A376469 Starts of runs of 3 consecutive integers in which each member of the run has at least one divisor of the form p^e with p <= e, where p is a prime.

71874, 109375, 156248, 181250, 228123, 265624, 409374, 446875, 493748, 518750, 565623, 603124, 746874, 784375, 831248, 856250, 903123, 940624, 1084374, 1121875, 1168748, 1193750, 1240623, 1278124, 1421874, 1459375, 1506248, 1531250, 1578123, 1615624, 1759374, 1796875

Offset

1,1

Comments

The start of the least run of 4 (and also 5) consecutive integers with this property is 3988418748.

The numbers of terms that do not exceed 10^k, for k = 5, 6, ..., are 1, 18, 178, 1783, 17845, 178458, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00001784... .

Links

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

Example

71874 = 2 * 3^3 * 11^3 is a term since it is divisible by 3^3, 71875 = 5^5 * 23 is divisible by 5^5, and 71876 = 2^2 * 7 * 17 * 151 is divisible by 2^2.

Mathematica

q[n_] := q[n] = AnyTrue[FactorInteger[n], First[#] <= Last[#] &]; Select[Range[2*10^6], q[#] && q[#+1] && q[#+2] &]

Prog

(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 = 0, is3); for(k = 3, kmax, is3 = is(k); if(is1 && is2 && is3, print1(k-2, ", ")); is1 = is2; is2 = is3); }

Crossrefs

Subsequence of A100716, A070258 and A129254.

Sequence in context: A025310 A187312 A080477 * A234330 A333684 A187137

Adjacent sequences: A376466 A376467 A376468 * A376470 A376471 A376472

Keyword

nonn

Author

Amiram Eldar, Sep 23 2024

Status

approved