A046308 -id:A046308 - cp's OEIS frontend

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

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

A112445 Pseudoprimes (base 2) equal to the product of 7 primes not necessarily distinct.

Original entry on oeis.org

370851481, 1310486905, 1452092005, 1553860945, 2719940041, 3328293745, 3860623585, 5394826801, 5612626041, 5659475185, 6295936465, 8857509661, 9234602385, 10780911505, 10975165201, 11718888181, 12254138065, 12416553721, 12452890681, 13577445505, 14795826661

Offset: 1

Shyam Sunder Gupta, Dec 11 2005

			a(1) = 370851481 = 7*11*13*17*19*31*37.

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

Intersection of A046308 and A001567.

s={}; Do[If[PrimeOmega[n] == 7 && PowerMod[2, n - 1, n] == 1, AppendTo[s, n]], {n, 1, 2*10^9}]; s (* Amiram Eldar, Nov 10 2019 *)

More terms from Amiram Eldar, Nov 10 2019

A114417 Records in 7-almost prime gaps, ordered by merit.

Original entry on oeis.org

64, 96, 112, 168, 210, 280

Offset: 1

Jonathan Vos Post, Nov 25 2005

			Records defined in terms of A114407 and A046308:
n A114407(n) A114407(n)/log(A046308(n))
1 64 64/log 128 = 30.371914
2 96 96/log 192 = 42.0443868
3 32 32/log 288 = 13.0113433
4 112 112/log 320 = 44.7079021
5 16 16/log 432 = 6.07099172
6 32 32/log 448 = 12.0696509
7 168 168/log 480 = 62.6575474
8 24 24/log 648 = 8.53614076

Cf. A046308, A065516, A111870, A111871, A114403-A114411, A114412-A114422.

a(n) = Records in A114417(n)/log(A046308(n)) = Records in (A046308(n+1) - A046308(n))/log(A046308(n)).

a(5)-a(6) from Donovan Johnson, Feb 17 2010

A114446 Indices of 7-almost prime pentagonal numbers.

Original entry on oeis.org

27, 43, 96, 107, 128, 147, 180, 187, 203, 224, 288, 312, 336, 352, 360, 387, 392, 395, 400, 411, 416, 475, 480, 486, 491, 495, 523, 539, 544, 560, 572, 587, 592, 600, 603, 619, 621, 627, 635, 704, 729, 735, 752, 763, 779, 795, 800, 810, 819, 840, 843, 882

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) = 27 because P(27) = PentagonalNumber(27) = 27*(3*27-1)/2 = 1080 = 2^3 * 3^3 * 5 is a 7-almost prime.
a(2) = 43 because P(43) = 43*(3*43-1)/2 = 2752 = 2^6 * 43 is a 7-almost prime.
a(7) = 180 because P(180) = 180*(3*180-1)/2 = 48510 = 2 * 3^2 * 5 x 7^2 * 11 is a 7-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, A046308.

Select[Range[2000],PrimeOmega[# (3#-1)/2]==7&] (* Harvey P. Dale, Jul 16 2011 *)

{a(n)} = {k such that A001222(A000326(k)) = 7}.
{a(n)} = {k such that k*(3*k-1)/2 has exactly 7 prime factors}.
{a(n)} = {k such that A000326(k) is an element of A046308}.

More terms from Harvey P. Dale, Jul 16 2011

A114505 Numbers k such that the k-th hexagonal number is a 7-almost prime.

Original entry on oeis.org

48, 64, 68, 72, 80, 88, 96, 104, 108, 122, 140, 162, 168, 188, 203, 208, 216, 228, 230, 240, 243, 264, 272, 280, 308, 312, 324, 360, 378, 380, 396, 408, 410, 424, 428, 438, 440, 446, 450, 473, 486, 513, 518, 527, 544, 564, 567, 572, 578, 620, 638, 662, 666, 675, 689, 696

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) = 48 because HexagonalNumber(48) = H(48) = 48*(2*48-1) = 4560 = 2^4 * 3 * 5 * 19 is a 7-almost prime.
a(2) = 64 because H(64) = 64*(2*64-1) = 8128 = 2^6 * 127 is a 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, Hexagonal Number.

Cf. A000384, A001222, A046308.

Select[Range[800],PrimeOmega[#(2#-1)]==7&] (* Harvey P. Dale, Jul 20 2013 *)
Position[PrimeOmega[PolygonalNumber[6,Range[700]]],7]//Flatten (* Harvey P. Dale, Jan 10 2024 *)

Numbers k such that hexagonal number A000384(k) is an element of A046308.
Numbers k such that A001222(A000384(k)) = 7.
Numbers k such that A001222(k*(2*k-1)) = 7.

A114559 Numbers k such that the k-th heptagonal number is 7-almost prime.

Original entry on oeis.org

60, 63, 72, 114, 144, 159, 167, 180, 183, 207, 216, 225, 247, 255, 275, 297, 312, 315, 320, 330, 343, 352, 360, 378, 387, 391, 399, 405, 408, 411, 416, 420, 429, 440, 447, 448, 450, 459, 465, 468, 483, 486, 504, 513, 520, 525, 531, 546, 588, 591, 594, 609

Offset: 1

Jonathan Vos Post, Feb 15 2006

			a(1) = 60 because Hep(60) = 60*(5*60-3)/2 = 8910 = 2 * 3^4 * 5 * 11 is 7-almost prime.
a(2) = 63 because Hep(63) = 63*(5*63-3)/2 = 9828 = 2^2 * 3^3 * 7 * 13 is 7-almost prime.
a(3) = 72 because Hep(72) = 72*(5*72-3)/2 = 12852 = 2^2 * 3^3 * 7 * 17 is 7-almost prime.
a(4) = 114 because Hep(114) = 114*(5*114-3)/2 = 32319 = 3^5 * 7 * 19 is 7-almost prime.

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, Heptagonal Number.

Cf. A000040, A000566, A001222, A001358, A046308.

Select[Range[250],PrimeOmega[(#(5#-3))/2]==7&] (* Harvey P. Dale, May 02 2019 *)

Numbers k such that Hep(k) = k*(5*k-3)/2 is 7-almost prime.
Numbers k such that A000566(k) is a term of A046308.
Numbers k such that A001222(A000566(k)) = 7.
Numbers k such that A001222(k*(5*k-3)/2) = 7.

More terms from Harvey P. Dale, May 02 2019

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

cp's OEIS Frontend

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

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A112445 Pseudoprimes (base 2) equal to the product of 7 primes not necessarily distinct.

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A114417 Records in 7-almost prime gaps, ordered by merit.

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A114446 Indices of 7-almost prime pentagonal numbers.

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A114505 Numbers k such that the k-th hexagonal number is a 7-almost prime.

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A114559 Numbers k such that the k-th heptagonal number is 7-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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