A309245 Least number k > 0 which is not a divisor of n such that k^2 + n is a nonsquare powerful number (A102834).

682, 5, 37, 11, 1879706, 463, 11, 10, 2046, 341881, 31, 74, 70, 5519, 793, 22, 1952785824219551870, 57, 559, 338, 4580728614212333152148, 503259461, 45, 926, 190, 109, 36, 62, 436, 832836278711, 63, 88, 2451448196948930, 7037029, 36, 33

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.

Example

a(1) = 682 since 682^2 + 1 = 465125 = 5^3 * 61^2 is a nonsquare powerful number and is the smallest k > 0 such that k^2 + 1 is not a nonsquare powerful number.

Mathematica

powerfulQ[n_] := Min@FactorInteger[n][[All, 2]] > 1; powerfulNonsquare[n_] := !IntegerQ[Sqrt[n]] && powerfulQ[n]; a[n_] := Module[{k=1}, While[Divisible[n, k] || !powerfulNonsquare[k^2 + n], k++]; k]; Table[a[n], {n, 1, 16}]

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, A140746, A309246.

Sequence in context: A185765 A206015 A202908 * A118509 A043691 A043577

Adjacent sequences: A309242 A309243 A309244 * A309246 A309247 A309248

Keyword

nonn

Author

Amiram Eldar, Jul 18 2019

Status

approved