A350372 -id:A350372 - cp's OEIS frontend

A350371 Numbers with exactly 4 semiprime divisors.

60, 84, 90, 120, 126, 132, 140, 150, 156, 168, 198, 204, 220, 228, 234, 240, 260, 264, 270, 276, 280, 294, 306, 308, 312, 315, 336, 340, 342, 348, 350, 364, 372, 378, 380, 408, 414, 440, 444, 456, 460, 476, 480, 490, 492, 495, 516, 520, 522, 525, 528, 532, 550, 552, 558

Offset: 1

Wesley Ivan Hurt, Dec 27 2021

nonn

Numbers with exactly k semiprime divisors: A346041 (k=1), A345381 (k=2), A345382 (k=3), this sequence (k=4), A350372 (k=5), A350373 (k=6), A350374 (k=7), A350375 (k=8).

q[n_] := DivisorSum[n, 1 &, PrimeOmega[#] == 2 &] == 4; Select[Range[600], q] (* Amiram Eldar, Dec 28 2021 *)
spd4Q[n_]:=Count[Divisors[n],?(PrimeOmega[#]==2&)]==4; Select[Range[600],spd4Q] (* _Harvey P. Dale, Apr 30 2023 *)

isok(k) = sumdiv(k, d, bigomega(d)==2) == 4; \\ Michel Marcus, Dec 28 2021

A350373 Numbers with exactly 6 semiprime divisors.

Original entry on oeis.org

210, 330, 390, 462, 510, 546, 570, 690, 714, 770, 798, 858, 870, 900, 910, 930, 966, 1110, 1122, 1155, 1190, 1218, 1230, 1254, 1290, 1302, 1326, 1330, 1365, 1410, 1430, 1482, 1518, 1554, 1590, 1610, 1722, 1764, 1770, 1785, 1794, 1800, 1806, 1830, 1870, 1914, 1938, 1974, 1995

Offset: 1

Wesley Ivan Hurt, Dec 27 2021

nonn

Numbers with exactly k semiprime divisors: A346041 (k=1), A345381 (k=2), A345382 (k=3), A350371 (k=4), A350372 (k=5), this sequence (k=6), A350374 (k=7), A350375 (k=8).

q[n_] := DivisorSum[n, 1 &, PrimeOmega[#] == 2 &] == 6; Select[Range[2000], q] (* Amiram Eldar, Dec 28 2021 *)

isok(k) = sumdiv(k, d, bigomega(d)==2) == 6; \\ Michel Marcus, Dec 28 2021

A350374 Numbers with exactly 7 semiprime divisors.

Original entry on oeis.org

420, 630, 660, 780, 840, 924, 990, 1020, 1050, 1092, 1140, 1170, 1320, 1380, 1386, 1428, 1470, 1530, 1540, 1560, 1596, 1638, 1650, 1680, 1710, 1716, 1740, 1820, 1848, 1860, 1890, 1932, 1950, 2040, 2070, 2142, 2184, 2220, 2244, 2280, 2380, 2394, 2436, 2460, 2508, 2550

Offset: 1

Wesley Ivan Hurt, Dec 27 2021

nonn

Numbers with exactly k semiprime divisors: A346041 (k=1), A345381 (k=2), A345382 (k=3), A350371 (k=4), A350372 (k=5), A350373 (k=6), this sequence (k=7), A350375 (k=8).

q[n_] := DivisorSum[n, 1 &, PrimeOmega[#] == 2 &] == 7; Select[Range[2500], q] (* Amiram Eldar, Dec 28 2021 *)

isok(k) = sumdiv(k, d, bigomega(d)==2) == 7; \\ Michel Marcus, Dec 28 2021

A350375 Numbers with exactly 8 semiprime divisors.

