A014614 -id:A014614 - cp's OEIS frontend

A046403 Numbers with exactly 5 distinct palindromic prime factors.

2310, 21210, 27510, 31710, 33330, 38010, 40110, 43230, 46662, 49830, 59730, 60522, 63030, 65730, 69762, 74130, 77770, 78330, 80430, 83622, 88242, 100870, 103290, 116270, 116490, 116655, 123090, 126390, 139370, 144606, 147070, 151305, 152670, 158970, 163086, 165270, 167370

Offset: 1

Patrick De Geest, Jun 15 1998

			144606 = 2 * 3 * 7 * 11 * 313 is in the sequence as it has exactly 5 prime factors each of which is a palindrome. - _David A. Corneth_, Aug 30 2020

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A002385, A014614, A046400, A046401, A046402, A046403, A046371.

Offset changed to 1 and more terms from David A. Corneth, Aug 30 2020

A211162 Sophie Germain 5-almost primes.

Original entry on oeis.org

688, 1552, 3496, 4360, 5008, 6352, 6952, 7546, 7672, 9256, 9625, 9712, 10062, 10300, 10840, 11632, 11875, 12112, 12136, 12460, 12712, 13432, 13648, 13744, 13912, 14152, 14812, 14920, 15484, 16562, 17050, 17104, 17272, 17608, 17752, 18130, 18232, 18616, 18952, 19062, 19624, 19792, 21100, 21136, 21352

Offset: 1

Jonathan Vos Post and Robert G. Wilson v, Jan 30 2013

nonn

Numbers n that are products of exactly 5 primes, such that 2*n + 1 are also products of exactly 5 primes. By analogy with A111153 Sophie Germain semiprimes: semiprimes n such that 2n+1 is also a semiprime; A111173 Sophie Germain 3-almost primes; A111176 Sophie Germain 4-almost primes.
From Zak Seidov, Jan 30 2013: (Start)
First integers n such that both n and 2n+1 are Sophie Germain 5-almost primes are: 54708, 103812, 111952, 113368, 117328, 134312, 159568, 160062, 165462, 199048, 205812.
First integers n such that n, 2n+1 and 4n+3 all are Sophie Germain 5-almost primes are: 159568, 301812, 431068, 444388, 564718, 1144468, 1420468, 1653162, 1687768, 1794568.
First integers n such that n, 2n+1, 4n+3 and 8n+7 all are Sophie Germain 5-almost primes are: 2991345, 4553367, 7760616, 9145318, 9332368, 12919266, 14283535, 14659746, 15144118.
First integers n such that n, 2n+1, 4n+3, 8n+7 and 16n+15 all are Sophie Germain 5-almost primes are: 15144118, 18515752, 41092024, 60406662, 71783890, 87353512, 94144212
First integers n such that n, 2n+1, 4n+3, 8n+7, 16n+15 and 32n+31 all are Sophie Germain 5-almost primes are: 211457337, 237572475, 245071092, 352015408, 415695462, 433833417.
First integers n such that n, 2n+1, 4n+3, 8n+7, 16n+15, 32n+31 and 64n+63 all are Sophie Germain 5-almost primes are: 433833417, 463078210, 648871975. (End)

			a(1) = 688 because 688 = 2^4 * 43, and 2*688 + 1 = 1377 = 3^4 * 17.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Cf. A014614, A111153, A111173, A111176.

Is5primes:=func; [n: n in [2..22000] | Is5primes(n) and Is5primes(2*n+1)]; // Bruno Berselli, Jan 30 2013

fQ[n_] := PrimeOmega[n] == 5 == PrimeOmega[2 n + 1]; Select[Range@ 100000, fQ] (* Robert G. Wilson v *)

is(n)=bigomega(n)==5 && bigomega(2*n+1)==5 \\ Charles R Greathouse IV, Feb 01 2017

{n in A014614 such that 2*n + 1 is in A014614}.

A046371 Numbers with exactly 5 palindromic prime factors (counted with multiplicity).

