cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A096156 Numbers with ordered prime signature (2,1).

Original entry on oeis.org

12, 20, 28, 44, 45, 52, 63, 68, 76, 92, 99, 116, 117, 124, 148, 153, 164, 171, 172, 175, 188, 207, 212, 236, 244, 261, 268, 275, 279, 284, 292, 316, 325, 332, 333, 356, 369, 387, 388, 404, 412, 423, 425, 428, 436, 452, 475, 477, 508, 524, 531, 539, 548, 549
Offset: 1

Views

Author

Alford Arnold, Jul 24 2004

Keywords

Comments

Numbers of the form p^2 * q where p and q are primes with p < q.
Also terms of A054753 that are not in A095990.
There are pairs that differ by 1, which is not the case in A095990, beginning with 44 and 45, 116 and 117, 171 and 172, 332 and 333, etc.

Examples

			a(2) = 20 because 20 = 2*2*5 and 2 < 5.
Note that 18 = 2*3^2 is not in the sequence, even though it has prime signature (2,1), because its ordered prime signature is (1,2) (A095990). Prime signatures correspond to partitions of Omega(n), while ordered prime signatures correspond to compositions of Omega(n).

Links

Crossrefs

Cf. A095990.
Subsequence of A054753, A097320, A325241, A345381.

Programs

  • Mathematica
    Take[ Sort[ Flatten[ Table[ Prime[p]^2 Prime[q], {q, 2, 33}, {p, q - 1}]]], 54] (* Robert G. Wilson v, Jul 28 2004 *)
Select[Range[10^5],FactorInteger[#][[All,2]]=={2,1}&] (* Enrique Pérez Herrero, Jun 27 2012 *)
  • PARI
    list(lim)=my(v=List()); forprime(q=3, lim\4, forprime(p=2, min(sqrtint(lim\q), q-1), listput(v, p^2*q))); Set(v) \\ Charles R Greathouse IV, Feb 26 2014
  • Python
    from sympy import factorint
def ok(n): return list(factorint(n).values()) == [2, 1]
print([k for k in range(550) if ok(k)]) # Michael S. Branicky, Dec 20 2021

Extensions

Edited and extended by Robert G. Wilson v and Rick L. Shepherd, Jul 27 2004

A348097 Numbers having equally many unitary and nonunitary prime divisors.

Original entry on oeis.org

1, 12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 63, 68, 75, 76, 80, 88, 92, 96, 98, 99, 104, 112, 116, 117, 124, 135, 136, 147, 148, 152, 153, 160, 162, 164, 171, 172, 175, 176, 184, 188, 189, 192, 207, 208, 212, 224, 232, 236, 242, 244, 245, 248, 250, 261
Offset: 1

Views

Author

Amiram Eldar, Sep 30 2021

Keywords

Comments

A prime divisor of k is unitary if its exponent in the prime factorization of k is 1, and is nonunitary otherwise.
Numbers k such that A056169(k) = A056170(k) = A001221(k)/2.
A345381 is a subsequence. After a(1) = 1, a(238) = 1260 is the next term that is not in A345381.

Examples

			12 = 2^2 * 3 is a term since it has one unitary prime divisor (3) and one nonunitary prime divisor (2).

Links

Crossrefs

Cf. A001221, A056169, A056170, A345381.
Subsequence of A030231.
Similar sequences: A016825 (odd and even divisors), A048109, A187039.

Programs

  • Mathematica
    q[n_] := n == 1 || Count[(e = FactorInteger[n][[;; , 2]]), 1] == Length[e]/2; Select[Range[300], q]

A350372 Numbers with exactly 5 semiprime divisors.

Original entry on oeis.org

180, 252, 300, 360, 396, 450, 468, 504, 540, 588, 600, 612, 684, 700, 720, 756, 792, 828, 882, 936, 980, 1008, 1044, 1080, 1100, 1116, 1176, 1188, 1200, 1224, 1300, 1332, 1350, 1368, 1400, 1404, 1440, 1452, 1476, 1500, 1512, 1548, 1575, 1584, 1620, 1656, 1692, 1700
Offset: 1

Views

Author

Wesley Ivan Hurt, Dec 27 2021

Keywords

Crossrefs

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

Programs

  • Mathematica
    q[n_] := DivisorSum[n, 1 &, PrimeOmega[#] == 2 &] == 5; Select[Range[1700], q] (* Amiram Eldar, Dec 28 2021 *)
  • PARI
    isok(k) = sumdiv(k, d, bigomega(d)==2) == 5; \\ Michel Marcus, Dec 28 2021

A350371 Numbers with exactly 4 semiprime divisors.

Original entry on oeis.org

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

Views

Author

Wesley Ivan Hurt, Dec 27 2021

Keywords

Crossrefs

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).

Programs

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

Views

Author

Wesley Ivan Hurt, Dec 27 2021

Keywords

Crossrefs

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).

Programs

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

Views

Author

Wesley Ivan Hurt, Dec 27 2021

Keywords

Crossrefs

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).

Programs

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

Views

Author

Wesley Ivan Hurt, Dec 27 2021

Keywords

Crossrefs

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).

Programs

  • Mathematica
    q[n_] := DivisorSum[n, 1 &, PrimeOmega[#] == 2 &] == 8; Select[Range[7500], q] (* Amiram Eldar, Dec 28 2021 *)
  • PARI
    isok(k) = sumdiv(k, d, bigomega(d)==2) == 8; \\ Michel Marcus, Dec 28 2021

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

Views

Author

Wesley Ivan Hurt, Dec 29 2021

Keywords

Comments

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

Examples

			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

Links

Crossrefs

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.

Programs

Showing 1-8 of 8 results.

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