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 1-3 of 3 results.

A369954 Numbers k that are neither squarefree nor prime powers and also coprime to 6.

Original entry on oeis.org

175, 245, 275, 325, 425, 475, 539, 575, 605, 637, 725, 775, 833, 845, 847, 875, 925, 931, 1025, 1075, 1127, 1175, 1183, 1225, 1325, 1375, 1421, 1445, 1475, 1519, 1525, 1573, 1625, 1675, 1715, 1775, 1805, 1813, 1825, 1859, 1925, 1975, 2009, 2023, 2057, 2075, 2107
Offset: 1

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Author

Michael De Vlieger, Mar 24 2024

Keywords

Comments

Define quality Q to signify a number k neither squarefree nor prime power, i.e., k is in A126706. For example, 12 has quality Q but numbers k = 1..11 do not.
Numbers k in this sequence have quality Q and are such that either (k-1) or (k+1) also have quality Q. Hence k also appears in A369276, but not in A369516.
Numbers k such that k mod 12 = 1 or k mod 12 = 5 imply (k-1) in A126706, since 4 divides (k-1).
Numbers k such that k mod 12 = 7 or k mod 12 = 11 imply (k+1) in A126706, since 4 divides (k+1).
Proper subset of A367455.
By definition these odd numbers are such that A053669(k) = 2, therefore A053669(k) < A003557(k), hence this sequence is a proper subset of A360765.

Links

Crossrefs

Cf. A003557, A007310, A008586, A013929, A024619, A053669, A126706, A356322, A369276, A360765, A367455, A369516.

Programs

  • Mathematica
    Select[Flatten[Array[6 # + {1, 5} &, 360]], Nor[PrimePowerQ[#], SquareFreeQ[#]] &]
  • PARI
    isok(k) = !issquarefree(k) && !isprimepower(k) && (gcd(k, 6)==1); \\ Michel Marcus, Mar 25 2024

Formula

Intersection of A007310 and A126706.
Intersection of A007310, A013929, and A024619.

A367018 Composite squarefree k that are not divisible by 6.

Original entry on oeis.org

10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 70, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 105, 106, 110, 111, 115, 118, 119, 122, 123, 129, 130, 133, 134, 141, 142, 143, 145, 146, 154, 155, 158, 159, 161, 165, 166, 170, 177, 178, 182
Offset: 1

Views

Author

Michael De Vlieger, Jan 15 2024

Keywords

Comments

The asymptotic density of this sequence is 1/(2*Pi^2). - Amiram Eldar, Jan 20 2024

Examples

			70 is in this sequence since it is composite and squarefree but not divisible by 6. It does not appear in A006881 since it is the product of 3 primes.

Links

Crossrefs

Cf. A047253, A120944, A367455.

Programs

  • Mathematica
    Select[Range[180], And[SquareFreeQ[#], CompositeQ[#], Mod[#, 6] != 0] &]

Formula

Intersection of A047253 and A120944 = { k : Omega(k) > omega(k) = 1, k mod 6 != 0 }.

A369150 Numbers k neither squarefree nor prime powers such that A053669(k) < k/rad(k) < A119288(k) that are not odd numbers of the form lpf(k)*rad(k), where lpf(k) = A020639(k) and rad(k) = A007947(k).

Original entry on oeis.org

40, 56, 88, 104, 136, 152, 176, 184, 208, 232, 248, 272, 280, 296, 297, 304, 328, 344, 351, 368, 376, 424, 440, 459, 464, 472, 488, 496, 513, 520, 536, 544, 568, 584, 592, 608, 616, 621, 632, 656, 664, 680, 688, 712, 728, 736, 752, 760, 776, 783, 808, 824, 837
Offset: 1

Views

Author

Michael De Vlieger, Jan 20 2024

Keywords

Comments

Numbers k neither squarefree nor prime powers such that the smallest nondivisor prime q < k/rad(k) < p, the second smallest prime factor of k where k/rad(k) != lpf(k).
Even k implies A053669(k) = 3, odd k implies A053669(k) = 2.
Sequence does not contain k divisible by 6; sequence does not meet A055932.
Proper subset of A367455.

Examples

			a(1) = 40 = 2^3 * 5, since 3 < 4 < 5 and 4 != 2.
a(2) = 56 = 2^3 * 7, since 3 < 4 < 7 and 4 != 2.
a(7) = 176 = 2^4 * 11, since 3 < 8 < 11 and 8 != 2.
a(15) = 297 = 3^3 * 11, since 2 < 9 < 11 and 9 != 3.
a(248) = 3625 = 5^3 * 29, since 2 < 25 < 29 and 25 != 5, etc.

Links

Crossrefs

Cf. A007947, A020639, A053669, A119288, A126706, A364997, A366460, A366825, A367455.

Programs

  • Mathematica
    s = Select[Range[1000], Nor[PrimePowerQ[#], SquareFreeQ[#]] &];
Select[s,
  And[#3 < #1 < #2, #1 != #4] & @@
  {#1/(Times @@ #2), #2[[2]], #3, First[#2]} & @@
  {#, FactorInteger[#][[All, 1]],
    If[OddQ[#], 2, q = 3; While[Divisible[#, q], q = NextPrime[q]]; q]} &]

Formula

This sequence is { A364997 \ A366460 } = { A364997 \ A366825 }.
Showing 1-3 of 3 results.

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