A365883 -id:A365883 - cp's OEIS frontend

A365884 Numbers k such that k and k+1 are both terms of A365883.

27, 188, 459, 620, 675, 836, 1107, 1268, 1323, 1484, 1755, 1916, 1971, 2132, 2403, 2564, 2619, 2780, 3051, 3124, 3212, 3267, 3428, 3699, 3860, 3915, 4076, 4347, 4508, 4563, 4724, 4995, 5156, 5211, 5372, 5643, 5804, 5859, 6020, 6291, 6452, 6507, 6668, 6939, 7100

Offset: 1

Amiram Eldar, Sep 22 2023

The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 1, 6, 63, 623, 6216, 62157, 621565, 6215645, 62156450, 621564494, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00621564... .

			27 = 3^3 is a term since its least prime factor, 3, is equal to its exponent, and also the least prime factor of 28 = 2^2 * 7, 2, is equal to its exponent.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A365883 and A365890.

A365885 is a subsequence.

q[n_] := Equal @@ 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[7200]

lista(kmax) = {my(q1 = 0, q2); for(k = 2, kmax, q2 = #Set(factor(k)[1,]) == 1; if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

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

Amiram Eldar, Sep 22 2023

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).

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A017101, A365883, A365884 and A365891.

q[n_] := Equal @@ FactorInteger[n][[1]]; Select[8*Range[125000] + 3, AllTrue[# + {0, 1, 2}, q] &]

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

Amiram Eldar, Sep 22 2023

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... .

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

q[n_] := Divisible @@ Reverse[FactorInteger[n][[1]]]; Select[Range[2, 400], q]

is(n) = {my(f = factor(n)); n > 1 && !(f[1, 2] % f[1, 1]);}

cp's OEIS Frontend

A365884 Numbers k such that k and k+1 are both terms of A365883.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI