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 1-6 of 6 results.

A348098 Number k such that k and k+1 both have an equal number of unitary and nonunitary prime divisors (A348097).

Original entry on oeis.org

44, 75, 98, 116, 135, 147, 152, 171, 175, 188, 207, 244, 296, 332, 351, 368, 375, 387, 404, 423, 424, 507, 548, 567, 603, 604, 639, 656, 711, 724, 775, 832, 844, 847, 872, 891, 908, 927, 931, 963, 1016, 1028, 1052, 1075, 1083, 1107, 1183, 1215, 1250, 1251, 1268
Offset: 1

Views

Author

Amiram Eldar, Sep 30 2021

Keywords

Examples

			44 is a term since 44 = 2^2 * 11 and 44 + 1 = 45 = 3^2 * 5 both have one unitary prime divisor and one nonunitary prime divisor.

Links

Crossrefs

Subsequence of A348097.
Cf. A335328.

Programs

  • Mathematica
    q[n_] := n == 1 || Count[(e = FactorInteger[n][[;; , 2]]), 1] == Length[e]/2; Select[Range[10^3], q[#] && q[# + 1] &]

A348099 Starts of runs of 3 consecutive numbers that have an equal number of unitary and nonunitary prime divisors (A348097).

Original entry on oeis.org

423, 603, 1250, 1375, 2007, 2523, 2527, 3175, 4075, 4203, 4374, 4923, 4948, 7442, 8991, 10375, 10467, 12591, 18027, 20402, 20575, 22023, 22687, 23823, 26071, 28375, 30231, 31507, 31850, 33271, 34623, 35574, 36162, 37348, 40023, 49975, 50274, 54475, 54511, 55323
Offset: 1

Views

Author

Amiram Eldar, Sep 30 2021

Keywords

Links

Crossrefs

Subsequence of A348097 and A348098.
Cf. A335397.

Programs

  • Mathematica
    q[n_] := n == 1 || Count[(e = FactorInteger[n][[;; , 2]]), 1] == Length[e]/2; v = q /@ Range[3]; seq = {}; Do[v = Append[Drop[v, 1], q[k]]; If[And @@ v, AppendTo[seq, k - 2]], {k, 4, 10^5}]; seq

Formula

423 is a term since 423 = 3^2 * 47, 423 + 1 = 424 = 2^3 * 53 and 423 + 2 = 425 = 5^2 * 17 all have one unitary prime divisor and one nonunitary prime divisor.

A348100 Starts of runs of 4 consecutive numbers that have an equal number of unitary and nonunitary prime divisors (A348097).

Original entry on oeis.org

56347, 906596, 906597, 1090250, 1141550, 1243275, 4972372, 5684822, 6288488, 6379748, 6486325, 6907974, 7480447, 8587249, 9129248, 11112173, 12133672, 12133673, 13852924, 14185448, 17519948, 19293208, 19293209, 19751750, 20738672, 21560848, 21721796, 21959350
Offset: 1

Views

Author

Amiram Eldar, Sep 30 2021

Keywords

Examples

			56347 is a term since 56347 = 29^2 * 67, 56347 + 1 = 56348 = 2^2 * 14087, 56347 + 2 = 56349 = 3^3 * 2087 and 56347 + 3 = 56350 = 2 * 5^2 * 7^2 * 23 all have the same number of unitary and nonunitary prime divisors.

Links

Crossrefs

Subsequence of A348097, A348098 and A348099.
Cf. A335398.

Programs

  • Mathematica
    q[n_] := n == 1 || Count[(e = FactorInteger[n][[;; , 2]]), 1] == Length[e]/2; v = q /@ Range[4]; seq = {}; Do[v = Append[Drop[v, 1], q[k]]; If[And @@ v, AppendTo[seq, k - 3]], {k, 5, 1.3*10^6}]; seq

A348101 Starts of runs of 5 consecutive numbers that have an equal number of unitary and nonunitary prime divisors (A348097).

