A117069 Leading terms in rows obtained by repeatedly computing consecutive absolute differences, starting with the squares of prime numbers.

5, 11, 3, 37, 21, 13, 5, 3, 13, 5, 3, 5, 11, 3, 5, 11, 5, 11, 5, 3, 5, 107, 91, 59, 43, 27, 11, 5, 11, 669, 11, 621, 13, 499, 13, 451, 13, 355, 13, 331, 11, 213, 13, 163, 11, 69, 13, 19, 13, 5, 11, 3, 5, 3, 5, 3, 5, 3, 5, 11, 5, 195, 19, 157, 19, 61, 19, 61, 19, 3, 5, 3, 13, 5, 3, 5, 11

Offset: 1

Author

Joseph L. Pe, Apr 17 2006

Keywords

• nonn

Comments

In the first million rows, only 70767 leading terms are composite.

It is conjectured that for any positive integer n, the number of prime leading elements in the first n rows is greater than the number of composite leading elements (Pe's conjecture).

Preliminary investigations have led me to make the following generalization of the Gilbreath's and Pe's conjectures: For a fixed positive integer n, let T(n) be the table of consecutive absolute differences of the n-th powers of primes. Then the number of k-almost prime leading elements, 0 < k < n, is greater than the number of leading elements that are not of this form. Recall that a number is called k-almost prime if the sum of the exponents in its prime factorization equals k. Thus a 0-almost prime equals 1, a 1-almost prime is a prime number and a 2-almost prime is a semiprime. If n = 1, we have a weak form of Gilbreath's conjecture and if n = 2, we have Pe's conjecture.

There is a more general conjecture due to Croft and others, mentioned in Guy's book, that the Gilbreath property will hold for any sequence of odd numbers (but with an initial term 2) that does not increase too fast. - N. J. A. Sloane, Apr 18 2006

Examples

			Start with the sequence of squares of primes:
4, 9, 25, 49, 121, ....
Take the absolute values of differences between consecutive terms:
5, 16, 24, 72, ....
Repeat this operation successively:
11, 8, 48, ....
3, 40, ....
....
a(n) consists of the leading terms of the rows of differences above.
		

References

• R. K. Guy, Unsolved Problems in Number Theory, Section A10.
• Joseph L. Pe, "On the Absolute Difference Table of Squares of Primes", Journal of Recreational Mathematics 33 (3) (2004/2005) 176-179.

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000
• Prime Puzzles, Pe's conjecture
• Joseph L. Pe, On the Absolute Difference Table of Squares of Primes
• Index entries for sequences related to Gilbreath conjecture and transform

Crossrefs

Cf. A001248 (1st row), A069482 (2nd row).

Programs

Mathematica

A117069[nmax_]:=Module[{d=Prime[Range[nmax+1]]^2},Table[First[d=Abs[Differences[d]]],nmax]];A117069[200] (* Paolo Xausa, May 14 2023 *)