A014614 -id:A014614 - cp's OEIS frontend

A120046 The 10^n-th 5-almost prime.

32, 176, 1272, 10374, 89896, 810220, 7475818, 70185558, 667561977, 6411296283, 62037096770, 603813941738

Offset: 0

Robert G. Wilson v, Feb 15 2006

Cf. A014614, A114453, A006988, A114125, A120044, A120045.

FiveAlmostPrimePi[n_] := Sum[ PrimePi[n/(Prime@i*Prime@j*Prime@k*Prime@l)] - l + 1, {i, PrimePi[n^(1/5)]}, {j, i, PrimePi[(n/Prime@i)^(1/4)]}, {k, j, PrimePi[(n/(Prime@i*Prime@j)^(1/3))]}, {l, k, PrimePi@Sqrt[(n/(Prime@i*Prime@j*Prime@k))]}];
FiveAlmostPrime[n_] := Block[{e = Floor[Log[2, n] +4], a, b}, a = 2^e; Do[b = 2^p; While[FiveAlmostPrimePi[a] < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; Do[ Print@FiveAlmostPrime[10^n], {n, 0, 13}]

lista(nmax) = {my(pow = 1, c = 0, n = 0); for(k = 1, oo, if(bigomega(k) == 5, c++; if(c == pow, print1(k, ", "); if(n == nmax, break); pow *= 10; n++)));} \\ Amiram Eldar, Apr 29 2024

a(n) = A014614(10^n). - Amiram Eldar, Apr 29 2024

a(6) corrected and a(7)-a(9) added by Amiram Eldar, Apr 29 2024

a(10)-a(11) from David A. Corneth, Apr 29 2024

A180151 Numbers k such that k and k + 2 are both divisible by exactly five primes (counted with multiplicity).

Original entry on oeis.org

270, 592, 700, 750, 918, 1168, 1240, 1638, 1648, 1672, 1710, 1750, 2070, 2310, 2392, 2548, 2550, 2608, 2728, 2860, 2862, 2896, 2898, 3184, 3330, 3568, 3630, 3822, 3848, 3850, 3942, 3976, 4230, 4264, 4648, 4662, 5070, 5080, 5236, 5238, 5390, 5550, 5560

Offset: 1

Jonathan Vos Post, Aug 12 2010

"5-almost primes" that keep that property when incremented by 2. This sequence is to 5 as 4 is to A180150, as 3 is to A180117, as A092207 is to 2, and as A001359 is to 1. That is, this sequence is the 5th row of the infinite array A[k,n] = n-th natural number m such that m and m+2 are both divisible by exactly k primes (counted with multiplicity). The first row is the lesser of twin primes. The second row is the sequence such that m and m+2 are both semiprimes.

			a(1) = 270 because 270 = 2 * 3^3 * 5 is divisible by exactly 5 primes (counted with multiplicity), and so is 270+2 = 272 = 2^4 * 17.

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

Cf. A000040, A001222, A001358, A001359, A014614, A033987, A101637, A180117, A180150.

f[n_] := Plus @@ (Last@# & /@ FactorInteger@n); fQ[n_] := f[n] == 5 == f[n + 2]; Select[ Range@ 10000, fQ] (* Robert G. Wilson v, Aug 15 2010 *)

for(x=2,10^4,if(bigomega(x)==5&&bigomega(x+2)==5,print1(x", "))) \\ Zak Seidov, Aug 12 2010

{m in A014614 and m+2 in A014614} = {m such that bigomega(m) = bigomega(m+2) = 5} = {m such that A001222(m) = A001222(m+2) = 5}.

Corrected and extended by Zak Seidov and R. J. Mathar, Aug 12 2010

A281926 Numbers that are the product of exactly 5 primes and are of the form prime(k) + prime(k + 1).

Original entry on oeis.org

112, 120, 162, 300, 396, 450, 456, 520, 630, 684, 696, 702, 752, 828, 882, 918, 924, 990, 1044, 1064, 1140, 1250, 1272, 1300, 1428, 1530, 1650, 1692, 1710, 1716, 1740, 1900, 2032, 2072, 2124, 2156

Offset: 1

Zak Seidov, Feb 02 2017

nonn

Note that there is no case of 2 primes.

Intersection of A001043 and A014614. - Bruno Berselli, Feb 02 2017

			112 = 2^4 * 7 = 53 + 59, 120 = 2^3 * 3 * 5 = 59 + 61, 162 = 2 * 3^4 = 79 + 83.

