A014613 -id:A014613 - cp's OEIS frontend

A118780 Semiprime(n)semiprime(n+3) - semiprime(n+1)semiprime(n+2), where semiprime(n) is the n-th semiprime.

-14, -6, -5, 0, -7, -87, -4, 76, -8, -212, 64, -4, 128, 68, -265, 31, -12, -177, 104, 109, -28, 103, -101, -40, -24, -348, -176, 253, 81, -285, -97, 928, 364, -841, -257, -361, -127, -3, -125, 603, 359, -675, 367, -8, -860, 139, -3, 995, 280, -1276, -167, 629, 145, 443, -365, -579, 171, -569

Offset: 1

Jonathan Vos Post, May 22 2006

Semiprime analog of A117301.

By construction, every entry is also the difference between two 4-almost primes: a(1) = A014613(4)-A014613(5); a(2) = A014613(9)-A014613(11); a(3) = A014613(16)-A014613(18); a(4) = A014613(27)-A014613(27); etc. - R. J. Mathar, Nov 27 2007

			a(1) = -14 because the determinant of the first block of 4 consecutive semiprimes is:
|4. 6.|
|9. 10|.
a(4) = 0 because the determinant of the 4th block of 4 semiprimes is the first of a presumably infinite number of singular matrices:
|10. 14.|
|15. 21.|.
a(8) = 76, the first positive value in the sequence:
|22. 25.|
|26. 33.|.

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

Cf. A001358, A067276, A117301, A118713.

A001358 := proc(n) option remember ; local a; if n = 1 then 4 ; else for a from A001358(n-1)+1 do if numtheory[bigomega](a)= 2 then RETURN(a) ; fi ; od: fi ; end: A118780 := proc(n) A001358(n)*A001358(n+3)-A001358(n+1)*A001358(n+2) ; end: seq(A118780(n),n=1..58) ; # R. J. Mathar, Nov 27 2007

nmax = 58; spmax = nmax; SP = {};
While[nmax+3 > Length[SP], spmax += nmax; SP = Select[Range[spmax], PrimeOmega[#] == 2&]];
a[n_] := SP[[n]] SP[[n+3]] - SP[[n+1]] SP[[n+2]];
Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Aug 01 2023 *)
#[[1]]#[[4]]-#[[2]]#[[3]]&/@Partition[Select[Range[300],PrimeOmega[#]==2&],4,1] (* Harvey P. Dale, Sep 08 2024 *)

a(n) = A001358(n)*A001358(n+3) - A001358(n+1)*A001358(n+2).

Better definition from Jens Kruse Andersen, May 03 2008

A128303 Indices of squares (of semiprimes) in the 4-almost primes.

Original entry on oeis.org

1, 3, 8, 12, 24, 29, 59, 66, 90, 97, 162, 172, 187, 224, 234, 335, 385, 412, 489, 531, 551, 630, 692, 791, 921, 997, 1128, 1223, 1256, 1285, 1420, 1484, 1518, 1549, 1937, 2146, 2315, 2441, 2483, 2556, 2606, 2651, 2915, 3124, 3175, 3542, 3587, 3645, 3751, 3800

Offset: 1

Rick L. Shepherd, Mar 04 2007

nonn

			a(5) = 24 as 196 = 14^2 = semiprime(5)^2, the 5th square in the 4-almost primes, is the 24th 4-almost prime.

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

Cf. A014613, A074985, A001358, A128302, A128304.

Position[Select[Range[3*10^4], PrimeOmega[#] == 4 &], ?(IntegerQ[Sqrt[#]] &)] // Flatten (* _Amiram Eldar, Apr 13 2025 *)

list(lim) = {my(f, c); for(k = 1, lim, f = factor(k); if(bigomega(f) == 4, c++; if(vecprod(f[,2]) == 4, print1(c, ", "))));} \\ Amiram Eldar, Apr 13 2025

A014613(a(n)) = A074985(n) = A001358(n)^2.

A128304 Indices of 4th powers (of primes) in the 4-almost primes.

Original entry on oeis.org

1, 8, 90, 385, 2556, 5138, 15590, 24646, 53993, 139199, 182476, 375363, 569617, 691012, 991150, 1613115, 2490040, 2849478, 4163793, 5263838, 5888203, 8100907, 9886861, 13102875, 18538021, 21816421, 23608907, 27525519, 29659283, 34290671

Offset: 1

Rick L. Shepherd, Mar 05 2007

nonn

			a(3) = 90 as 625 = 5^4 = prime(3)^4, the third 4th power in the 4-almost primes, is the 90th 4-almost prime.

