A046306 -id:A046306 - cp's OEIS frontend

A114447 Indices of 6-almost prime pentagonal numbers.

32, 48, 64, 72, 81, 91, 99, 108, 112, 117, 123, 135, 139, 144, 152, 155, 160, 162, 176, 195, 207, 208, 216, 219, 240, 252, 264, 272, 275, 279, 292, 297, 300, 323, 324, 327, 331, 342, 347, 351, 355, 375, 376, 378, 399, 405, 417, 425, 435, 444, 450, 451, 455, 464

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) = 32 because P(32) = PentagonalNumber(32) = 32*(3*32-1)/2 = 1520 = 2^4 * 5 * 19 is a 6-almost prime.
a(3) = 64 because P(64) = 64*(3*64-1)/2 = 6112 = 2^5 * 191 is a 6-almost prime.
a(15) = 144 because P(144) = 144*(3*144-1)/2 = 31032 = 2^3 * 3^2 * 431 is a 6-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, A046306.

Select[Range[500], PrimeOmega[PolygonalNumber[5, #]] == 6 &] (* Amiram Eldar, Oct 05 2024 *)

{a(n)} = {k such that A001222(A000326(k)) = 6}.
{a(n)} = {k such that k*(3*k-1)/2 has exactly 6 prime factors}.
{a(n)} = {k such that A000326(k) is an element of A046306}.

82 removed by R. J. Mathar, Dec 22 2010

More terms from Amiram Eldar, Oct 05 2024

A114504 Numbers k such that the k-th hexagonal number is a 6-almost prime.

Original entry on oeis.org

50, 56, 60, 63, 81, 95, 98, 112, 116, 120, 138, 150, 152, 158, 172, 180, 182, 189, 196, 198, 204, 212, 221, 238, 242, 252, 260, 266, 275, 276, 296, 300, 304, 306, 315, 328, 332, 333, 340, 344, 348, 350, 356, 363, 374, 375, 388, 390, 405, 413, 420, 423, 434, 452, 455, 456, 459, 462, 472

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) = 50 because HexagonalNumber(50) = H(50) = 50*(2*50-1) = 4950 = 2 * 3^2 * 5^2 * 11 is a 6-almost prime.
a(2) = 56 because H(56) = 56*(2*56-1) = 6216 = 2^3 * 3 * 7 * 37 is a 6-almost prime.
a(5) = 81 because H(81) = 81*(2*81-1) = 13041 = 3^4 * 7 * 23 is a 6-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, Hexagonal Number.

Cf. A000384, A001222, A046306.

Select[Range[500], PrimeOmega[PolygonalNumber[6, #]] == 6 &] (* Amiram Eldar, Oct 06 2024 *)

Numbers k such that hexagonal number A000384(k) is an element of A046306.
Numbers k such that A001222(A000384(k)) = 6.
Numbers k such that A001222(k*(2*k-1)) = 6.

199 replaced by 198 by R. J. Mathar, Dec 22 2010

A114558 Numbers k such that the k-th heptagonal number is 6-almost prime.

Original entry on oeis.org

15, 24, 27, 33, 48, 51, 55, 64, 71, 75, 81, 99, 105, 108, 111, 119, 120, 123, 126, 132, 141, 147, 150, 156, 160, 162, 171, 175, 177, 189, 198, 199, 204, 208, 215, 219, 222, 224, 249, 252, 258, 261, 263, 264, 267, 270, 272, 280, 285, 291, 294, 300, 304, 335

Offset: 1

Jonathan Vos Post, Feb 15 2006

			a(1) = 15 because Hep(15) = 15*(5*15-3)/2 = 540 = 2^2 * 3^3 * 5 is 6-almost prime.
a(2) = 24 because Hep(24) = 24*(5*24-3)/2 = 1404 = 2^2 * 3^3 * 13.
a(7) = 55 because Hep(55) = 55*(5*55-3)/2 = 7480 = 2^3 * 5 * 11 * 17 is 6-almost prime [also 7480 = Hep(55) = Hep(Hep(5)) is an iterated heptagonal number].
a(11) = 81 because Hep(81) = 81*(5*81-3)/2 = 16281 = 3^5 * 67 [also 16281 = Hep(81) = Hep(Hep(6)) is an iterated heptagonal number].
a(24) = 156 because Hep(156) = 156*(5*156-3)/2 = 60606 = 2 * 3^2 * 7 * 13 * 37 is 6-almost prime (and a palindrome).
a(30) = 189 because Hep(189) = 189*(5*189-3)/2 = 89019 = 3^4 * 7 * 157 is 6-almost prime [also 89019 = Hep(189) = Hep(Hep(9)) is an iterated heptagonal number].