Original entry on oeis.org

32, 48, 72, 80, 108, 112, 120, 162, 168, 176, 180, 200, 243, 252, 264, 270, 280, 300, 378, 392, 396, 405, 420, 440, 450, 500, 567, 588, 594, 616, 630, 660, 675, 700, 750, 882, 891, 924, 945, 968, 980, 990, 1050, 1100, 1125, 1250, 1323, 1372, 1386, 1452

Offset: 1

Patrick De Geest, Jun 15 1998

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

Intersection of A033620 and A014614.

Cf. A046403.

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

A114415 Records in 5-almost prime gaps ordered by merit.

Original entry on oeis.org

16, 24, 28, 42, 56, 70

Offset: 1

Jonathan Vos Post, Nov 25 2005

Next term, if it exists, is associated with indices above 100000 in A114405 and A014614. - R. J. Mathar, May 10 2007

			Records defined in terms of A114405 and A014614:
  n  A114405(n)  A114405(n)/log_10(A014614(n))
  =  ==========  =============================
  1      16      16/log_10(32)  = 10.6301699
  2      24      24/log_10(48)  = 14.2751673
  3      8       8/log_10(72)   = 4.30725248
  4      28      28/log_10(80)  = 14.7129144
  5      4       4/log_10(108)  = 1.96712564
  6      8       8/log_10(112)  = 3.90392819
  7      42      42/log_10(120) = 20.2002592
  8      6       6/log_10(168)  = 2.69625443
  ...
  22     56      56/log_10(312) = 22.4524976

Cf. A014614, A065516, A111870, A111871, A114403-A114411, A114412-A114422.

A014614 := proc(nmax) local a,i; a := [] ; i := 1 ; while nops(a) < nmax do if numtheory[bigomega](i) = 5 then a := [op(a),i] ; fi ; i := i+1 ; end: a ; end: A114405 := proc(a014614) local a,i; a := [] ; for i from 2 to nops(a014614) do a := [op(a), op(i,a014614)-op(i-1,a014614)] ; od ; a ; end: a014614 := A014614(100000) : a114405 := A114405(a014614) : Digits := 30 : rec := -1 : for i from 1 to nops(a114405) do if evalf(a114405[i]/log(a014614[i])) > rec then printf("%d, ",a114405[i]) ; rec := evalf(a114405[i]/log(a014614[i])) ; fi ; od ; # R. J. Mathar, May 10 2007

a(n) = records in A114405(n)/log_10(A014614(n)) = records in (A014614(n+1) - A014614(n))/log_10(A014614(n)).

a(6) from R. J. Mathar, May 10 2007

A114621 Numbers k such that the k-th octagonal number is 5-almost prime.

Original entry on oeis.org

8, 10, 12, 20, 26, 28, 45, 58, 63, 68, 76, 81, 82, 92, 99, 106, 115, 116, 129, 146, 159, 165, 171, 172, 188, 195, 202, 212, 213, 218, 225, 236, 255, 259, 261, 268, 273, 279, 298, 309, 325, 339, 343, 351, 362, 375, 387, 395, 399

Offset: 1

Jonathan Vos Post, Feb 17 2006

It is necessary but not sufficient that k must be prime (A000040), semiprime (A001358), 3-almost prime (A014612), or 4-almost prime (A014613).

			a(1) = 8 because OctagonalNumber(8) = Oct(8) = 8*(3*8-2) = 176 = 2^4 * 11 has exactly 5 prime factors (four are all equally 2; factors need not be distinct). Also, 176 = Oct(8) = Oct(Oct(2)), an iterated octagonal number. Also, 176 is a pentagonal number, hence a term of A046189 octagonal pentagonal numbers.
a(2) = 10 because Oct(10) = 10*(3*10-2) = 280 = 2^3 * 5 * 7 is 5-almost prime.
a(4) = 20 because Oct(20) = 20*(3*20-2) = 1160 = 2^3 * 5 * 29.
a(5) = 26 because Oct(26) = 26*(3*26-2) = 1976 = 2^3 * 13 * 19.
a(19) = 129 because Oct(129) = 129*(3*129-2) = 49665 = 3 * 5 * 7 * 11 * 43 is 5-almost prime (in this case, the 5 prime factors are distinct).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Octagonal Number.