Amiram Eldar, Table of n, a(n) for n = 1..99 (terms 1..40 from Zak Seidov)

Cf. A000040, A014613, A030514, A128303, A128305.

Position[Select[Range[10^6], PrimeOmega[#] == 4 &], ?(PrimeNu[#] == 1 &)] // Flatten (* _Amiram Eldar, Apr 13 2025 *)

list(lim) = {my(f, c); for(k = 1, lim, f = factor(k); if(bigomega(f) == 4, c++; if(omega(f) == 1, print1(c, ", "))));} \\ Amiram Eldar, Apr 13 2025

A014613(a(n)) = A030514(n) = A000040(n)^4.

A166718 Numbers with at most 4 prime factors (counted with multiplicity).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76

Offset: 1

Michael B. Porter, Oct 20 2009

Complement of A046304, A001222(a(n)) <= 4.
Maynard shows there are infinitely many integers n such that the interval [n,n+90] contains 2 primes and a number with at most 4 prime factors [Jonathan Vos Post, May 23 2012]
Subset of the 5-free numbers (numbers where each exponent in the prime factorization is <=4). - R. J. Mathar, Aug 08 2012

			88 = 2*2*2*11 is in the sequence since it has 4 prime factors
72 = 2*2*2*3*3 is not in the sequence since it has 5 prime factors

G. C. Greubel, Table of n, a(n) for n = 1..10000
James Maynard, Bounded length intervals containing two primes and an almost-prime, arXiv:1205.5020v1 [math.NT], May 22 2012

Cf. A046304, A001222

For numbers with at most n prime factors: n=1: A000040, n=2: A037143, n=3: A037144, n=5: A166719

Select[Range[100],PrimeOmega[#]<= 4 &] (* G. C. Greubel, May 24 2016 *)

PARI
```
isA166718(n) = (bigomega(n) <= 4)
```

UNION of A000040, A001358, A014612, and A014613. - R. J. Mathar, Aug 08 2012

A215405 Largest prime factor of the n-th n-almost prime.

Original entry on oeis.org

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

Offset: 1

Juri-Stepan Gerasimov, Aug 09 2012

nonn

Technically, the prime numbers are "1-almost prime."
Prime(m) (m>=1) occurs first at index n = 1, 2, 4, 6, 10, 13, 18, 21, 24, 34, 36, 43, 49, 54, 61, 66, 75, 79, 91, 97, 101, 107, 113, 124, 138, 144, 148, 157, 162, 167, 187, 194, 202, 207, 224, 229,... in the sequence. - R. J. Mathar, Aug 09 2012
n <= a(n) at 1, 2, 3, 4, 6, 10, 13,...
n < 2*a(n) at n = 1, 2, 3, 4, 6, 7, 9, 10, 13, 16, 18, 21, 22, 24, 29, 33, 34, 36, 40, 43, 49, 54, 55, 59, 61, 66, 69,...
Also largest prime factor of A122943(n) for n>1. - Eric Desbiaux, Mar 20 2016

			a(2) = 3 because the 2nd 2-almost prime (semiprime, A001358) is 6 = 2 * 3, the largest prime factor there being 3.
a(3) = 3 because the 3rd 3-almost prime (A014612) is 18 = 2 * 3^2, the largest prime factor there being 3.
a(4) = 5 because the 4th 4-almost prime (A014613) is 40 = 2^3 * 5, the largest prime factor there being 5.

Cf. A078841, A101695.

A215405 := proc(n)
        A006530(A101695(n)) ;
end proc: # R. J. Mathar, Aug 09 2012

Corrected by R. J. Mathar, Aug 09 2012

A334069 Number of numbers <= 2^n that are the product of exactly four primes, not necessarily distinct.

Original entry on oeis.org

0, 0, 0, 1, 2, 7, 14, 34, 71, 152, 325, 669, 1405, 2866, 5931, 12139, 24782, 50444, 102458, 207945, 420511, 850518, 1716168, 3460304, 6968639, 14022029, 28189833, 56631732, 113697179, 228115641, 457456902, 916899721, 1836996851, 3678943569, 7365141297, 14740076678, 29490954290

Offset: 1

Robert G. Wilson v, Apr 13 2020

nonn

			a(6) = 7 because
  16 = 2 * 2 * 2 * 2,
  24 = 2 * 2 * 2 * 3,
  36 = 2 * 2 * 3 * 3,
  40 = 2 * 2 * 2 * 5,
  54 = 2 * 3 * 3 * 3,
  56 = 2 * 2 * 2 * 7, and
  60 = 2 * 2 * 3 * 5
are the seven numbers less than 2^6 = 64 that are each the product of four primes.

