A065516 -id:A065516 - cp's OEIS frontend

A166237 Differences between consecutive products of two distinct primes: a(n) = A006881(n+1) - A006881(n).

4, 4, 1, 6, 1, 4, 7, 1, 1, 3, 1, 7, 5, 4, 2, 1, 4, 3, 4, 5, 3, 5, 3, 1, 1, 4, 2, 1, 1, 11, 5, 4, 3, 1, 3, 1, 6, 4, 1, 7, 1, 1, 2, 1, 9, 3, 1, 2, 5, 11, 1, 5, 2, 2, 7, 7, 1, 1, 2, 1, 3, 4, 1, 1, 2, 1, 1, 2, 5, 9, 2, 10, 2, 4, 1, 5, 3, 3, 2, 7, 4, 9, 4, 4, 3, 1, 2, 1, 1, 2, 4, 5, 5, 2, 2, 3, 1, 2, 5, 1, 4, 2, 5, 9, 3

Offset: 1

Vladimir Joseph Stephan Orlovsky, Oct 09 2009

nonn

Goldston, Graham, Pintz & Yıldırım (2005) prove that a(n+1) - a(n) <= 26 infinitely often. - Charles R Greathouse IV, Dec 26 2020

Michael De Vlieger, Table of n, a(n) for n = 1..10000
D. A. Goldston, S. W. Graham, J. Pintz and C. Y. Yıldırım, Small gaps between primes and almost primes, arXiv:math/0506067 [math.NT], 2005; Proceedings of the London Mathematical Society 98:3 (May 2009), pp. 741-774.
Yang Liu, Peter S. Park, and Zhuo Qun Song, Bounded gaps between products of distinct primes, arXiv:1607.03887 [math.NT], 2016-2017; Research in Number Theory 3:26 (2017).
Keiju Sono, Small gaps between the set of products of at most two primes, arXiv:1605.02920 [math.NT], 2016-2018; Journal of the Mathematical Society of Japan 72:1 (2020), pp. 81-118.

Cf. A006881 (products of two distinct primes), A001358 (semiprimes: products of two primes), A065516 (differences between products of two primes), A001223 (differences between consecutive primes).

T:=[ n: n in [1..360] | #PrimeDivisors(n) eq 2 and &*[ d[2]: d in Factorization(n) ] eq 1 ]; [ T[j+1]-T[j]: j in [1..#T-1] ]; // Klaus Brockhaus, Oct 13 2009

f[n_]:=Last/@FactorInteger[n]=={1,1}; a=6;lst={};Do[If[f[n],AppendTo[lst,n-a];a=n],{n,9,6!}];lst

{m=106; v=vector(m); n=0; c=0; while(cKlaus Brockhaus, Oct 13 2009

Edited by Klaus Brockhaus, Oct 13 2009

Added formula to clarify the definition. - N. J. A. Sloane, Jul 19 2022

A338903 Number of integer partitions of the n-th squarefree semiprime into squarefree semiprimes.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 5, 4, 6, 5, 12, 14, 19, 22, 27, 36, 38, 51, 77, 86, 128, 141, 163, 163, 207, 233, 259, 260, 514, 657, 813, 983, 1010, 1215, 1255, 1720, 2112, 2256, 3171, 3370, 3499, 3864, 4103, 6292, 7313, 7620, 8374, 10650, 17579, 18462, 23034, 25180

Offset: 1

Gus Wiseman, Nov 24 2020

nonn

A squarefree semiprime (A006881) is a product of any two distinct prime numbers.

			The a(n) partitions for n = 1, 5, 7, 9, 10, 11, 13:
  6  21    26       34          35        38           46
     15,6  14,6,6   22,6,6      21,14     26,6,6       34,6,6
           10,10,6  14,14,6     15,14,6   22,10,6      26,14,6
                    14,10,10    15,10,10  14,14,10     21,15,10
                    10,6,6,6,6            14,6,6,6,6   22,14,10
                                          10,10,6,6,6  26,10,10
                                                       15,15,10,6
                                                       22,6,6,6,6
                                                       14,14,6,6,6
                                                       14,10,10,6,6
                                                       10,10,10,10,6
                                                       10,6,6,6,6,6,6

