A303693 a(n) begins the first run of least n consecutive numbers whose sum of divisors has the same set of distinct prime divisors.

1, 5, 33, 3777, 20154, 13141793, 11022353993

Offset: 1

Author

Amiram Eldar, Apr 28 2018

Keywords

• nonn
• more

Comments

a(n) is the least k such that rad(sigma(k)) = rad(sigma(k+1)) = ... = rad(sigma(k+n-1)), where rad(n) is the squarefree kernel of n (A007947) and sigma(n) is the sum of the divisors of n (A000203).

Examples

			a(4) = 3777 since it is the least number such that
sigma(3777) =  5040 = 2^4 * 3^2 * 5 * 7,
sigma(3778) =  5670 = 2^1 * 3^4 * 5 * 7,
sigma(3779) =  3780 = 2^2 * 3^3 * 5 * 7,
sigma(3780) = 13440 = 2^7 * 3^1 * 5 * 7,
all having the same set of prime divisors: 2, 3, 5, 7.
		

Crossrefs

Cf. A000203, A007947.

Programs

Mathematica

rad[n_] := Times @@ (First@# & /@ FactorInteger@n); radsig[n_] := rad[ DivisorSigma[1, n] ]; Seq[n_, q_] := Map[rsig, Range[n, n + q - 1]];
findConsec[q_, nmin_, nmax_] := Module[{}, s = Seq[1, q]; n = q + 1;   Do[If[CountDistinct[s] == 1, Break[]]; s = Rest[AppendTo[s, radsig[n]]]; n++, {k, nmin, nmax}]; n - q]; seq = {1}; nmax = 10^10; Do[n1 = Last[ seq ]; s1 = findConsec[m, n1, nmax]; AppendTo[seq, s1], {m, 2, 6}]; seq

Extensions

a(7) from Giovanni Resta, May 04 2018