Original entry on oeis.org

906596, 12133672, 19293208, 23542000, 25793449, 70289224, 77449300, 130397524, 316377124, 359762848, 371355172, 395284372, 415670200, 527032924, 600284788, 642788072, 730243348, 746696248, 754642996, 792007675, 1153139048, 1153702448, 1338997372, 1359156472
Offset: 1

Views

Author

Amiram Eldar, Sep 30 2021

Keywords

Comments

1744218747 is the least start of a run of 6 consecutive numbers. The next such run starts with 73840265847.

Examples

			906596 is a term since 906596 = 2^2 * 226649, 906596 + 1 = 906597 = 3^2 * 100733, 906596 + 2 = 906598 = 2 * 7^2 * 11 * 29^2, 906596 + 3 = 906599 = 71 * 113^2 and 906596 + 4 = 906600 = 2^3 * 3 * 5^2 * 1511 all have the same number of unitary and nonunitary prime divisors.

Links

Crossrefs

Subsequence of A348097, A348098, A348099 and A348100.
Cf. A335399.

Programs

  • Mathematica
    q[n_] := n == 1 || Count[(e = FactorInteger[n][[;; , 2]]), 1] == Length[e]/2; v = q /@ Range[5]; seq = {}; Do[v = Append[Drop[v, 1], q[k]]; If[And @@ v, AppendTo[seq, k - 4]], {k, 6, 3*10^7}]; seq

A356413 Numbers with an equal sum of the even and odd exponents in their prime factorizations.

Original entry on oeis.org

1, 60, 84, 90, 126, 132, 140, 150, 156, 198, 204, 220, 228, 234, 260, 276, 294, 306, 308, 315, 340, 342, 348, 350, 364, 372, 380, 414, 444, 460, 476, 490, 492, 495, 516, 522, 525, 532, 550, 558, 564, 572, 580, 585, 620, 636, 644, 650, 666, 693, 708, 726, 732, 735
Offset: 1

Views

Author

Amiram Eldar, Aug 06 2022

Keywords

Comments

Numbers k such that A350386(k) = A350387(k).
A085987 is a subsequence. Terms that are not in A085987 are 1, 2160, 3024, ...

Examples

			60 is a term since A350386(60) = A350387(60) = 2.

Links

Crossrefs

Cf. A350386, A350387.
Subsequence of A028260.
Subsequences: A085987, A179698, A190109, A190110.
Similar sequences: A048109, A187039, A348097.

Programs

  • Mathematica
    f[p_, e_] := (-1)^e*e; q[1] = True; q[n_] := Plus @@ f @@@ FactorInteger[n] == 0; Select[Range[1000], q]
  • PARI
    isok(n) = {my(f = factor(n)); sum(i = 1, #f~, (-1)^f[i,2]*f[i,2]) == 0};

A348121 Numbers having more nonunitary than unitary prime divisors.

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 180, 196, 200, 216, 225, 243, 252, 256, 288, 289, 300, 324, 343, 360, 361, 392, 396, 400, 432, 441, 450, 468, 484, 500, 504, 512, 529, 540, 576, 588, 600, 612, 625, 648, 675, 676, 684
Offset: 1

Views

Author

Amiram Eldar, Oct 01 2021

Keywords

Comments

First differs from A080366 at n = 20.
The first 19 terms are also the first 19 powerful numbers (A001694) above 1. a(20) = 180 is the least nonpowerful term.
Numbers k such that A056169(k) < A056170(k).

Examples

			4 = 2^2 is a term since it has 1 nonunitary prime divisor, 2, and no unitary prime divisors.
180 = 2^2 * 3^2 * 5 is a term since it has 2 nonunitary prime divisors, 2 and 3, and one unitary prime divisor, 5.

Links

Crossrefs

Cf. A056169, A056170, A348097.
Subsequence of A013929.
A001694 is a subsequence.

Programs

  • Mathematica
    q[n_] := 2*Count[(e = FactorInteger[n][[;; , 2]]), 1] < Length[e]; Select[Range[700], q]
Showing 1-6 of 6 results.

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

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