A002100 counts partitions into squarefree semiprimes.
A056768 uses primes instead of squarefree semiprimes.
A101048 counts partitions into semiprimes.
A338902 is the not necessarily squarefree version.
A339113 includes the Heinz numbers of these partitions.
A001358 lists semiprimes, with odd and even terms A046315 and A100484.
A006881 lists squarefree semiprimes, with odd and even terms A046388 and A100484.
A320656 counts factorizations into squarefree semiprimes.
A338898/A338912/A338913 give prime indices of semiprimes, with sum/difference/product A176504/A176506/A087794.
A338899, A270650, and A270652 give the prime indices of squarefree semiprimes.
Cf. A000041, A000607, A065516, A115392, A128301, A320655, A320732, A320892, A320912, A338915, A338916.

nn=100;
sqs=Select[Range[nn],SquareFreeQ[#]&&PrimeOmega[#]==2&];
Table[Length[IntegerPartitions[n,All,sqs]],{n,sqs}]

a(n) = A002100(A006881(n)).

A088707 Semiprimes + 1.

Original entry on oeis.org

5, 7, 10, 11, 15, 16, 22, 23, 26, 27, 34, 35, 36, 39, 40, 47, 50, 52, 56, 58, 59, 63, 66, 70, 75, 78, 83, 86, 87, 88, 92, 94, 95, 96, 107, 112, 116, 119, 120, 122, 123, 124, 130, 134, 135, 142, 143, 144, 146, 147, 156, 159, 160, 162, 167, 170, 178, 179, 184, 186

Offset: 1

Reinhard Zumkeller, Oct 11 2003

nonn

a(n) = A001358(n) + 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Semiprime

Cf. A005385 (primes), A109373 (semiprimes), A065516 (first differences).

a088707 = (+ 1) . a001358  -- Reinhard Zumkeller, Feb 20 2012

A130533 a(n) = smallest k such that A001358(n+1) = A001358(n) + (A001358(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 2, 6, 13, 9, 2, 19, 2, 19, 2, 3, 4, 37, 8, 43, 47, 47, 53, 2, 6, 59, 61, 8, 71, 6, 79, 2, 5, 83, 89, 2, 3, 12, 101, 107, 4, 3, 3, 2, 11

Offset: 1

Rémi Eismann, Aug 16 2007 - Jan 20 2011

nonn

a(n) is the "weight" of semiprimes.

The decomposition of semiprimes into weight * level + gap is A001358(n) = a(n) * A184729(n) + A065516(n) if a(n) > 0.

			For n = 1 we have A001358(n) = 4, A001358(n+1) = 6; there is no k such that 6 - 4 = 2 = (4 mod k), hence a(1) = 0.
For n = 3 we have A001358(n) = 9, A001358(n+1) = 10; 2 is the smallest k such that 10 - 9 = 1 = (9 mod k), hence a(3) = 2.
For n = 19 we have A001358(n) = 55, A001358(n+1) = 57; 53 is the smallest k such that 57 - 55 = 2 = (55 mod k), hence a(19) = 53.

Remi Eismann, Table of n, a(n) for n=1..9999

Cf. A001358, A065516, A184729, A184728, A117078, A117563, A001223, A118534.

A136196 Numbers n such that n and n+2 are consecutive semiprimes.

Original entry on oeis.org

4, 49, 55, 91, 119, 143, 159, 183, 185, 203, 215, 219, 235, 247, 265, 287, 289, 299, 303, 319, 321, 327, 339, 391, 411, 413, 415, 451, 469, 471, 515, 517, 527, 533, 535, 543, 551, 579, 581, 589, 667, 669, 679, 685, 687, 695, 721, 753, 779, 789, 791, 813, 815

Offset: 1

Zak Seidov, Dec 21 2007

nonn

			4 and 6 are 1st and 2nd semiprimes,
49 and 51 are 17th and 18th semiprimes,
55 and 57 are 19th and 20th semiprimes.

Zak Seidov, Table of n, a(n) for n = 1..1000.

Cf. A001358, A065516.

