A217579 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).

1, 4, 9, 8, 25, 12, 49, 16, 27, 25, 121, 24, 169, 49, 45, 32, 289, 36, 361, 40, 63, 121, 529, 48, 125, 169, 81, 56, 841, 60, 961, 64, 121, 289, 175, 72, 1369, 361, 169, 80, 1681, 84, 1849, 121, 135, 529, 2209, 96, 343, 125, 289, 169, 2809, 108, 275, 112, 361

Offset

1,2

Comments

Function considered by Schinzel and Szekeres in connection with a sieve problem.

Links

Altug Alkan, Table of n, a(n) for n = 1..10000

Pierre Mazet and Eric Saias, Etude du graphe divisoriel 4, Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Vol. 29, No. 4 (2020), pp. 971-975. See p. 973. arXiv preprint, arXiv:1803.10073 [math.NT], 2018. See p. 3.

A. Schinzel and G. Szekeres, Sur un problème de M. Paul Erdős, Acta Sci. Math. Szeged 20 (1959), 221-229.

Gérald Tenenbaum, Sur un problème de crible et ses applications, Annales scientifiques de l'École Normale Supérieure, 4ème série, tome 19, n°1, (1986), pp .1-30.

Gérald Tenenbaum, Sur un problème de crible et ses applications. II. Corrigendum et étude du graphe divisoriel, Annales scientifiques de l'École Normale Supérieure, Série 4 : Tome 28 (1995) no. 2 , pp. 115-127.

Andreas Weingartner, Integers with dense divisors, Journal of Number Theory, Volume 108, Issue 1, September 2004, Pages 1-17.

Andreas Weingartner, The Schinzel-Szekeres function, arXiv:2310.13038 [math.NT], 2023.

Formula

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

Example

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.

Mathematica

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 *)

Prog

(PARI) spf(n) = vecmin(factor(n)[, 1]);
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

Crossrefs

Cf. A020639.

Sequence in context: A318279 A065642 A285109 * A118585 A067666 A355012

Adjacent sequences: A217576 A217577 A217578 * A217580 A217581 A217582

Keyword

nonn,look

Author

Michel Marcus, Oct 12 2012

Status

approved