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Heptagonal Number.

Cf. A000040, A000566, A001222, A001358, A046306.

Select[Range[400],Total[Transpose[FactorInteger[# (5#-3)/2]][[2]]]==6&] (* Harvey P. Dale, May 15 2011 *)

Numbers k such that Hep(k) = k*(5*k-3)/2 is 6-almost prime.
Numbers k such that A000566(k) is a term of A046306.
Numbers k such that A001222(A000566(k)) = 6.
Numbers k such that A001222(k*(5*k-3)/2) = 6.

More terms from Harvey P. Dale, May 15 2011

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

A213063 Balanced numbers (of order one): k-almost primes that are the average of three successive k-almost primes.

Original entry on oeis.org

5, 34, 53, 68, 86, 94, 102, 122, 142, 157, 171, 173, 185, 188, 194, 202, 204, 211, 214, 218, 245, 257, 258, 262, 263, 285, 289, 302, 314, 321, 338, 342, 358, 366, 371, 373, 394, 404, 407, 413, 415, 422, 429, 435, 446, 471, 489, 490, 493, 497, 507, 513, 517, 524, 535, 562

Offset: 1

Gerasimov Sergey, Jun 03 2012

nonn

Balanced numbers of order one: defined by the union of balanced primes A006562, balanced semiprimes A213025, balanced 3-almost primes (68, 102, 171, 188, 245, 258, 285, 338, 366, 404, 429, 435, 507, 524,..), balanced 4-almost primes (204, 342, 490, 513,..),.., balanced k-almost primes - all of order one.

Balanced numbers of order two are 79, 119, 148, 205, 218, 281, 299, 302, 339, 349, 410, 439, 493,.., defined by the union of balanced primes of order two of A082077, balanced semiprimes of order two (119, 205, 218, 299, 302, 339, 493,..), balanced 3-almost primes of order two (148, 410, 604, 609, 642..),.., balanced k-almost primes of order two.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001222, A006562, A014612, A014613, A014614, A046306, A046308, A213025.

list(lim)={
lim=lim\1+.5;
my(v=List(),L=log(lim)\log(2),left=vector(L),middle=vector(L),t);
for(n=3,2*lim,
t=bigomega(n);
if(t>L,next);
if(middle[t],
if(2*middle[t] == left[t] + n,
if(middle[t] < lim,
listput(v,middle[t])
,
if(vecmin(middle) > lim, return(vecsort(Vec(v))))
)
);
left[t]=middle[t];
middle[t]=n
,
if(left[t],middle[t]=n,left[t]=n)
)
)
}; \\ Charles R Greathouse IV, Jun 14 2012

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

A342246 Numbers k such that k-1, k and k+1 are all composite with four, five and six (not necessarily distinct) prime factors respectively.

Original entry on oeis.org

11151, 13455, 23375, 26271, 31311, 33776, 36125, 40375, 45495, 46375, 48411, 49049, 49167, 61335, 63125, 74151, 77895, 78111, 78351, 80271, 82575, 83511, 84591, 86031, 87375, 88749, 90207

Offset: 1

Sean Lestrange, Mar 07 2021

nonn

			For k=11151 we have 11150 = 2 * 5^2 * 223 which is composite with four prime factors, 11151 = 3^3 * 7 * 59 which is composite with five prime factors and 11152 = 2^4 * 17 * 41 which is composite with six prime factors.

Dumitru Damian, Table of n, a(n) for n = 1..10456

Subsequence of A342258.

Cf. A001222, A014613, A014614, A046306.

SequencePosition[PrimeOmega[Range[100000]],{4,5,6}][[;;,1]]+1 (* Harvey P. Dale, Jul 30 2024 *)

for(n=3,100000,if(bigomega(n-1)==4&&bigomega(n)==5&&bigomega(n+1)==6,print1(n,", "))) \\ Hugo Pfoertner, Mar 07 2021

# The following SageMath algorithm will generate all terms up to 100000
L=[]
for n in [1..100000]:
    sum1, sum2, sum3 = 0,0,0
    for f in list(factor(n)):
        sum1+=f[1]
    for f in list(factor(n+1)):
        sum2+=f[1]
    for f in list(factor(n+2)):
        sum3+=f[1]
    if sum1==4 and sum2==5:
        if sum3==6:
            L.append(n+1)
print(L)

A386972 Numbers that are the product of a semiprime and the square of another semiprime.