Position[Partition[PrimeOmega[Range[900]],3,1],?(#[[1]]==#[[3]]==2 && #[[2]]!=2&),1,Heads->False]//Flatten (* _Harvey P. Dale, Oct 15 2021 *)

isok(n) = (bigomega(n) == 2) && (bigomega(n+2) == 2) && (bigomega(n+1) != 2); \\ Michel Marcus, Oct 13 2013

A338902 Number of integer partitions of the n-th semiprime into semiprimes.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 4, 7, 7, 10, 17, 25, 21, 34, 34, 73, 87, 103, 149, 176, 206, 281, 344, 479, 725, 881, 1311, 1597, 1742, 1841, 2445, 2808, 3052, 3222, 6784, 9298, 11989, 14533, 15384, 17414, 18581, 19680, 28284, 35862, 38125, 57095, 60582, 64010, 71730, 76016

Offset: 1

Gus Wiseman, Nov 24 2020

nonn

A semiprime (A001358) is a product of any two prime numbers.

			The a(1) = 1 through a(33) = 17 partitions of 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, where A-Z = 10-35:
  4  6  9  A   E    F   L     M      P      Q       X
           64  A4   96  F6    994    FA     M4      EA9
               644      966   A66    L4     AA6     F99
                        9444  E44    A96    E66     FE4
                              6664   F64    9944    L66
                              A444   9664   A664    P44
                              64444  94444  E444    9996
                                            66644   AA94
                                            A4444   E964
                                            644444  F666
                                                    FA44
                                                    L444
                                                    96666
                                                    A9644
                                                    F6444
                                                    966444
                                                    9444444

A002100 counts partitions into squarefree semiprimes.
A056768 uses primes instead of semiprimes.
A101048 counts partitions into semiprimes.
A338903 is the squarefree version.
A339112 includes the Heinz numbers of these partitions.
A001358 lists semiprimes, with odd and even terms A046315 and A100484.
A037143 lists primes and semiprimes.
A084126 and A084127 give the prime factors of semiprimes.
A320655 counts factorizations into semiprimes.
A338898/A338912/A338913 give prime indices of semiprimes, with sum/difference/product A176504/A176506/A087794.
A338899/A270650/A270652 give prime indices of squarefree semiprimes.
Cf. A000041, A000607, A006881, A065516, A115392, A128301, A320656, A320732, A320892, A320912, A338915, A338916, A339113.

nn=100;Table[Length[IntegerPartitions[n,All,Select[Range[nn],PrimeOmega[#]==2&]]],{n,Select[Range[nn],PrimeOmega[#]==2&]}]

a(n) = A101048(A001358(n)).

A338909 Numbers of the form prime(x) * prime(y) where x and y have a common divisor > 1.

Original entry on oeis.org

9, 21, 25, 39, 49, 57, 65, 87, 91, 111, 115, 121, 129, 133, 159, 169, 183, 185, 203, 213, 235, 237, 247, 259, 267, 289, 299, 301, 303, 305, 319, 321, 339, 361, 365, 371, 377, 393, 417, 427, 445, 453, 481, 489, 497, 515, 517, 519, 529, 543, 551, 553, 559, 565

Offset: 1

Gus Wiseman, Nov 20 2020

nonn

			The sequence of terms together with their prime indices begins:
      9: {2,2}     169: {6,6}     319: {5,10}
     21: {2,4}     183: {2,18}    321: {2,28}
     25: {3,3}     185: {3,12}    339: {2,30}
     39: {2,6}     203: {4,10}    361: {8,8}
     49: {4,4}     213: {2,20}    365: {3,21}
     57: {2,8}     235: {3,15}    371: {4,16}
     65: {3,6}     237: {2,22}    377: {6,10}
     87: {2,10}    247: {6,8}     393: {2,32}
     91: {4,6}     259: {4,12}    417: {2,34}
    111: {2,12}    267: {2,24}    427: {4,18}
    115: {3,9}     289: {7,7}     445: {3,24}
    121: {5,5}     299: {6,9}     453: {2,36}
    129: {2,14}    301: {4,14}    481: {6,12}
    133: {4,8}     303: {2,26}    489: {2,38}
    159: {2,16}    305: {3,18}    497: {4,20}

A082023 counts partitions with these as Heinz numbers, complement A023022.
A300912 is the complement in A001358.
A339002 is the squarefree case.
A001221 counts distinct prime indices.
A001222 counts prime indices.
A001358 lists semiprimes, with odds A046315 and evens A100484.
A004526 counts 2-part partitions, with strict case A140106 (shifted left).
A006881 lists squarefree semiprimes, with odds A046388 and evens A100484.
A176504/A176506/A087794 give sum/difference/product of semiprime indices.
A318990 lists semiprimes with divisible indices.
A320655 counts factorizations into semiprimes.
A338898, A338912, and A338913 give semiprime indices.
A338899, A270650, and A270652 give squarefree semiprime indices.
A338910 lists semiprimes with odd indices.
A338911 lists semiprimes with even indices.
Cf. A005117, A037143, A055684, A056239, A065516, A112798, A115392, A128301, A289182, A338900, A338904.

