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.

A369541 Numbers k neither squarefree nor prime powers that are products of primorials such that A119288(k) <= k/A007947(k) < A053669(k).

Original entry on oeis.org

24, 120, 180, 840, 1260, 1680, 9240, 13860, 18480, 27720, 120120, 180180, 240240, 360360, 480480, 2042040, 3063060, 4084080, 6126120, 8168160, 38798760, 58198140, 77597520, 116396280, 155195040, 892371480, 1338557220, 1784742960, 2677114440, 3569485920, 5354228880
Offset: 1

Views

Author

Michael De Vlieger, Jan 28 2024

Keywords

Comments

Proper subset of A369540, itself contained in A060735, which in turn is a subset of A055932.

Examples

			Seen as an irregular triangle T(n,k) of rows n where P(n) | T(n,k)
2:      24;
3:     120,     180;
4:     840,    1260,    1680;
5:    9240,   13860,   18480,   27720;
6:  120120,  180180,  240240,  360360,  480480;
7: 2042040, 3063060, 4084080, 6126120, 8168160;
   ...

Links

Crossrefs

Cf. A002110, A007947, A025487, A053669, A055932, A060735, A119288, A364998, A369540.

Programs

  • Mathematica
    P = 2; nn = 10;
 s = Select[Range[4, Prime[nn], 2],
   Or[IntegerQ@ Log2[#],
     And[Union@ Differences@ PrimePi[#1] == {1},
        AllTrue[Differences[#2], # <= 0 &]] & @@
        Transpose@ FactorInteger[#]] &];
 Table[P *= Prime[n];
   P*TakeWhile[s, # <= Prime[n + 1] &], {n, 2, nn}] // Flatten

Formula

{a(n)} = { m × P(n) : 3 <= m < q, n >= 2, m not in A025487 }.
Intersection of A364998 and A025487.

A369419 Numbers k that are neither squarefree nor prime powers such that A119288(k) <= k/A007947(k) < A053669(k) and A007947(k) is not a primorial.

Original entry on oeis.org

18, 90, 150, 630, 1050, 1470, 1890, 2100, 6930, 11550, 16170, 20790, 23100, 25410, 90090, 150150, 210210, 270270, 300300, 330330, 390390, 420420, 450450, 1531530, 2552550, 3573570, 4594590, 5105100, 5615610, 6636630, 7147140, 7657650, 8678670, 9189180, 29099070
Offset: 1

Views

Author

Michael De Vlieger, Mar 10 2024

Keywords

Examples

			Seen as an irregular triangle T(n,k) of rows n where T(n,k) = P(n)*k, and k < prime(n+1) is in A369361.
n\k    3       5       7       9      10      11
------------------------------------------------
2:    18;
3:    90,    150;
4:   630,   1050,   1470,   1890,   2100;
5:  6930,  11550,  16170,  20790,  23100,  25410;
    ...

Links

Crossrefs

Cf. A002110, A003557, A007947, A025487, A053669, A055932, A056808, A060735, A119288, A364998, A369361, A369540, A369541.

Programs

  • Mathematica
    P = 2; nn = 8;
s = Select[Range[3, Prime[nn+1]],
  Nor[IntegerQ@ Log2[#],
      And[EvenQ[#1], Union@ Differences@ PrimePi[#2[[All, 1]]] == {1},
          AllTrue[Differences@ #2[[All, -1]], # <= 0 &]]] & @@
    {#, FactorInteger[#]} &];
Table[P *= Prime[n]; P*TakeWhile[s, # < Prime[n + 1] &], {n, 2, nn}]

Formula

This sequence is { k = m*P(i) : 3 <= m < prime(i), i > 1, m in A369361 }.
Intersection of A364998 and A056808.

A380473 Numbers k neither squarefree nor prime power (i.e., in A126706) such that A119288(k) <= A003557(k) < A053669(k) < A006530(k).

Original entry on oeis.org

126, 168, 198, 234, 264, 306, 312, 342, 408, 414, 456, 522, 552, 558, 666, 696, 738, 744, 774, 846, 888, 954, 984, 990, 1032, 1062, 1098, 1128, 1170, 1206, 1272, 1278, 1314, 1320, 1386, 1416, 1422, 1464, 1494, 1530, 1560, 1602, 1608, 1638, 1650, 1704, 1710, 1746
Offset: 1

Views

Author

Michael De Vlieger, Jul 22 2025

Keywords

Comments

Let rad = A007947, p = A119288, q = A053669, g = A006530, and r = A003557.
Numbers k in A126706 such that p <= r < q < g.
Terms are products k of a number s in A033845 and a number t in A007310 with at least one prime power factor p^m | k such that m > 1.

Examples

			Table of n, a(n) for select n:
   n    a(n)                       r   q
  --------------------------------------
   1    126 = 2 * 3^2 * 7          3   5
   2    168 = 2^3 * 3 * 7          4   5
   3    198 = 2 * 3^2 * 11         3   5
   4    234 = 2 * 3^2 * 13         3   5
   5    264 = 2^3 * 3 * 11         4   5
   6    306 = 2 * 3^2 * 17         3   5
   7    312 = 2^3 * 3 * 13         4   5
  24    990 = 2 * 3^2 * 5 * 11     3   7
  29   1170 = 2 * 3^2 * 5 * 13     3   7
  45   1650 = 2 * 3 * 5^2 * 11     5   7
  57   1980 = 2^2 * 3^2 * 5 * 11   6   7
  68   2340 = 2^2 * 3^2 * 5 * 13   6   7

Links

Crossrefs

Cf. A003557, A007310, A007947, A033845, A053669, A055932, A080259, A119288, A126706, A364998, A369540.

Programs

  • Mathematica
    a053669[x_] := Block[{q = 2}, While[Divisible[x, q], q = NextPrime[q] ]; q];
s = Select[Range[2^12], Nor[SquareFreeQ[#], PrimePowerQ[#]] &];
Select[s, And[#3 < #4 < #2[[-1, 1]], #2[[2, 1]] <= #3] & @@
  {#1, #2, #1/Apply[Times, #2[[All, 1]]], a053669[#1]} & @@
  {#, FactorInteger[#]} &]

Formula

Intersection of A364998 and A080259 = A364998 \ A055932 = A364998 \ A369540.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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