A046310 -id:A046310 - cp's OEIS frontend

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

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

A268469 Integers k such that k and k+1 are products of 8 primes.

Original entry on oeis.org

50624, 78975, 156735, 176175, 194480, 245024, 257984, 309375, 390624, 439424, 540224, 543104, 620864, 631071, 693279, 705375, 809919, 854144, 916352, 998000, 1087424, 1143800, 1147040, 1159839, 1165184, 1188999, 1266111, 1274048, 1276479, 1347920, 1389375

Offset: 1

Zak Seidov, Feb 08 2016

nonn

Primes counted with multiplicity. - Harvey P. Dale, Jun 12 2025

			50624 = 2^6*7*113; 50625 = 3^4*5^4.

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

Intersection of A046310 and A045920.

SequencePosition[Table[If[PrimeOmega[n]==8,1,0],{n,139*10^4}],{1,1}][[;;,1]] (* Harvey P. Dale, Jun 12 2025 *)

is(n)=bigomega(n)==8 && bigomega(n+1)==8 \\ Charles R Greathouse IV, Feb 08 2016

A288507 Least number k such that both prime(k+1) -/+ prime(k) are products of n prime factors (counting multiplicity).

Original entry on oeis.org

24, 319, 738, 57360, 1077529, 116552943

Offset: 3

Zak Seidov, Jun 10 2017

Prime(k) + prime(k+1) cannot be semiprime, so the offset is 3.

For n=3 to 8, all terms k happen to satisfy prime(k+1) - prime(k) = 2^n. - Michel Marcus, Jul 24 2017

			n = 8: k = 116552943, p = prime(k) = 2394261637, q = prime(k+1) = 2394261893; both q-p = 2^8 and  p+q = 2*3^2*5*7^3*155119 are 8-almost primes (A046310).

Cf. A001043, A001223, A046310, A251600.

a(n) = my(k = 1, p = 2, q = nextprime(p+1)); while((bigomega(p+q)!= n) || (bigomega(p-q)!= n), k++; p = q; q = nextprime(p+1)); k; \\ Michel Marcus, Jul 24 2017

from sympy import factorint, nextprime
def A288507(n):
    k, p, q = 1, 2, 3
    while True:
        if sum(factorint(q-p).values()) == n and sum(factorint(q+p).values()) == n:
            return k
        k += 1
        p, q = q, nextprime(q) # Chai Wah Wu, Jul 23 2017

A288517 Least integer k such that A001358(k) + A001358(k+1) is the product of exactly n prime factors (counting multiplicity).

Original entry on oeis.org

3, 1, 28, 4, 19, 39, 48, 89, 120, 551, 447, 589, 3707, 10137, 21644, 28456, 22998, 44494, 86132, 166930, 703448, 628371, 1220814, 1608668, 11153853, 6091437, 56676014, 268389220, 146153797, 193010987, 916382785, 738246947, 4702317172, 2830095027, 12627951809

Offset: 1

Zak Seidov, Jun 10 2017

nonn

			n=1: k=3, A001358(3) + A001358(4) = 9 + 10 = 19 = A000040(8) (8th prime),
n=2: k=1, A001358(1)+A001358(2) = 4+6 = 10 = 2*5 = A001358(4) (4th semiprime),
n=11: k=447, A001358(447)+A001358(448) = 1535+1537 = 3072 = 2^10*3 = A069272(2) (2nd 11-almost prime).

Charles R Greathouse IV, Table of n, a(n) for n = 1..50

Cf. A000040, A001358, A014612, A014614, A046306, A046308, A046310, A046312, A046314, A069272.

a(21)-a(35) from Charles R Greathouse IV, Jun 10 2017

A321169 a(n) is the smallest prime p such that p + 2 is a product of n primes (counted with multiplicity).

Original entry on oeis.org

3, 2, 43, 79, 241, 727, 3643, 15307, 19681, 164023, 1673053, 885733, 2657203, 18600433, 23914843, 100442347, 358722673, 645700813, 4519905703, 18983603959, 48427561123, 31381059607, 261508830073, 1307544150373, 3295011258943, 24006510600883, 12709329141643, 53379182394907, 190639937124673, 2615579937350539

Offset: 1

Amiram Eldar and Zak Seidov, Jan 10 2019

nonn

a(n) ~ c * 3^n. - David A. Corneth, Jan 11 2019

			a(1) = 3 as 3 + 2 = 5 (prime),
a(2) = 2 as 2 + 2 = 4 = 2*2 (semiprime),
a(3) = 43 as 43 + 2 = 45 = 3*3*5  (3-almost prime),
a(4) = 79 as 79 + 2 = 81 = 3*3*3*3 (4-almost prime).

