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.

A365885 Starts of run of 3 consecutive integers that are terms of A365883.

Original entry on oeis.org

228123, 903123, 1121875, 2253123, 2928123, 3146875, 3821875, 4278123, 5846875, 6303123, 6978123, 7196875, 7871875, 9003123, 9221875, 9896875, 10353123, 11028123, 11246875, 12378123, 13053123, 13271875, 13946875, 14403123, 15971875, 16428123, 17103123, 17321875
Offset: 1

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Author

Amiram Eldar, Sep 22 2023

Keywords

Comments

Numbers of the form 4*k+2 are not terms of A365883. Therefore there are no runs of 4 or more consecutive integers.
Since the middle integer in each triple is not divisible by 8, all the terms of this sequence are of the form 8*k+3.
The numbers of terms not exceeding 10^k, for k = 6, 7, ..., are 2, 16, 158, 1585, 15853, 158540, ... . Apparently, the asymptotic density of this sequence exists and equals 1.585...*10^(-6).

Examples

			228123 = 3^3 * 7 * 17 * 71 is a term since its least prime factor, 3, is equal to its exponent, the least prime factor of 228123 = 2^2 * 13 * 41 * 107, 2, is equal to its exponent, and the least prime factor of 228125 = 5^5 * 73, 5, is also equal to its exponent.

Links

Crossrefs

Subsequence of A017101, A365883, A365884 and A365891.

Programs

  • Mathematica
    q[n_] := Equal @@ FactorInteger[n][[1]]; Select[8*Range[125000] + 3, AllTrue[# + {0, 1, 2}, q] &]
  • PARI
    is(n) = #Set(factor(n)[1,]) == 1;
lista(kmax) = forstep(k = 3, kmax, 8, if(is(k) && is(k+1) && is(k+2), print1(k, ", ")));

A365889 Numbers k whose least prime divisor divides its exponent in the prime factorization of k.

Original entry on oeis.org

4, 12, 16, 20, 27, 28, 36, 44, 48, 52, 60, 64, 68, 76, 80, 84, 92, 100, 108, 112, 116, 124, 132, 135, 140, 144, 148, 156, 164, 172, 176, 180, 188, 189, 192, 196, 204, 208, 212, 220, 228, 236, 240, 244, 252, 256, 260, 268, 272, 276, 284, 292, 297, 300, 304, 308
Offset: 1

Views

Author

Amiram Eldar, Sep 22 2023

Keywords

Comments

Numbers k such that A020639(k) | A051904(k).
The asymptotic density of terms with least prime factor prime(n) (within all the positive integers) is d(n) = ((prime(n)-1)/(prime(n)*(prime(n)^prime(n)-1))) * Product_{k=1..(n-1)} (1-1/prime(k)). For example, for n = 1, 2, 3, 4 and 5, d(n) = 1/6, 1/78, 1/11715, 4/14411985 and 8/10984499318485.
The asymptotic density of this sequence is Sum_{n>=1} d(n) = 0.17957281768342725732... .

Examples

			4 = 2^2 is a term since its least prime factor, 2, divides its exponent, 2.
16 = 2^4 is a term since its least prime factor, 2, divides its exponent, 4.

Links

Crossrefs

Cf. A020639, A051904.
Subsequence of A342090.
Subsequences: A365883, A365890, A365891.

Programs

  • Mathematica
    q[n_] := Divisible @@ Reverse[FactorInteger[n][[1]]]; Select[Range[2, 400], q]
  • PARI
    is(n) = {my(f = factor(n)); n > 1 && !(f[1, 2] % f[1, 1]);}

A365890 Numbers k such that k and k+1 are both terms of A365889.

Original entry on oeis.org

27, 188, 459, 620, 675, 783, 836, 944, 1107, 1268, 1323, 1484, 1647, 1755, 1808, 1916, 1971, 2132, 2240, 2403, 2564, 2619, 2780, 3051, 3124, 3212, 3267, 3375, 3428, 3536, 3644, 3699, 3860, 3915, 4076, 4239, 4347, 4400, 4508, 4563, 4671, 4724, 4995, 5103, 5156
Offset: 1

Views

Author

Amiram Eldar, Sep 22 2023

Keywords

Comments

The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 1, 8, 88, 862, 8607, 86044, 860407, 8604097, 86041005, 860410068, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00860410... .

Examples

			27 = 3^3 is a term since its least prime factor, 3, divides its exponent, 3, and the least prime factor of 28 = 2^2 * 7, 2, also divides its exponent, 2.
783 = 3^3 * 29 is a term since its least prime factor, 3, divides its exponent, 3, and the least prime factor of 784 = 2^4 * 7^2, 2, also divides its exponent, 4.

Links

Crossrefs

Subsequence of A365889.
Subsequences: A365884, A365891.

Programs

  • Mathematica
    q[n_] := Divisible @@ Reverse[FactorInteger[n][[1]]]; consec[kmax_] := Module[{m = 1, c = Table[False, {2}], s = {}}, Do[c = Join[Rest[c], {q[k]}]; If[And @@ c, AppendTo[s, k - 1]], {k, 1, kmax}]; s]; consec[6000]
  • PARI
    is(n) = {my(f = factor(n)); n > 1 && !(f[1, 2] % f[1, 1]);}
lista(kmax) = {my(q1 = 0, q2); for(k = 2, kmax, q2 = is(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}
Showing 1-3 of 3 results.

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

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