A309246 Least number k which is not a divisor of n such that k^2 - n is a nonsquare powerful number.

3, 11427, 15503069909027, 6, 73, 62531004125, 85227106679780, 20, 15, 71457130044805582612325294634331, 56, 47, 16, 33017, 1138, 68, 23, 19, 762488, 146, 1552808, 47, 6234, 32, 45, 2537, 51700, 54, 426, 83, 34, 40, 3601, 948281, 531783519104, 42, 73, 16493

Offset

1,1

Comments

De Koninck et al. calculated the first 50 terms of this sequence.

McDaniel and Vanden Eynden proved that a(n) exists for all n > 0. - Lingling Tong, Jun 04 2026

Links

Amiram Eldar, Table of n, a(n) for n = 1..50

Charles Vanden Eynden, Differences between squares and powerful numbers, The Fibonacci Quarterly, Vol. 24, No. 4 (1986), pp. 347-348.

Jean-Marie De Koninck, Nicolas Doyon, Florian Luca, and Michoacán Morelia, Powerful values of quadratic polynomials, Journal of Integer Sequences, Vol. 14, No. 3 (2011), Article 11.3.3.

Wayne L. McDaniel, Representations of every integer as the difference of powerful numbers, The Fibonacci Quarterly, Vol. 20, No. 1 (1982), pp. 85-87.

Formula

a(2) = 11427 since 11427^2 - 2 = 130576327 = 7^3 * 617^2 is a nonsquare powerful number, and k^2 - 2 is not a nonsquare powerful number for all k < 11427.

Prog

(PARI) is_a102834(n) = ispowerful(n) && !issquare(n) \\ after Charles R Greathouse IV in A102834
a(n) = for(k=1, oo, if(n%k!=0 && is_a102834(k^2-n), return(k))) \\ Felix Fröhlich, Jul 19 2019

Crossrefs

Cf. A001694, A102834, A309245.

Sequence in context: A171363 A280655 A262651 * A055379 A047777 A195834

Adjacent sequences: A309243 A309244 A309245 * A309247 A309248 A309249

Keyword

nonn,changed

Author

Amiram Eldar, Jul 18 2019

Status

approved