A129254 Numbers k such that both k and k+1 have at least one divisor of the form p^e with p<=e, p prime.

27, 80, 135, 188, 243, 296, 351, 404, 459, 512, 567, 620, 675, 728, 783, 836, 891, 944, 999, 1052, 1107, 1160, 1215, 1268, 1323, 1376, 1431, 1484, 1539, 1592, 1647, 1700, 1755, 1808, 1863, 1916, 1971, 2024, 2079, 2132, 2187, 2240, 2295, 2348, 2403, 2456

Offset: 1

Author

Reinhard Zumkeller, Apr 07 2007

Keywords

• nonn

Comments

From Amiram Eldar, Sep 23 2024: (Start)

This sequence is infinite: For example, if k is even then (2*k+1)*27 is a term, and if k is odd then (2*k+1)*27-1 is a term.

The numbers of terms that do not exceed 10^k, for k = 2, 3, ..., are 2, 19, 187, 1868, 18686, 186851, 1868507, 18685075, 186850742, ... . Apparently, the asymptotic density of this sequence exists and equals 0.01868507... . (End)

Examples

			135 = 5*3^3 and 135+1 = 136 = 17*2^3, therefore 135 is a term: a(3) = 135.
188 = 47*2^2 and 188+1 = 189 = 7*3^3, therefore 188 is a term: a(4) = 188.
		

Links

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

Crossrefs

Subsequence of A068781 and A100716.

Programs

Mathematica

SequencePosition[Table[If[AnyTrue[FactorInteger[n],#[[2]]>=#[[1]]&],1,0],{n,2500}],{1,1}][[All,1]] (* Harvey P. Dale, Sep 14 2019 *)

PARI

is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i,1] <= f[i,2], return(1))); 0;}
lista(kmax) = {my(is1 = 0, is2); for(k = 2, kmax, is2 = is(k); if(is1 && is2, print1(k-1, ", ")); is1 = is2);} \\ Amiram Eldar, Sep 23 2024

Formula

A129251(a(n)) > 0, A129251(a(n)+1) > 0.

If A100716(k) = a(n) then: A100716(k+1) = a(n) + 1.