A174926 -id:A174926 - cp's OEIS frontend

A174979 Smallest prime p which is a concatenation of n^3 and the cubic digits 0, 1, 8.

11, 181, 127, 641, 11251, 2161, 10343, 15121, 10729, 1000081, 81331, 117281, 12197, 1274401, 33751, 40961, 84913, 58321, 106859, 180001, 89261, 1064801, 812167, 138241, 8156251, 10175761, 196831, 2195201, 2438911, 270001, 297911

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 03 2010

There are three decimal digits which are cubes: 0 = 0^3, 1 = 1^3, 8 = 2^3. It is conjectured that sequence is infinite.

See comments in A174926.

			41^3 = 68921, and 1689211 is the smallest prime which can be produced by concatenating 68921 with some combination of the digits 0, 1, and 8.

J.-P. Allouche, J. Shallit: Automatic Sequences, Theory, Applications, Generalizations, Cambridge University Press, 2003
C. Dumitrescu and V. Seleacu: Some Notions and Questions in Number Theory, Glendale, Arizona, Erhus University Press, 1994
O. Oystein: Number Theory and its History, Dover Classics of Science and Mathematics, 1988

Cf. A000040, A000578, A174926.

Corrected and edited by D. S. McNeil, Nov 21 2010

A176185 Numbers n with property that concatenation (2*n+1)//n of the decimals is a square.

Original entry on oeis.org

29, 76, 2289, 3796, 6369, 8756, 16736, 19696, 24900, 28484, 77529, 83761, 94169, 222889, 887556, 22228889, 88875556, 112594641, 368762025, 651177616

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 11 2010

Sequence is infinite; two infinite "families" of such numbers n are:
(a) n = 8_(k)75_(k)6, 2 * n + 1 = 17_(k)51_(k)3, N = 2 * 6_(k+1)16_(k-1)7,
(b) n = 2_(k+1)8_(k)9, 2 * n + 1 = 4_(k)57_(k)9, N = 6_(k)76_(k)7, (k = 1, 2, ...)
List of (2*n+1)//n = N^2:
59//29 = 7^2 x 11^2, 153//76 = 2^4 x 31^2, 4579//2289 = 67^2 x 101^2,
7593//3796 = 2^2 x 4357^2, 12739//6369 = 11287^2, 17513//8756 = 2^2 x 13^2 x 509^2,
33473//16736 = 2^18 x 113^2, 39393//19696 = 2^4 x 13^2 x 17^2 x 71^2, 49801//24900,
56969//28484 = 2^2 x 13^2 x 2903^2, 155059//77529 = 7^2 x 17789^2, 167523//83761 = 347^2 x 373^2,
188339//94169 = 19^2 x 31^2 x 233^2, 445779//222889 = 7^2 x 11^2 x 13^2 x 23^2 x 29^2,
1775113//887556 = 2^2 x 666167^2, 44457779//22228889 = 59^2 x 73^2 x 113^2 x 137^2,
177751113//88875556 = 2^2 x 66661667 ^ 2, 225189283//112594641 = 23^2 x 83^2 x 331^2 x 751^2,
737524051//368762025 = 5^2 x 2161^2 x 79481^2, 1302355233//651177616 = 2^4 x 285301949^2

			n = 29 is a term: 2 * n + 1 = 59, 5929 = 59//29 = 77^2 is a perfect square.
n = 6369 is a term: 2 * n + 1 = 12739. 12739//6369 = 11287^2 is a perfect square.

J. Buchmann, U. Vollmer: Binary Quadratic Forms, Springer, Berlin, 2007
L. E. Dickson: History of the Theory of numbers, vol. 2: Diophantine Analysis, Dover Publications, 2005

Cf. A174370, A174454, A174926, A176130

isA176185 := proc(n)
    digcat2(2*n+1,n) ; # of oeis.org/transforms.txt
    issqr(%)  ;
end proc:
for n from 1 do
    if isA176185(n) then
        print(n) ;
    end if;
end do: # R. J. Mathar, May 21 2025

Select[Range[6512*10^5],IntegerQ[Sqrt[(2 #+1)10^IntegerLength[#]+#]]&] (* Harvey P. Dale, Mar 05 2022 *)

cp's OEIS Frontend

A174979 Smallest prime p which is a concatenation of n^3 and the cubic digits 0, 1, 8.

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