David A. Corneth, Table of n, a(n) for n = 1..801

Cf. A001222, A001358, A014612, A014613, A014614, A046306, A046308, A046310, A046312, A046314, A073919, A118883.

ptns[n_, 0] := If[n == 0, {{}}, {}]; ptns[n_, k_] := Module[{r}, If[n < k, Return[{}]]; ptns[n, k] = 1 + Union @@ Table[PadRight[#, k] & /@ ptns[n - k, r], {r, 0, k}]]; a[n_] := Module[{i, l, v}, v = Infinity; For[i = n, True, i++, l = (Times @@ Prime /@ # &) /@ ptns[i, n]; If[Min @@ l > v, Return[v]]; minp = Min @@ Select[l - 2, PrimeQ]; If[minp < v, v = minp]]] ; Array[a, 10] (* after Amarnath Murthy at A073919 *)

a(n) = forprime(p=2, , if (bigomega(p+2) == n, return (p))); \\ Michel Marcus, Jan 10 2019

a(n) = {my(p3 = 3^n, u, c); if(n <= 2, return(4 - n)); if(isprime(p3 - 2), return(p3 - 2)); forprime(p = 5, oo, if(isprime(p3 / 3 * p - 2), u = p3 / 3 * p - 2; break ) ); for(i = 2, n, if(p3 * (5/3)^i > u, return(u)); for(j = 1, oo, if(p3 * j \ 3^i > u, next(2)); if(bigomega(j) == i, if(isprime(p3 / 3^(i) * j - 2), u = p3 / 3^(i) * j - 2; next(2) ) ) ) ); return(u) } \\ David A. Corneth, Jan 11 2019

A369898 Numbers k such that k and k + 1 each have 9 prime divisors, counted with multiplicity.

Original entry on oeis.org

203391, 698624, 1245375, 1942784, 2176064, 2282175, 2536191, 2858624, 2953664, 3282687, 3560192, 3655935, 3914000, 4068224, 4135616, 4205600, 4244967, 4586624, 4695488, 4744575, 4991679, 5055615, 5450624, 5475519, 5519744, 6141824, 6246800, 6410096, 6655040, 6660224, 6753375, 6816879, 6862400

Offset: 1

Robert Israel, Feb 04 2024

nonn

Numbers k such that k and k + 1 are in A046312.

If a and b are coprime terms of A046310, one of them even, then Dickson's conjecture implies there are infinitely many terms k where k/a and (k+1)/b are primes.

			a(3) = 1245375 is a term because 1245375 = 3^5 * 5^3 * 41 and 1245376 = 2^6 * 11 * 29 * 61 each have 9 prime factors, counted with multiplicity.

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

Cf. A001222, A046310, A046312, A115186, A369897.

with(priqueue):
R:= NULL:  count:= 0:
initialize(Q); r:= 0:
insert([-2^9, [2$9]], Q);
while count < 40 do
  T:= extract(Q);
  if -T[1] = r + 1 then
    R:= R, r; count:= count+1;
  fi;
  r:= -T[1];
  p:= T[2][-1];
  q:= nextprime(p);
  for i from 9 to 1 by -1 while T[2][i] = p do
    insert([-r*(q/p)^(10-i), [op(T[2][1..i-1]), q$(10-i)]], Q);
  od
od:
R;

A028261 Numbers whose total number of prime factors (counting multiplicity) is squarefree.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Daniel Asimov (dan(AT)research.att.com)

nonn

The complement is 1, 16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90, 100.. and contains all entries of A014613, all of A046310, all of A046312 etc. - R. J. Mathar, Dec 15 2015

Cf. A001222, A005117.

isok(n) = issquarefree(bigomega(n)); \\ Michel Marcus, Aug 31 2013

A123118 Partial products of A101695.