Select[Range[100],PrimeOmega[#]==2&&GCD@@PrimePi/@First/@FactorInteger[#]>1&]

Equals A001358 \ A300912.

Equals A339002 \/ (A001248 \ {4}).

A339362 Sum of prime indices of the n-th squarefree semiprime.

Original entry on oeis.org

3, 4, 5, 5, 6, 6, 7, 7, 8, 7, 9, 8, 10, 9, 8, 10, 11, 12, 9, 11, 13, 9, 14, 10, 15, 12, 10, 13, 16, 11, 17, 14, 12, 18, 11, 19, 15, 16, 12, 20, 17, 21, 11, 13, 22, 14, 23, 18, 13, 24, 19, 25, 20, 15, 12, 26, 21, 27, 14, 16, 28, 13, 22, 29, 17, 15, 30, 23, 13

Offset: 1

Gus Wiseman, Dec 06 2020

nonn

A squarefree semiprime (A006881) is a product of any two distinct prime numbers. A prime index of n is a number m such that the m-th prime number divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of all squarefree semiprimes together with the sums of their prime indices begins:
   6: 1 + 2 = 3
  10: 1 + 3 = 4
  14: 1 + 4 = 5
  15: 2 + 3 = 5
  21: 2 + 4 = 6
  22: 1 + 5 = 6
  26: 1 + 6 = 7
  33: 2 + 5 = 7
  34: 1 + 7 = 8
  35: 3 + 4 = 7

A001358 lists semiprimes.
A003963 gives the product of prime indices of n.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes.
A025129 gives the sum of squarefree semiprimes of weight n.
A056239 (weight) gives the sum of prime indices of n.
A332765/A339114 give the greatest/least squarefree semiprime of weight n.
A338898/A338912/A338913 give the prime indices of semiprimes, with product/sum/difference A087794/A176504/A176506.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes, with product/sum/difference A339361/A339362/A338900.
A338904 groups semiprimes by weight.
A338905 groups squarefree semiprimes by weight.
A338907/A338908 list squarefree semiprimes of odd/even weight.
A339116 groups squarefree semiprimes by greater prime factor.
Cf. A001221, A046388, A065516, A112798, A115392, A166237, A320656, A338901, A339003, A339004.

Table[Plus@@PrimePi/@First/@FactorInteger[n],{n,Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&]}]

a(n) = A056239(A006881(n)).

a(n) = A270650(n) + A270652(n).

A215231 Increasing gaps between semiprimes.

Original entry on oeis.org

2, 3, 4, 6, 7, 11, 14, 19, 20, 24, 25, 28, 30, 32, 38, 47, 54, 55, 70, 74, 76, 82, 85, 87, 88, 95, 98, 107, 110, 112, 120, 123, 126, 146, 163, 166, 171, 174

Offset: 1

T. D. Noe, Aug 07 2012

See A215232 and A217851 for the semiprimes that begin and end the gaps.
Records in A065516. - R. J. Mathar, Aug 09 2012
How long can these gaps be? In the Cramér model, with x = A215232(n), they are of length log(x)^2/log(log(x))(1 + o(1)) with probability 1. - Charles R Greathouse IV, Sep 07 2012
a(n) = A065516(A085809(n)). - Reinhard Zumkeller, Mar 23 2014

			4 is here because the difference between 10 and 14 is 4, and there is no smaller semiprimes with this property.