Robert G. Wilson v, Table of n, a(n) for n = 1..53
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A007053, A014613, A082996, A125527, A127396, A126279.

Partial sums of A120035.

FourAlmostPrimePi[n_] := Sum[ PrimePi[n/(Prime@i*Prime@j*Prime@k)] - k + 1, {i, PrimePi[n^(1/4)]}, {j, i, PrimePi[(n/Prime@i)^(1/3)]}, {k, j, PrimePi@Sqrt[n/(Prime@i*Prime@j)]}]; Array[FourAlmostPrimePi[2^#] &, 37]

a(n) = A082996(2^n).

A036093 Centered cube numbers: (n+1)^15 + n^15.

Original entry on oeis.org

1, 32769, 14381675, 1088090731, 31591319949, 500702562701, 5217746494519, 39931933598775, 241075504183481, 1205891132094649, 5177248169415651, 19584269744002019, 66592914588677125, 206753988571902981

Offset: 0

N. J. A. Sloane

Never prime nor semiprime, nor triprime, as a(n) = (2n+1) * (n^2 + n + 1) * (n^4 + 2n^3 + 4n^2 + 3n + 1) * (n^8 + 4n^7 + 30n^6 + 76n^5 + 99n^4 + 76n^3 + 35n^2 + 9n + 1). Has the nontrivial minimum 4 prime factors when n is in {1, 5, 105, ...}. - Jonathan Vos Post, Aug 27 2011

			1^15 + (1+1)^15 = 32769 = 3^2 * 11 * 331 which has the nontrivial minimum 4 prime factors (see A014613).

B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Cf. A036091, A036092.

[(n+1)^15+n^15: n in [0..20]]; // Vincenzo Librandi, Aug 28 2011

Total/@Partition[Range[0,20]^15,2,1] (* Harvey P. Dale, Apr 18 2013 *)

a(n)=(n+1)^15+n^15 \\ Charles R Greathouse IV, Aug 30 2017

A046370 Numbers with exactly 4 palindromic prime factors (counted with multiplicity).

Original entry on oeis.org

16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90, 100, 126, 132, 135, 140, 150, 189, 196, 198, 210, 220, 225, 250, 294, 297, 308, 315, 330, 350, 375, 441, 462, 484, 490, 495, 525, 550, 625, 686, 693, 726, 735, 770, 808, 825, 875, 1029, 1048, 1078, 1089, 1155, 1208

Offset: 1

Patrick De Geest, Jun 15 1998

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Intersection of A033620 and A014613.

Cf. A046402.

Offset changed by Andrew Howroyd, Aug 14 2024

A101696 a(n) = sum(i=1,n)(i-th i-almost prime). Cumulative sums of A101695.

Original entry on oeis.org

2, 8, 26, 66, 174, 398, 878, 2174, 4862, 10494, 22014, 45054, 98302, 222718, 480766, 1021438, 2127358, 4355582, 8943102, 18773502, 38696446, 79590910, 175142398, 368080382, 764442110, 1586525694, 3247470078, 6644856318, 13489960446

Offset: 1

Jonathan Vos Post, Dec 12 2004, Sep 28 2006

nonn

It seems that this sum can never be a prime after a(1) = 2, since the n-th n-almost prime is always even. The number of prime factors (with multiplicity) of a(n) is 1, 3, 2, 3, 3, 2, 2, 2, 4, 5, 4, 4, 3, 3, 5, 4, 3, 4, 7, 4, 2, 5, 5, 2, 3, 7, 4, 3, 4.

This is the diagonalization of the set of sequences {j-almost prime(k)}. The cumulative sums of this sequence are in A101696. a(1)=2 is prime. a(2)=8 is a 3-almost prime. a(3)=26 is a semiprime. a(4)=66 is a 3-almost prime. a(5)= 174 is a 3-almost prime. a(6)=398 is a semiprime. a(7)=878 is a semiprime. a(8)=2174 is a semiprime. a(9)=4862 is a 4-almost prime. a(10)=10494 is a 5-almost prime. a(11)=22014 is a 4-almost prime. a(12)=45054 is a 3-almost prime. a(13)=98302 is a 3-almost prime. a(14)=222718 is a 3-almost prime. a(15)=480766 is a 5-almost prime. a(16)=1021438 is a 4-almost prime. a(17)=2127358 is a 3-almost prime. a(18)=4355582 is a 4-almost prime. a(19)=8943102 is a 7-almost prime. a(20)=18773502 is a 4-almost prime. 21-almost numbers are not yet listed in the OEIS.