Original entry on oeis.org

1260, 1980, 2100, 2340, 2520, 2772, 2940, 3060, 3150, 3276, 3300, 3420, 3780, 3900, 3960, 4140, 4200, 4284, 4410, 4680, 4788, 4950, 5040, 5100, 5148, 5220, 5544, 5580, 5700, 5796, 5850, 5880, 5940, 6120, 6468, 6552, 6600, 6660, 6732, 6840, 6900, 7020, 7260, 7308

Offset: 1

Wesley Ivan Hurt, Dec 27 2021

nonn

Numbers with exactly k semiprime divisors: A346041 (k=1), A345381 (k=2), A345382 (k=3), A350371 (k=4), A350372 (k=5), A350373 (k=6), A350374 (k=7), this sequence (k=8).

q[n_] := DivisorSum[n, 1 &, PrimeOmega[#] == 2 &] == 8; Select[Range[7500], q] (* Amiram Eldar, Dec 28 2021 *)

isok(k) = sumdiv(k, d, bigomega(d)==2) == 8; \\ Michel Marcus, Dec 28 2021

A378984 Squares in A378769.

Original entry on oeis.org

32400, 63504, 90000, 129600, 156816, 202500, 219024, 254016, 291600, 345744, 360000, 374544, 467856, 490000, 518400, 571536, 627264, 685584, 777924, 810000, 876096, 960400, 1016064, 1089936, 1166400, 1210000, 1245456, 1382976, 1411344, 1440000, 1498176, 1587600

Offset: 1

Michael De Vlieger, Dec 15 2024

Let omega = A001221, bigomega = A001222, and rad = A007947.
Numbers k that have all types of divisor pairs (d, k/d), d | k, that are listed in both A378769 and A378900. These are listed below:
Type A*: (Nontrivial) unitary divisor pairs, i.e., d coprime to k/d. The rest of the types are in cototient.
Type B*: gcd(d, k/d) > 1, rad(d) !| k/d, rad(k/d) !| d. These exist for k in A375055.
Type C: d < k/d, d | k/d but rad(k/d) !| d. Implies rad(k/d) = rad(k) and omega(d) < omega(k/d). These exist for k in A126706.
Type D: Either rad(d) | k/d, rad(k/d) !| d or vice versa. These exist for k in A378767.
Type E*: d = k/d = sqrt(k).
Type F: rad(d) = rad(k/d) = rad(k), d < k/d, d | k/d. These exist for k in A320966.
Type G*: rad(d) = rad(k/d) = rad(k), neither d | k/d nor k/d | d. These exist for k in A376936.
Asterisks denote symmetric types.
Since numbers d and k/d are either coprime or not, and if not, the squarefree kernel of one either divides the other or not, and if so, d divides k/d or not, and if so, d = k/d or not, there are no other types.
Smallest odd term is a(45) = 2480625.
Square roots not A350372: sqrt(810000) = 900 is not in A350372.

			a(1) = 32400 = 2^4 * 3^4 * 5^2 has the following divisor pair types:
  Type A: 16 * 2025, Type B: 48 * 675, Type C: 2 * 16200, Type D: 8 * 4050
  Type E: 180 * 180, Type F: 30 * 1080, Type G: 120 * 270.
a(2) = 63504 = 2^4 * 3^4 * 7^2 has the following divisor pair types:
  Type A: 16 * 3969, Type B: 48 * 1323, Type C: 2 * 31752, Type D: 8 * 7938
  Type E: 252 * 252, Type F: 42 * 1512, Type G: 168 * 378.
a(3) = 90000 = 2^4 * 3^2 * 5^4 has the following divisor pair types:
  Type A: 9 * 10000, Type B: 18 * 5000, Type C: 2 * 45000, Type D: 8 * 11250
  Type E: 300 * 300, Type F: 30 * 3000, Type G: 120 * 750, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Diagram listing all divisor pairs for a(n), n = 1..8, showing type A in white, type B in light gray, type C in green or red, type D in blue or gold, type E in dark gray, type F in orange or purple, and type G in black.
Michael De Vlieger, Diagram listing divisor pairs (d, k/d) for k = a(n), n = 1..60, showing only those with the smallest d and using the same color scheme as above, for each type and its reversal if the type is nonsymmetric.

Cf. A000290, A001221, A001222, A007947, A126706, A320966, A350372, A375055, A376936, A378767, A378769, A378900.

