A303693 - OEIS

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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

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OFFSET

1,2

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).

LINKS

Table of n, a(n) for n=1..7.

EXAMPLE

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.

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

CROSSREFS

Cf. A000203, A007947.

Sequence in context: A145505 A276126 A193325 * A256373 A124936 A213063

Adjacent sequences: A303690 A303691 A303692 * A303694 A303695 A303696

KEYWORD

nonn,more

AUTHOR

Amiram Eldar, Apr 28 2018

EXTENSIONS

a(7) from Giovanni Resta, May 04 2018

STATUS

approved