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

Cf. A001043, A014614.

Cf. A105936 (products of 3 primes), A281925 (products of 4 primes).

Total[#] & /@ Select[Partition[Prime[Range[1000]], 2, 1], 5 == PrimeOmega[Total[#]] &]

list(lim)=my(v=List()); forprime(p=2,lim\16, forprime(q=2,min(lim\(8*p),p), forprime(r=2,min(lim\(4*p*q),q), forprime(s=2,min(lim\(2*p*q*r),r), my(t=2*p*q*r*s); if(nextprime(t/2)+precprime(t/2)==t, listput(v,t)))))); Set(v) \\ Charles R Greathouse IV, Feb 05 2017

A337112 Smallest term of A337081 that has exactly n prime factors, or 0 if no such term exists.

Original entry on oeis.org

0, 4, 0, 90, 675, 1134, 6318, 4374, 32805, 255879, 1003833, 531441, 327544803, 20751953125, 225830078125, 91552734375, 1068115234375, 23651123046875, 316619873046875, 1697540283203125, 13256072998046875, 85353851318359375, 541210174560546875, 4518032073974609375, 58233737945556640625

Offset: 1

Matej Veselovac, Aug 16 2020

nonn

a(n) is the smallest product of n primes that has unordered factorizations whose sums of factors are the same (is a term of A337080) and all of whose proper divisors have the complementary property: that every unordered factorization has a distinct sum of factors (i.e., all proper divisors are terms of A337037).

Cf. A337113 (factors of terms).
Cf. A337037, A337080, A337081.
Cf. A056472 (all factorizations of n).
Cf. r-almost primes: A000040 (r = 1), A001358 (r = 2), A014612 (r = 3), A014613 (r = 4), A014614 (r = 5), A046306 (r = 6), A046308 (r = 7), A046310 (r = 8), A046312 (r = 9), A046314 (r = 10), A069272 (r = 11), A069273 (r = 12), A069274 (r = 13), A069275 (r = 14), A069276 (r = 15), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20).

a(14) onward from David A. Corneth, Aug 26 2020

A112443 Pseudoprimes (base-2) equal to the product of 5 primes not necessarily distinct.

Original entry on oeis.org

825265, 1050985, 1275681, 2113665, 2503501, 2615977, 2882265, 3370641, 3755521, 4670029, 4698001, 4895065, 5034601, 6242685, 6973057, 7428421, 8322945, 9223401, 9224391, 9890881, 10877581, 12067705, 12945745, 13757653, 13823601, 13992265, 16778881, 17698241, 18007345

Offset: 1

Shyam Sunder Gupta, Dec 11 2005

			a(1) = 825265 = 5*7*17*19*73.

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

Intersection of A014614 and A001567.

s={}; Do[If[PrimeOmega[n] == 5 && PowerMod[2, n - 1, n] == 1, AppendTo[s, n]], {n, 1, 10^7}]; s (* Amiram Eldar, Nov 10 2019 *)

More terms from Amiram Eldar, Nov 10 2019

A112452 Strong pseudoprimes (base-2) equal to product of 5 primes not necessarily distinct.

Original entry on oeis.org

293609485, 440707345, 606057985, 831807145, 958970545, 1816572745, 2395916965, 2708826841, 2907393385, 3246238801, 4340265931, 4953963781, 5949820045, 6845182669, 9580649065, 10121349421, 11360308765, 11892462985, 13560708421, 16034618701, 16720656121, 16765381165

Offset: 1

Shyam Sunder Gupta, Dec 12 2005

nonn

			a(1) = 293609485 = 5*13*29*109*1429.

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

Intersection of A014614 and A001262.

More terms from Amiram Eldar, Nov 10 2019

A114445 Indices of 5-almost prime pentagonal numbers.