			a(1) = first 1-almost prime = first prime = A000040(1) = 2.
a(2) = 2 + 2nd 2-almost prime = 2 + A001358(2) = 2+ 6 = 8.
a(3) = a(2) + 3rd 3-almost prime = 8+A014612(3) = 8+18 = 26.
a(4) = a(3) + 4th 4-almost prime = 26+A014613(4) = 26+40 = 66.
a(5) = a(4) + 5th 5-almost prime = 66+A014614(5) = 66+108=174.
...
a(12) = a(11) + 12th 12-almost prime = 22014 + 23040 = 45054 (the first nontrivial palindrome in the sequence).

Eric Weisstein's World of Mathematics, Almost Prime.

Cf. A000040, A001358, A014612, A014613, A046314, A046306, A046308, A046310, A046312, A046314, A069272, A069273, A069274, A069275, A069276, A069277, A069278, A069279, A069280, A069281, A101637, A101638, A101605, A101606, A101695.

a(1) = first 1-almost prime = first prime = A000040(1). a(2) = a(1) + 2nd 2-almost prime = a(1) + 2nd semiprime = A000040(1)+A001358(2). a(3) = a(2) + 3rd 3-almost prime = a(2) + A014612(3). a(4) = a(3) + 4th 4-almost prime = a(3) + A014613(4)... a(n) = a(n-1) + n-th n-almost prime.

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar

A109636 Let T(n,k) be the n-th k-almost prime. Then a(n) = T(n,k) such that k is minimal and for all m>0, T(n,k+m) >= 2^m * T(n,k).

Original entry on oeis.org

2, 3, 9, 10, 27, 28, 30, 81, 84, 88, 90, 100, 104, 243, 252, 264, 270, 272, 280, 300, 304, 312, 729, 736, 756, 784, 792, 810, 816, 840, 880, 900, 912, 928, 936, 992, 1000, 1040, 2187, 2208, 2268, 2352, 2368, 2376, 2430, 2448, 2464, 2520, 2624

Offset: 1

Yury V. Shlapak (shlapak(AT)imp.kiev.ua), Aug 04 2005

nonn

If one writes the k-almost primes in rows (one row for each k), one observes that there exists a P_{k_0}(n) such that P_{k_0+1}(n) = 2P_{k_0}(n) and for each k>=k_0, P_{k+1}(n)=2P_{k}(n). Then a(n) = P_{k_0}(n). In other words in the columns the values double from row k_0 on. - Peter Pein (petsie(AT)dordos.net), Mar 16 2007

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Wikipedia, k-almost prime numbers.

Cf. A000040, A001358, A014612, A014613, A014614, A078840.

a[n_] := Module[{p = Prime[Range[n]], pal}, pal = Transpose /@ Partition[NestList[Take[Union[Flatten[Outer[Times, #1, p]]], Length[#1]] &, p, n], 2, 1]; Complement @@ Transpose[Cases[pal, {k_, kk_} /; kk == 2*k, {2}]]] ; a[50] (* Peter Pein, Nov 10 2007 *)

from itertools import count
# uses function A078840_T from A078840
def A109636(n):
    a = A078840_T(1,n)
    for k in count(2):
        b = A078840_T(k,n)
        if b==(a<<1):
            return a
        a = b # Chai Wah Wu, Mar 30 2025

Edited by Max Alekseyev, Mar 16 2007
More terms from Peter Pein, Mar 16 2007
Definition corrected by Chai Wah Wu, Mar 30 2025

cp's OEIS Frontend

A118780 Semiprime(n)*semiprime(n+3) - semiprime(n+1)*semiprime(n+2), where semiprime(n) is the n-th semiprime.

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A128303 Indices of squares (of semiprimes) in the 4-almost primes.

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A128304 Indices of 4th powers (of primes) in the 4-almost primes.

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A166718 Numbers with at most 4 prime factors (counted with multiplicity).

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A215405 Largest prime factor of the n-th n-almost prime.

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A334069 Number of numbers <= 2^n that are the product of exactly four primes, not necessarily distinct.

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A036093 Centered cube numbers: (n+1)^15 + n^15.

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A118780 Semiprime(n)semiprime(n+3) - semiprime(n+1)semiprime(n+2), where semiprime(n) is the n-th semiprime.