Cf. A000040, A000567, A001222, A001358, A014612, A014613, A014614.

Select[Range[500], PrimeOmega[PolygonalNumber[8, #]] == 5 &] (* Amiram Eldar, Oct 07 2024 *)

Numbers k such that k*(3*k-2) has exactly five prime factors (with multiplicity).
Numbers k such that A000567(k) is a term of A014614.
Numbers k such that A001222(A000567(k)) = 5.
Numbers k such that A001222(k) + A001222(3*k-2) = 5.
Numbers k such that [(3*k-2)*(3*k-1)*(3*k)]/[(3*k-2)+(3*k-1)+(3*k)] is a term of A014614.

12, 63, 99 inserted and 117 removed by R. J. Mathar, Dec 22 2010

A114634 Numbers k such that the k-th octagonal number is 6-almost prime.

Original entry on oeis.org

6, 14, 16, 18, 34, 36, 40, 42, 44, 46, 50, 52, 56, 60, 62, 74, 88, 98, 100, 122, 124, 130, 132, 135, 138, 142, 148, 152, 156, 158, 170, 178, 186, 189, 194, 196, 209, 226, 232, 242, 243, 244, 258, 260, 266, 274, 282, 292, 296, 297, 302, 308, 314, 315, 316, 322

Offset: 1

Jonathan Vos Post, Feb 17 2006

It is necessary but not sufficient that k must be prime (A000040), semiprime (A001358), 3-almost prime (A014612), 4-almost prime (A014613), or 5-almost prime (A014614).

			a(1) = 6 because OctagonalNumber(6) = Oct(6) = 6*(3*6-2) = 96 = 2^5 * 3 has exactly 6 prime factors (five are all equally 2; factors need not be distinct).
a(2) = 14 because Oct(14) = 14*(3*14-2) = 560 = 2^4 * 5 * 7 is 6-almost prime.
a(3) = 16 because Oct(16) = 16*(3*16-2) = 736 = 2^5 * 23.
a(7) = 40 because Oct(40) = 40*(3*40-2) = 4720 = 2^4 * 5 * 59 [also, 4720 = Oct(40) = Oct(Oct(4)), an iterated octagonal number].
a(19) = 100 because Oct(100) = 100*(3*100-2) = 29800 = 2^3 * 5^2 * 149.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Octagonal Number.

Cf. A000040, A000567, A001222, A001358, A014612, A014613, A014614, A046306.

Flatten[Position[Table[n(3n-2),{n,400}],?(PrimeOmega[#]==6&)]] (* _Harvey P. Dale, Jun 17 2013 *)
Select[Range[400],PrimeOmega[PolygonalNumber[8,#]]==6&] (* Harvey P. Dale, Feb 23 2022 *)

is(n)=my(t=bigomega(3*n-2)); t<6 && (t<5 || !isprime(n)) && t+bigomega(n)==6 \\ Charles R Greathouse IV, Feb 01 2017

Numbers k such that k*(3*k-2) has exactly six prime factors (with multiplicity).
Numbers k such that A000567(k) is a term of A046306.
Numbers k such that A001222(A000567(k)) = 6.
Numbers k such that A001222(k) + A001222(3*k-2) = 6.
Numbers k such that [(3*k-2)*(3*k-1)*(3*k)]/[(3*k-2)+(3*k-1)+(3*k)] is a term of A046306.

A114635 Numbers k such that the k-th octagonal number is 7-almost prime.