Original entry on oeis.org

11, 35, 40, 42, 51, 54, 59, 63, 67, 80, 87, 92, 100, 115, 120, 125, 126, 131, 132, 136, 159, 165, 167, 168, 175, 184, 189, 200, 204, 210, 215, 217, 222, 225, 227, 231, 232, 242, 247, 250, 251, 255, 259, 260, 261, 270, 280, 282, 283, 285, 287, 291, 295, 304, 306

Offset: 1

Jonathan Vos Post, Feb 14 2006

P(2) = 5 is the only prime pentagonal number, all other factor as P(k) = (k/2)*(3*k-1) or k*((3*k-1)/2) and thus have at least 2 prime factors. P(k) is semiprime iff [k prime and (3*k-1)/2 prime] or [k/2 prime and 3*k-1 prime].

			a(1) = 11 because P(11) = PentagonalNumber(11) = 11*(3*11-1)/2 = 176 = 2^4 * 11 is a 4-almost prime (the prime factors need not be distinct).
a(2) = 35 because P(35) = 35*(3*35-1)/2 = 1820 = 2^2 * 5 * 7 * 13 is a 5-almost prime.
a(13) = 100 because P(100) = 100*(3*100-1)/2 = 14950 = 2 * 5^2 * 13 * 23 is a 5-almost prime.

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, Pentagonal Number.

Cf. A000326, A001222, A014614.

Select[Range[300], PrimeOmega[PolygonalNumber[5, #]] == 5 &] (* Amiram Eldar, Oct 05 2024 *)

{a(n)} = {k such that A001222(A000326(k)) = 5}.
{a(n)} = {k such that k*(3*k-1)/2 has exactly 5 prime factors}.
{a(n)} = {k such that A000326(k) is an element of A014614}.

More terms from Amiram Eldar, Oct 05 2024

A114456 Numbers k such that the k-th hexagonal number is a 5-almost prime.

Original entry on oeis.org

8, 14, 16, 18, 20, 24, 28, 36, 38, 40, 41, 44, 54, 74, 77, 78, 84, 86, 90, 92, 100, 102, 105, 110, 113, 123, 124, 125, 126, 130, 132, 135, 136, 143, 148, 149, 153, 156, 164, 165, 170, 171, 184, 185, 186, 194, 207, 210, 213, 215, 218, 220, 225, 232, 234, 236

Offset: 1

Jonathan Vos Post, Feb 14 2006

There are no prime hexagonal numbers. The k-th hexagonal number A000384(k) = k*(2*k-1) is semiprime iff both k and 2*k-1 are primes iff A000384(k) is an element of A001358 iff k is an element of A005382.

			a(1) = 8 because HexagonalNumber(8) = H(8) = 8*(2*8-1) = 120 = 2^3 * 3 * 5 is a 5-almost prime.
a(2) = 14 because H(14) = 14*(2*14-1) = 378 = 2 * 3^3 * 7 is a 5-almost prime.
a(3) = 18 because H(18) = 18*(2*18-1) = 630 = 2 * 3^2 * 5 * 7 is a 5-almost prime.
a(20) = 100 because H(100) = 100*(2*100-1) = 19900 = 2^2 * 5^2 * 199 is a 5-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, Hexagonal Number.

Cf. A000384, A001222, A014614.

Select[Range[300], PrimeOmega[#*(2*# - 1)] == 5 &] (* Giovanni Resta, Jun 14 2016 *)
Select[Range[300],PrimeOmega[PolygonalNumber[6,#]]==5&] (* Harvey P. Dale, Jan 15 2023 *)

Numbers k such that hexagonal number A000384(k) is an element of A014614.
Numbers k such that A001222(A000384(k)) = 5.
Numbers k such that A001222(k*(2*k-1)) = 5.

Missing a(3)=16 and more terms from Giovanni Resta, Jun 14 2016

A114556 Numbers k such that the k-th heptagonal number is 5-almost prime.

Original entry on oeis.org

7, 16, 23, 30, 32, 36, 42, 45, 54, 69, 78, 79, 80, 84, 88, 90, 93, 95, 100, 102, 104, 112, 115, 117, 140, 143, 151, 153, 165, 170, 174, 176, 184, 186, 191, 200, 203, 210, 213, 228, 232, 234, 245, 250, 259, 271, 273, 282, 287, 296, 306, 308, 310, 311, 318, 319

Offset: 1

Jonathan Vos Post, Feb 15 2006

			a(1) = 7 because Hep(7) = 7*(5*7-3)/2 = 112 = 2^4 * 7 is 5-almost prime [also 112 = Hep(7) = Hep(Hep(2)) is an iterated heptagonal number].
a(2) = 16 because Hep(16) = 16*(5*16-3)/2 = 616 = 2^3 * 7 * 11 is 5-almost prime.
a(3) = 23 because Hep(23) = 23*(5*23-3)/2 = 1288 = 2^3 * 7 * 23.
a(18) = 100 because Hep(100) = 100*(5*100-3)/2 = 24850 = 2 * 5^2 * 7 * 71.
a(21) = 112 because Hep(112) = 112*(5*112-3)/2 = 31192 = 2^3 * 7 * 557 [also 31192 = Hep(112) = Hep(Hep(7)) = Hep(Hep(Hep(2))) is an iterated heptagonal number].

