%I #53 Sep 27 2024 08:34:01
%S 1,4,9,8,25,12,49,16,27,25,121,24,169,49,45,32,289,36,361,40,63,121,
%T 529,48,125,169,81,56,841,60,961,64,121,289,175,72,1369,361,169,80,
%U 1681,84,1849,121,135,529,2209,96,343,125,289,169,2809,108,275,112,361
%N a(1) = 1; for n > 1, a(n) = max(d*lpf(d) : d|n, d > 1), where lpf is the least prime factor function (A020639).
%C Function considered by Schinzel and Szekeres in connection with a sieve problem.
%H Altug Alkan, <a href="/A217579/b217579.txt">Table of n, a(n) for n = 1..10000</a>
%H Pierre Mazet and Eric Saias, <a href="https://doi.org/10.5802/afst.1652">Etude du graphe divisoriel 4</a>, Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Vol. 29, No. 4 (2020), pp. 971-975. See p. 973. <a href="https://arxiv.org/abs/1803.10073">arXiv preprint</a>, arXiv:1803.10073 [math.NT], 2018. See p. 3.
%H A. Schinzel and G. Szekeres, <a href="http://pub.acta.hu/acta/showCustomerArticle.action?id=6751&dataObjectType=article&returnAction=showCustomerVolume&sessionDataSetId=5661e069c630ba5b&style=">Sur un problème de M. Paul Erdős</a>, Acta Sci. Math. Szeged 20 (1959), 221-229.
%H Gérald Tenenbaum, <a href="https://doi.org/10.24033/asens.1502">Sur un problème de crible et ses applications</a>, Annales scientifiques de l'École Normale Supérieure, 4ème série, tome 19, n°1, (1986), pp .1-30.
%H Gérald Tenenbaum, <a href="https://doi.org/10.24033/asens.1710">Sur un problème de crible et ses applications. II. Corrigendum et étude du graphe divisoriel</a>, Annales scientifiques de l'École Normale Supérieure, Série 4 : Tome 28 (1995) no. 2 , pp. 115-127.
%H Andreas Weingartner, <a href="http://dx.doi.org/10.1016/j.jnt.2004.05.008">Integers with dense divisors</a>, Journal of Number Theory, Volume 108, Issue 1, September 2004, Pages 1-17.
%H Andreas Weingartner, <a href="https://arxiv.org/pdf/2310.13038">The Schinzel-Szekeres function</a>, arXiv:2310.13038 [math.NT], 2023.
%F a(n) = n * max_{i=1..k-1} d_{i+1}/d_{i}, where d_1 < d_2 < ... < d_k are the divisors of n (Tenenbaum, 1986, Lemma 2.2, p. 8). - _Amiram Eldar_, Sep 27 2024
%e The divisors of 6 greater than 1 are : 2, 3, 6. The maximum of (2*A020639(2), 3*A020639(3), 6*A020639(6)) is max (2*2, 3*3, 6*2) is 6*2=12, so a(6)=12.
%t a[1]=1; a[n_] := Table[d FactorInteger[d][[1, 1]], {d, Rest[Divisors[n]]}] // Max; Array[a, 60] (* _Jean-François Alcover_, Sep 26 2018 *)
%o (PARI) spf(n) = vecmin(factor(n)[,1]);
%o a(n) = if (n==1, 1, d = divisors(n); vecmax(vector(#d-1, k, d[k+1]*spf(d[k+1])))); \\ _Michel Marcus_, Mar 28 2018
%Y Cf. A020639.
%K nonn,look
%O 1,2
%A _Michel Marcus_, Oct 12 2012