A051200 Except for initial term, primes of form "n 3's followed by 1".

3, 31, 331, 3331, 33331, 333331, 3333331, 33333331, 333333333333333331, 3333333333333333333333333333333333333331, 33333333333333333333333333333333333333333333333331

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

"A remarkable pattern that is entirely accidental and leads nowhere" - M. Gardner, referring to the first 8 terms.

a(2)*a(3)*a(4) = 34179391, a Zeisel number (A051015) with coefficients (10,21).

a(2)*a(3)*a(4)*a(5) = 1139233281421, a Zeisel number with coefficients (10,21).

a(2)*a(3)*..*a(6) = 379741768929343351, a Zeisel number with coefficients (10,21).

a(2)*a(3)*..*a(7) = 1265805010367017001532181, a Zeisel number with coefficients (10,21).

a(2)*a(3)*..*a(8) = 42193497392022209194699696424911, a Zeisel number with coefficients (10,21).

Besides first 3, primes of the form (10^n-7)/3, n>1. See A123568. - Vincenzo Librandi, Aug 06 2010

The integer lengths of the terms of the sequence are 1, 2, 3, 4, 5, 6, 7, 8, 18, 40, 50, 60, 78, 101, 151, 319, 382, etc. - Harvey P. Dale, Dec 01 2018

References

• Martin Gardner, The Last Recreations, Chapter 12: Strong Laws of Small Primes, Springer-Verlag, 1997, pp. 191-205, especially p. 194.
• W. Sierpiński, 200 Zadan z Elementarnej Teorii Liczb, Warsaw, 1964; Problem 88 [in English: 200 Problems from the Elementary Theory of Numbers]
• W. Sierpiński, 250 Problems in Elementary Number Theory. New York: American Elsevier, Warsaw, 1970, pp. 8, 56-57.
• F. Smarandache, Properties of numbers, University of Craiova, 1973

Links

• R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
• Eric Weisstein's World of Mathematics, 3.

Crossrefs

Cf. A055520, A089017, A089018, A093671, A056698, A105427, A105428, A033175, A123568.

Programs

Mathematica

Join[{3},Select[Rest[FromDigits/@Table[PadLeft[{1},n,3], {n,50}]], PrimeQ]]  (* Harvey P. Dale, Apr 20 2011 *)

Formula

Union of 3 and A123568.

Extensions

More terms from James Sellers

Cross reference added by Harvey P. Dale, May 21 2014