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, Heptagonal Number.

Cf. A000040, A000566, A001222, A001358, A014614.

Select[Range[400], PrimeOmega[# (5 # - 3)/2] == 5 &] (* Giovanni Resta, Jun 14 2016 *)
Select[Range[400],PrimeOmega[PolygonalNumber[7,#]]==5&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 22 2020 *)

Numbers k such that Hep(k) = k*(5*k-3)/2 is 5-almost prime.
Numbers k such that A000566(k) is a term of A014614.
Numbers k such that A001222(A000566(k)) = 5.
Numbers k such that A001222(k*(5*k-3)/2) = 5.

Corrected and extended by Giovanni Resta, Jun 14 2016

A114828 Numbers k such that the k-th octagonal number has 9 prime factors counted with multiplicity.

Original entry on oeis.org

64, 96, 128, 144, 162, 182, 198, 216, 224, 234, 246, 270, 278, 288, 304, 310, 320, 324, 352, 390, 414, 416, 432, 438, 480, 504, 528, 544, 550, 558, 584, 594, 600, 646, 648, 654, 662, 684, 694, 702, 710, 729, 750, 752, 756, 798, 810, 834, 850, 870, 888, 900

Offset: 1

Jonathan Vos Post, Feb 19 2006

k has at most 8 prime factors counted with multiplicity.

			a(1) = 64 because OctagonalNumber(64) = Oct(64) = 64*(3*64-2) = 12160 = 2^7 * 5 * 19 has exactly 9 prime factors (seven are all equally 2; factors need not be distinct).
a(2) = 96 because Oct(96) = 96*(3*96-2) = 27456 = 2^6 * 3 * 11 * 13 is 9-almost prime [also 27456 = Oct(96) = Oct(Oct(6)) is an iterated octagonal number].
a(3) = 128 because Oct(128) = 128*(3*128-2) = 48896 = 2^8 * 191.

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, A046306, A046308, A046310, A046312, A088878, A114606, A114618, A114621, A114634, A114635, A114636.

A000567:=func< n | n*(3*n-2) >; Is9almostprime:=func< n | &+[k[2]: k in Factorization(n)] eq 9 >; [ n: n in [2..1000] | Is9almostprime(A000567(n)) ]; // Klaus Brockhaus, Dec 22 2010

Select[Range[900],PrimeOmega[PolygonalNumber[8,#]]==9&] (* James C. McMahon, Jul 30 2024 *)

isok(k) = bigomega(k*(3*k-2)) == 9; \\ Michel Marcus, Aug 02 2024

Integers k such that k*(3*k-2) has exactly nine prime factors (with multiplicity).
Integers k such that A000567(k) is a term of A046312.
Integers k such that A001222(A000567(k)) = 9.
Integers k such that A001222(k) + A001222(3*k-2) = 9.
Integers k such that (3*k-2)*(3*k-1)*(3*k)/((3*k-2)+(3*k-1)+(3*k)) is in A046310.

Missing terms inserted by R. J. Mathar, Dec 22 2010
a(40)-a(52) from James C. McMahon, Jul 30 2024
Name edited by David A. Corneth, Jul 31 2024

cp's OEIS Frontend

A120046 The 10^n-th 5-almost prime.

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A180151 Numbers k such that k and k + 2 are both divisible by exactly five primes (counted with multiplicity).

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A281926 Numbers that are the product of exactly 5 primes and are of the form prime(k) + prime(k + 1).

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A337112 Smallest term of A337081 that has exactly n prime factors, or 0 if no such term exists.

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A112443 Pseudoprimes (base-2) equal to the product of 5 primes not necessarily distinct.

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A112452 Strong pseudoprimes (base-2) equal to product of 5 primes not necessarily distinct.

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A114445 Indices of 5-almost prime pentagonal numbers.

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A114456 Numbers k such that the k-th hexagonal number is a 5-almost prime.

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