Cf. A013661, A082020.

s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[#/b^3]}] &[2^21],  IntegerQ@ Sqrt[#] &];
t = Select[s, Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &];
Select[t, PrimeOmega[#] > PrimeNu[#] > 2 &]

This sequence is { k = s^2 : rad(k)^2 | k,
  bigomega(k) > omega(k) > 2, p^3 | k and q^3 | k for distinct primes p, q }.
Intersection of A378769 and A378900.
Intersection of A000290, A375055, and A376936.
Sum_{n>=1} = Pi^2/6 - (15/Pi^2) * (1 + Sum_{p prime} (1/(p^4-1))) - ((Sum_{p prime} (1/(p^2*(p^2-1))))^2 - Sum_{p prime} (1/(p^4*(p^2-1)^2)))/2 = 0.00015490158528995570146... . - Amiram Eldar, Dec 21 2024

A350416 Numbers with exactly 9 semiprime divisors.

Original entry on oeis.org

6300, 8820, 9900, 11700, 12600, 14700, 15300, 17100, 17640, 18900, 19404, 19800, 20700, 21780, 22050, 22932, 23400, 25200, 26100, 26460, 27900, 29400, 29700, 29988, 30420, 30492, 30600, 31500, 33300, 33516, 34200, 35100, 35280, 36300, 36900, 37800, 38700, 38808

Offset: 1

Wesley Ivan Hurt, Dec 29 2021

Numbers with exactly four distinct prime divisors (cf. A033993), one of which has multiplicity 1 and the others at least 2. - David A. Corneth, Jun 10 2022

			6300 is in the sequence as 4, 6, 9, 10, 14, 15, 21, 25, 35 are the exactly 9 of its semiprime divisors. - _David A. Corneth_, Jun 10 2022

David A. Corneth, Table of n, a(n) for n = 1..10000

Numbers with exactly k semiprime divisors: A346041 (k=1), A345381 (k=2), A345382 (k=3), A350371 (k=4), A350372 (k=5), A350373 (k=6), A350374 (k=7), A350375 (k=8), this sequence (k=9).

Cf. A033993, A220264.

q[n_] := DivisorSum[n, 1 &, PrimeOmega[#] == 2 &] == 9; Select[Range[40000], q] (* Amiram Eldar, Dec 30 2021 *)
spd9Q[n_]:=Count[Divisors[n],?(PrimeOmega[#]==2&)]==9; Select[Range[ 40000],spd9Q] (* _Harvey P. Dale, Jun 09 2022 *)

isok(k) = sumdiv(k, d, bigomega(d)==2) == 9; \\ Michel Marcus, Dec 30 2021

is(n)= if(n==1, return(0)); my(f = vecsort(factor(n)[,2])); #f == 4 && f[1] == 1 && f[2]>=2 \\ David A. Corneth, Jun 10 2022

A378430 a(n) = Sqrt(A378984(n)).

Original entry on oeis.org

180, 252, 300, 360, 396, 450, 468, 504, 540, 588, 600, 612, 684, 700, 720, 756, 792, 828, 882, 900, 936, 980, 1008, 1044, 1080, 1100, 1116, 1176, 1188, 1200, 1224, 1260, 1300, 1332, 1350, 1368, 1400, 1404, 1440, 1452, 1476, 1500, 1512, 1548, 1575, 1584, 1620, 1656

Offset: 1

Michael De Vlieger, Dec 23 2024

Distinct from A350372; a(20) = 900 is not in A350372.

Proper subset of A126706, intersects A286708.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A126706, A286708, A350372, A378984.

s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[#/b^3]}] &[2^21],  IntegerQ@ Sqrt[#] &];
t = Select[s, Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &];
Map[Sqrt, Select[t, PrimeOmega[#] > PrimeNu[#] > 2 &] ]

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A350371 Numbers with exactly 4 semiprime divisors.

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A350373 Numbers with exactly 6 semiprime divisors.

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A350374 Numbers with exactly 7 semiprime divisors.

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A350375 Numbers with exactly 8 semiprime divisors.

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A378984 Squares in A378769.

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A350416 Numbers with exactly 9 semiprime divisors.

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A378430 a(n) = Sqrt(A378984(n)).

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