A046306 -id:A046306 - cp's OEIS frontend

A114839 Indices of Fibonacci numbers with 6 distinct prime factors.

40, 48, 54, 56, 64, 78, 92, 95, 99, 102, 116, 117, 129, 133, 155, 159, 175, 177, 188, 194, 205, 206, 219, 237, 245, 265, 278, 314, 323, 327, 339, 341, 343, 346, 356, 358, 361, 362, 394, 407, 411, 417, 422, 427, 437, 446, 454, 466, 482, 502, 503, 505, 514, 515, 527, 535, 542, 545, 551, 562, 573, 577, 583, 593, 607, 614, 622, 623, 625, 634, 655, 662, 667, 674, 713, 727, 731, 769, 781, 789, 791, 803, 809, 821, 835, 893, 917, 919, 974, 977, 982, 993, 995, 1013, 1039, 1047, 1057, 1081, 1097, 1103, 1121, 1138, 1151, 1165, 1172, 1202, 1203

Offset: 1

Shyam Sunder Gupta, Feb 19 2006

nonn

Numbers n such that A000045(n) is in A046306.

			a(1) = 40 because 40th Fibonacci number consists of 6 distinct prime factors (i.e., 102334155 = 3 x 5 x 7 x 11 x 41 x 2161).
a(31) = 341 because F(341)= 89 * 557 * 2417 * 761227665342913 * 197907695243868721 * 4558282384863830955384586674337 has exactly 6 prime factors.

Amiram Eldar, Table of n, a(n) for n = 1..123
Blair Kelly, Fibonacci and Lucas Factorizations.

Cf. A114823-A114826, A114836-A114841.

Column k=6 of A303217.

n=1;while(n<330,if(omega(fibonacci(n))==6,print1(n,", "));n++)

More terms from Jonathan Vos Post, Mar 22 2006
Corrected by Ryan Propper, Apr 26 2006
a(55)-a(107) from Max Alekseyev, Aug 18 2013

A046319 Odd numbers divisible by exactly 6 primes (counted with multiplicity).

Original entry on oeis.org

729, 1215, 1701, 2025, 2673, 2835, 3159, 3375, 3969, 4131, 4455, 4617, 4725, 5265, 5589, 5625, 6237, 6615, 6885, 7047, 7371, 7425, 7533, 7695, 7875, 8775, 8991, 9261, 9315, 9375, 9639, 9801, 9963, 10395, 10449, 10773, 11025, 11421, 11475, 11583

Offset: 1

Patrick De Geest, Jun 15 1998

nonn

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A046306.

Select[Range[1,12000,2],PrimeOmega[#]==6&] (* Harvey P. Dale, Nov 26 2011 *)

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

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.

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

A386977 Products of 3 distinct semiprimes.

Original entry on oeis.org

216, 240, 336, 360, 504, 528, 540, 560, 600, 624, 756, 792, 810, 816, 840, 880, 900, 912, 936, 1000, 1040, 1104, 1134, 1176, 1188, 1224, 1232, 1260, 1320, 1350, 1360, 1368, 1392, 1400, 1404, 1456, 1488, 1500, 1520, 1560, 1656, 1764, 1776, 1782, 1836, 1840, 1848

Offset: 1

Ian Hahus, Aug 11 2025

nonn

			216 = 4 * 6 * 9.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A001222, A001358 (semiprimes), A007304 (sphenic numbers).

Subsequence of A046306.

q:= n-> (l-> l=[3$2] or nops(l)>2 and add(i, i=l)=6)(ifactors(n)[2][..., 2]):
select(q, [$1..2000])[];  # Alois P. Heinz, Aug 12 2025

Select[Range@ 2000, MemberQ[{{3,3}, {1,1,4}, {1,2,3}, {2,2,2}, {1,1,1,3}, {1,1,2,2}, {1,1,1,1,2}, {1,1,1,1,1,1}}, Sort[Last /@ FactorInteger@ #]] &] (* Giovanni Resta, Aug 12 2025 *)

A112444 Pseudoprimes (base-2) equal to the product of 6 primes not necessarily distinct.

Original entry on oeis.org

45593065, 68800501, 81722145, 110851741, 192112921, 226018585, 238018341, 250385401, 265729465, 272479285, 321197185, 329379505, 332633665, 353987481, 357339745, 358466361, 413631505, 415702105, 416392005, 417241045, 421121701, 464790781, 476177905, 483532105

Offset: 1

Shyam Sunder Gupta, Dec 11 2005

			a(1) = 45593065 = 5*7*17*19*37*109.

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

Intersection of A046306 and A001567.

s={}; Do[If[PrimeOmega[n] == 6 && PowerMod[2, n - 1, n] == 1, AppendTo[s, n]], {n, 1, 10^8}]; s (* Amiram Eldar, Nov 10 2019 *)

More terms from Amiram Eldar, Nov 10 2019

A112453 Strong pseudoprimes (base-2) equal to product of 6 primes not necessarily distinct.

Original entry on oeis.org

10761055201, 26244332101, 49430153305, 53125756201, 247173316831, 276228879031, 360187750105, 516311394481, 558417883465, 605526139765, 765771364801, 823307740165, 877094650081, 904455227845, 1113985668601, 1237898798461, 1324214598565, 1353668162185, 1370368908805

Offset: 1

Shyam Sunder Gupta, Dec 12 2005

nonn

			a(1) = 10761055201 = 13*29*41*61*101*113.

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

Intersection of A046306 and A001262.

More terms from Amiram Eldar, Nov 10 2019

cp's OEIS Frontend

A114839 Indices of Fibonacci numbers with 6 distinct prime factors.

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A046319 Odd numbers divisible by exactly 6 primes (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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A114634 Numbers k such that the k-th octagonal number is 6-almost prime.

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

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

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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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A386977 Products of 3 distinct semiprimes.

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