Original entry on oeis.org

2, 12, 216, 8640, 933120, 209018880, 100329062400, 130026464870400, 349511137571635200, 1968446726803449446400, 22676506292775737622528000, 522466704985552994823045120000, 27820307107070725868337506549760000

Offset: 1

Jonathan Vos Post, Sep 28 2006

nonn

The number of prime factors (with multiplicity) of a(n) is T(n) = A000217(n) = n*(n+1)/2.

			a(1) = 2 = prime(1).
a(2) = 12 = 2 * 6 = prime(1) * semiprime(2) = 2^2 * 3.
a(3) = 216 = 2 * 6 * 18 = prime(1) * semiprime(2) * 3-almostprime(3) = 2^3 * 3^3.
a(4) = 8640 = 2 * 6 * 18 * 40 = prime(1) * semiprime(2) * 3-almostprime(3) * 4-almostprime(4) = 2^6 * 3^3 * 5.
a(15) = 893179304874387947794472921245209518407680000 = 2 * 6 * 18 * 40 * 108 * 224 * 480 * 1296 * 2688 * 5632 * 11520 * 23040 * 53248 * 124416 * 258048 = 2^88 * 3^23 * 5^4 * 7^3 * 11 * 13.

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(n) = Prod[i=1..n] i-th i-almost prime = Prod[i=1..n] A101695(i).

A321590 Smallest number m that is a product of exactly n primes and is such that m-1 and m+1 are products of exactly n-1 primes.

Original entry on oeis.org

4, 50, 189, 1863, 10449, 447849, 4449249, 5745249, 3606422049, 16554218751, 105265530369, 1957645712385

Offset: 2

Zak Seidov, Nov 13 2018

From Jon E. Schoenfield, Nov 15 2018: (Start)
If a(11) is odd, it is 16554218751.
If a(12) is odd, it is 105265530369.
If a(13) is odd, it is 1957645712385. (End)
a(11), a(12), and a(13) are indeed odd. - Giovanni Resta, Jan 04 2019
10^13 < a(14) <= 240455334218751, a(15) <= 2992278212890624. - Giovanni Resta, Jan 06 2019

			For n = 3, 50 = 2*5*5, and the numbers before and after 50 are 49 = 7*7 and 51 = 3*17.

Cf. A078840.

Sequences listing r-almost primes, that is, the n such that A001222(n) = r: 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[n_] := Module[{o={0,0,0}, k=1}, While[o!={n-1,n,n-1}, o=Rest[AppendTo[o,PrimeOmega[k]]]; k++]; k-2]; Array[a,7,2] (* Amiram Eldar, Nov 14 2018 *)

{for(n=2,10,for(k=2^n,10^12,if(n==bigomega(k) &&
n-1==bigomega(k-1) && n-1==bigomega(k+1),print1(k", ");break())))}

a(10) from Jon E. Schoenfield, Nov 14 2018

a(11)-a(13) from Giovanni Resta, Jan 04 2019

A334583 Numbers m such that m, m + 1 and m + 2 each have exactly eight prime factors, not necessarily distinct.

Original entry on oeis.org

40909374, 71410624, 87278750, 126237375, 152439488, 161590624, 166450624, 209140623, 227929624, 243409374, 267308990, 267639470, 290696768, 291513248, 292088510, 295644734, 307885374, 310314158, 319874750, 321890750, 331690624, 336958622, 343030624, 352749248, 354109374, 356269374, 366681248, 391390623, 401375168, 407590623

Offset: 1

Zak Seidov, May 06 2020

nonn

			40909374 = 2 * 3^4 * 11^2 * 2087, 40909375 = 5^5 * 13 * 19 * 53, and 40909376 = 2^6 * 179 * 3571.

Intersection of A045939 and A046310.

Cf. A001222, A113752, A113789.

list(lim)=my(v=List(), k, o); forfactored(n=40909374, lim\1+2, o=bigomega(n); if(o==8, if(k++>2, listput(v, n[1]-2)), k=0)); Vec(v) \\ Charles R Greathouse IV, May 07 2020

A001222(a(n)+i) = 8 for i in {0,1,2}.

cp's OEIS Frontend

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

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A268469 Integers k such that k and k+1 are products of 8 primes.

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A288507 Least number k such that both prime(k+1) -/+ prime(k) are products of n prime factors (counting multiplicity).

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A288517 Least integer k such that A001358(k) + A001358(k+1) is the product of exactly n prime factors (counting multiplicity).

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A321169 a(n) is the smallest prime p such that p + 2 is a product of n primes (counted with multiplicity).

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A369898 Numbers k such that k and k + 1 each have 9 prime divisors, counted with multiplicity.

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A028261 Numbers whose total number of prime factors (counting multiplicity) is squarefree.

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A123118 Partial products of A101695.

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