation of sigma(p^2) is equal to p.

p^2 is a term iff p is in A048161, and this subsequence of p^2 is A263951. - Bernard Schott, Jan 10 2021

			a(1) = 9 = 3*3 is the first number for which SRS(a(1)) consists of three regions ( 5, 3, 5 ).
a(6) = 85 = 5*17, both (1+85)/2 = 43 and (5+17)/2 = 11 are primes, and SRS(a(6)) consists of the 4 regions ( 43, 11, 11, 43 ).

Union of A128283 and A263951.
Subsequence of A046315 (all odd semiprimes).
Cf. A048161, A070552, A128281, A237591, A237593, A249223, A262045, A340482.

dQ[s_] := Module[{d=Divisors[s]}, AllTrue[Map[(d[[#]]+d[[-#]])/2&, Range[Length[d]/2]], PrimeQ]]
a340482[n_] := Select[Range[n], PrimeOmega[#]==2&&dQ[#]&]
a340482[2700]

isok(m) = if ((m % 2) && (bigomega(m)==2), if (issquare(m), isprime((m+1)/2), my(p=factor(m)[1,1], q=factor(m)[2,1]); isprime((p*q+1)/2) && isprime((p+q)/2))); \\ Michel Marcus, Jan 10 2021

A086097 Numbers n such that n and n+1 are semiprimes with a semiprime number of 1's in their binary representation.

Original entry on oeis.org

57, 85, 141, 177, 201, 393, 537, 553, 633, 697, 717, 745, 921, 933, 1141, 1285, 1345, 1477, 1641, 1713, 1761, 1981, 1982, 2041, 2181, 2361, 2433, 2558, 2577, 2605, 2761, 2941, 2977, 3062, 3117, 3241, 3273, 3326, 3397, 3578, 3721, 3777, 3901, 3902, 3957

Offset: 1

Jason Earls, Jul 09 2003

3721 is the first square entry. Conjecture: sequence is infinite.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A070552 and of A086096.

Cf. A001358, A000120.

binWt[n_] := DigitCount[n, 2, 1]; seqQ[n_] := PrimeOmega[n] == 2 && PrimeOmega[binWt[n]] == 2; Select[Range[4000], seqQ[#] && seqQ[#+1] &] (* Amiram Eldar, Dec 14 2019 *)
SequencePosition[Table[If[PrimeOmega[n]==PrimeOmega[DigitCount[n,2,1]]==2,1,0],{n,4000}],{1,1}][[All,1]] (* Harvey P. Dale, Dec 03 2021 *)

issemi(n)=bigomega(n)==2
is(n)=issemi(hammingweight(n))&&issemi(hammingweight(n+1))&&issemi(n)&&issemi(n+1) \\ Charles R Greathouse IV, Mar 28 2013

cp's OEIS Frontend

A092128 Numbers n such that n, n+2, n+4, n+6, n+8, n+10 are semiprimes.

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A092127 Numbers n such that n, n+2, n+4, n+6, n+8 are semiprimes.

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A109373 Semiprimes of the form semiprime + 1.

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

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A123017 Semiprimes k such that k+3 is also a semiprime.

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A124570 Array read by antidiagonals: T(d,k) (k >= 1, d = 1,2,3,4,5,6,...) = smallest semiprime s of k (not necessarily consecutive) semiprimes in arithmetic progression with common difference d, or 0 if there is no such arithmetic progression.

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A123255 Numbers k such that 4k+1, 4k+2, and 4k+3 are all semiprimes.

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Magma

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A126436 Number of composites between successive values of A014612.

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A340482 Numbers that are the product of two not necessarily distinct odd primes pq with the property that (pq+1)/2 and (p+q)/2 are primes.