Cf. A001358 (semiprimes), A131109, A215232, A217851.
Cf. A005250 (increasing gaps between primes).
Cf. A239673 (increasing gaps between sphenic numbers).

a215231 n = a215231_list !! (n-1)
(a215231_list, a085809_list) = unzip $ (2, 1) : f 1 2 a065516_list where
   f i v (q:qs) | q > v = (q, i) : f (i + 1) q qs
                | otherwise = f (i + 1) v qs
-- Reinhard Zumkeller, Mar 23 2014

SemiPrimeQ[n_Integer] := If[Abs[n] < 2, False, (2 == Plus @@ Transpose[FactorInteger[Abs[n]]][[2]])]; nextSemiprime[n_] := Module[{m = n + 1}, While[! SemiPrimeQ[m], m++]; m]; t = {{0, 0}}; s1 = nextSemiprime[1]; While[s1 < 10^7, s2 = nextSemiprime[s1]; d = s2 - s1; If[d > t[[-1, 1]], AppendTo[t, {d, s1}]; Print[{d, s1}]]; s1 = s2]; t = Rest[t]; Transpose[t][[1]]

a(27)-a(31) from Donovan Johnson, Aug 07 2012

a(32)-a(38) from Donovan Johnson, Sep 20 2012

A213025 Balanced semiprimes (of order one): semiprimes which are the average of the previous semiprime and the following semiprime.

Original entry on oeis.org

34, 86, 94, 122, 142, 185, 194, 202, 214, 218, 262, 289, 302, 314, 321, 358, 371, 394, 407, 413, 415, 422, 446, 471, 489, 493, 497, 517, 535, 562, 581, 586, 626, 634, 669, 687, 698, 734, 785, 791, 815, 838, 842, 922, 982, 989, 1042, 1057, 1079, 1135, 1138

Offset: 1

Gerasimov Sergey, Jun 03 2012

nonn

Semiprimes that are the average of three successive semiprimes.

First term not also in A086005 is 185. - Alonso del Arte, Jun 04 2012

			194 is in the sequence because 194 = (187 + 194 + 201)/3 = (A001358(61) + A001358(62) + A001358(63))/3.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Semiprime

Cf. A086005 (subsequence), A001358, A006562, A065516, A212820.

a213025 n = a213025_list !! (n-1)
a213025_list = f a001358_list where
   f (x:sps'@(y:z:sps)) | 2 * y == (x + z) = y : f sps'
                        | otherwise        = f sps'
-- Reinhard Zumkeller, Jun 10 2012

with(numtheory):
prevsp:= proc(n) local k; for k from n-1 by -1
           while isprime(k) or bigomega(k)<>2 do od; k end:
nextsp:= proc(n) local k; for k from n+1
           while isprime(k) or bigomega(k)<>2 do od; k end:
a:= proc(n) option remember; local s;
      s:= `if`(n=1, 4, a(n-1));
      do s:= nextsp(s);
         if s=(prevsp(s)+nextsp(s))/2 then break fi
      od; s
    end:
seq (a(n), n=1..100);  # Alois P. Heinz, Jun 03 2012

bspQ[{a_,b_,c_}]:=b==(a+c)/2; With[{sp=Select[Range[1200],PrimeOmega[#] == 2&]}, Transpose[Select[Partition[sp,3,1],bspQ]][[2]]] (* Harvey P. Dale, Nov 18 2012 *)
Select[Partition[Select[Range[1200],PrimeOmega[#]==2&],3,1],Mean[#]==#[[2]]&][[;;,2]] (* Harvey P. Dale, Jul 31 2025 *)

2*sp_(n) = sp_(n - 1) + sp_(n + 1).

a(n) = (1/3) * (sp(i) + sp(i + 1) + sp(i + 2)), for some i(n).

cp's OEIS Frontend

A166237 Differences between consecutive products of two distinct primes: a(n) = A006881(n+1) - A006881(n).

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A338903 Number of integer partitions of the n-th squarefree semiprime into squarefree semiprimes.

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A088707 Semiprimes + 1.

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A130533 a(n) = smallest k such that A001358(n+1) = A001358(n) + (A001358(n) mod k), or 0 if no such k exists.

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A136196 Numbers n such that n and n+2 are consecutive semiprimes.

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A338902 Number of integer partitions of the n-th semiprime into semiprimes.

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A338909 Numbers of the form prime(x) * prime(y) where x and y have a common divisor > 1.

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A339362 Sum of prime indices of the n-th squarefree semiprime.

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A215231 Increasing gaps between semiprimes.