Original entry on oeis.org

96, 144, 160, 224, 240, 324, 336, 352, 360, 400, 416, 486, 504, 528, 540, 544, 560, 600, 608, 624, 736, 756, 784, 792, 810, 816, 880, 900, 912, 928, 936, 992, 1040, 1104, 1134, 1176, 1184, 1188, 1215, 1224, 1232, 1260, 1312, 1350, 1360, 1368, 1376, 1392, 1400, 1404, 1456, 1488, 1500

Offset: 1

Ian Hahus, Aug 11 2025

nonn

Numbers with prime signature [5, 1], [4, 2], [4, 1, 1], [3, 2, 1], [2, 2, 2] or [2, 2, 1, 1]. So, necessarily but not sufficiently, terms t have bigomega(t) = 6. - David A. Corneth, Aug 11 2025

			96 = 6 * 4^2;
144 = 9 * 4^2 or 4 * 6^2.

David A. Corneth, First 1000000 terms colorcoded by prime signature

Cf. A001358 (semiprimes), A046306, A054753, A386977.

M:= 2000: # for terms <= M
P:= select(isprime, [2,seq(i,i=3..M/8,2)]): nP:= nops(P):
S:= {}:
for i1 from 1 to nP do
  p1:= P[i1];
  if p1^2*4^2 > M then break fi;
  for i2 from i1 to nP do
    p2:= P[i2];
    if p1*p2*4^2 > M then break fi;
    for i3 from 1 to nP do
      p3:= P[i3];
      if p1*p2*p3^4 > M then break fi;
      for i4 from i3 to nP do
        p4:= P[i4];
        v:= p1*p2*(p3*p4)^2;
        if v > M then break fi;
        if p1*p2 = p3*p4 then next fi;
        S:= S union {v}
od od od od:
sort(convert(S,list)); # Robert Israel, Aug 11 2025

Select[Range@ 1500, MemberQ[{{1,5}, {2,4}, {1,1,4}, {1,2,3}, {2,2,2}, {1,1,2,2}}, Sort[ Last /@ FactorInteger[#]]] &] (* Giovanni Resta, Aug 12 2025 *)

is(n) = {my(f = factor(n), b = bigomega(f)); if(b != 6, return(0)); f = vecsort(f[,2]~); #setminus(Set([f]), Set([[1, 5], [2, 4], [1, 1, 4], [1, 2, 3], [2, 2, 2], [1, 1, 2, 2]])) == 0} \\ David A. Corneth, Aug 12 2025

A111730 6-almost primes with semiprime digits (digits 4, 6, 9 only).

Original entry on oeis.org

64, 96, 4644, 4944, 6664, 6966, 9464, 9996, 44464, 44944, 46496, 46644, 49644, 49696, 64449, 64496, 66444, 66696, 69444, 69496, 69966, 94496, 94644, 94696, 96496, 96944, 99666, 99944, 444496, 444664, 444696, 444996, 446664, 446944, 446964, 449469, 449694, 449964, 464496, 464646, 464664, 464994, 469464, 494494, 494944, 494949, 494964, 496464, 499446, 499944, 644464, 644944

Offset: 1

Jonathan Vos Post, Nov 18 2005

			64 = 2^6
96 = 2^5 * 3
4644 = 2^2 * 3^3 * 43
4944 = 2^4 * 3 * 103
6664 = 2^3 * 7^2 * 17
6966 = 2 * 3^4 * 43
9464 = 2^3 * 7 * 13^2
9996 = 2^2 * 3 * 7^2 * 17
44464 = 2^4 * 7 * 397
44944 = 2^4 * 53^2 = 212^2
46496 = 2^5 * 1453

Intersection of A046306 and A107665.

Cf. A001358, A107342, A111494, A111496, A111697.

Select[Range[645000],ContainsOnly[IntegerDigits[#],{4,6,9}]&&PrimeOmega[#]==6&] (* James C. McMahon, Jun 05 2024 *)

isok(k) = (bigomega(k) == 6) && (#setminus(Set(digits(k)), Set([4,6,9])) == 0); \\ Michel Marcus, Apr 13 2022

Missing a(1)=64 prepended and several terms corrected by Georg Fischer and Michel Marcus, Apr 13 2022

cp's OEIS Frontend

A114447 Indices of 6-almost prime pentagonal numbers.

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A114504 Numbers k such that the k-th hexagonal number is a 6-almost prime.

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A114558 Numbers k such that the k-th heptagonal number is 6-almost prime.

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

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A213063 Balanced numbers (of order one): k-almost primes that are the average of three successive k-almost primes.

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PARI

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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Mathematica

PARI

PARI

A342246 Numbers k such that k-1, k and k+1 are all composite with four, five and six (not necessarily distinct) prime factors respectively.