Original entry on oeis.org

24, 30, 32, 38, 48, 66, 72, 78, 90, 94, 104, 110, 112, 114, 120, 136, 140, 154, 164, 166, 168, 176, 180, 190, 204, 206, 208, 210, 220, 222, 228, 238, 248, 254, 276, 280, 284, 286, 290, 300, 306, 312, 326, 338, 344

Offset: 1

Jonathan Vos Post, Feb 18 2006

It is necessary but not sufficient that k must be prime (A000040), semiprime (A001358), 3-almost prime (A014612), 4-almost prime (A014613), 5-almost prime (A014614), or 6-almost prime (A046308).

			a(1) = 24 because OctagonalNumber(24) = Oct(24) = 24*(3*24-2) = 96 = 1680 = 2^4 * 3 * 5 * 7 has exactly 7 prime factors (four are all equally 2; factors need not be distinct).
a(2) = 30 because Oct(30) = 30*(3*30-2) = 2640 = 2^4 * 3 * 5 * 11 is 7-almost prime.
a(3) = 32 because Oct(32) = 32*(3*32-2) = 3008 = 2^6 * 47 is 7-almost prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Octagonal Number.

Cf. A000040, A000567, A001222, A001358, A014612, A014613, A014614, A046306, A046308.

Select[Range[400],PrimeOmega[PolygonalNumber[8,#]]==7&] (* Harvey P. Dale, Aug 13 2021 *)

Numbers k such that k*(3*k-2) has exactly seven prime factors (with multiplicity).
Numbers k such that A000567(k) is a term of A046308.
Numbers k such that A001222(A000567(k)) = 7.
Numbers k such that A001222(k) + A001222(3*k-2) = 7.
Numbers k such that [(3*k-2)*(3*k-1)*(3*k)]/[(3*k-2)+(3*k-1)+(3*k)] is a term of A046308.

A114636 Numbers k such that the k-th octagonal number is 8-almost prime.

Original entry on oeis.org

22, 70, 80, 84, 102, 108, 118, 126, 134, 160, 174, 184, 200, 230, 240, 250, 252, 262, 264, 272, 318, 330, 334, 336, 350, 368, 378, 400, 408, 420, 430, 434, 444, 450, 454, 459, 462, 464, 484, 494, 500, 502, 510, 518, 520, 522, 540, 560, 564, 566, 570, 574, 582

Offset: 1

Jonathan Vos Post, Feb 18 2006

It is necessary but not sufficient that k must be prime (A000040), semiprime (A001358), 3-almost prime (A014612), 4-almost prime (A014613), 5-almost prime (A014614), 6-almost prime (A046306), or 7-almost prime (A046308).

			a(1) = 22 because OctagonalNumber(22) = Oct(22) = 22*(3*22-2) = 1408 = 2^7 * 11 has exactly 8 prime factors (seven are all equally 2; factors need not be distinct).
a(2) = 70 because Oct(70) = 70*(3*70-2) = 14560 = 2^5 * 5 * 7 * 13 is 8-almost prime.
a(3) = 80 because Oct(80) = 80*(3*80-2) = 19040 = 2^5 * 5 * 7 * 17.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2000 from Harvey P. Dale)
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Octagonal Number.

Cf. A000040, A000567, A001222, A001358, A014612, A014613, A014614, A046306, A046308, A046310.

Select[Range[400],PrimeOmega[PolygonalNumber[8,#]]==8&] (* Harvey P. Dale, Aug 31 2020 *)

Numbers k such that k*(3*k-2) has exactly eight prime factors (with multiplicity).
Numbers k such that A000567(k) is a term of A046310.
Numbers k such that A001222(A000567(k)) = 8.
Numbers k such that A001222(k) + A001222(3*k-2) = 8.
Numbers k such that [(3*k-2)*(3*k-1)*(3*k)]/[(3*k-2)+(3*k-1)+(3*k)] is a term of A046310.

cp's OEIS Frontend

A046403 Numbers with exactly 5 distinct palindromic prime factors.

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A211162 Sophie Germain 5-almost primes.

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A046371 Numbers with exactly 5 palindromic prime factors (counted with multiplicity).

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

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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).

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A114415 Records in 5-almost prime gaps ordered by merit.

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A114621 Numbers k such that the k-th octagonal number is 5-almost prime.

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A114634 Numbers k such that the k-th octagonal number is 6-almost prime.

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