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

%I A117069 #23 Sep 27 2023 00:54:48
%S A117069 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,
%T A117069 5,11,669,11,621,13,499,13,451,13,355,13,331,11,213,13,163,11,69,13,
%U A117069 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
%N A117069 Leading terms in rows obtained by repeatedly computing consecutive absolute differences, starting with the squares of prime numbers.
%C A117069 In the first million rows, only 70767 leading terms are composite.
%C A117069 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).
%C A117069 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.
%C A117069 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
%D A117069 R. K. Guy, Unsolved Problems in Number Theory, Section A10.
%D A117069 Joseph L. Pe, "On the Absolute Difference Table of Squares of Primes", Journal of Recreational Mathematics 33 (3) (2004/2005) 176-179.
%H A117069 Paolo Xausa, <a href="/A117069/b117069.txt">Table of n, a(n) for n = 1..10000</a>
%H A117069 Prime Puzzles, <a href="http://www.primepuzzles.net/conjectures/conj_041.htm">Pe's conjecture</a>
%H A117069 Joseph L. Pe, <a href="http://numeratus.net/enlightened/psd.html">On the Absolute Difference Table of Squares of Primes</a>
%H A117069 <a href="/index/Ge#Gilbreath">Index entries for sequences related to Gilbreath conjecture and transform</a>
%e A117069 Start with the sequence of squares of primes:
%e A117069 4, 9, 25, 49, 121, ....
%e A117069 Take the absolute values of differences between consecutive terms:
%e A117069 5, 16, 24, 72, ....
%e A117069 Repeat this operation successively:
%e A117069 11, 8, 48, ....
%e A117069 3, 40, ....
%e A117069 ....
%e A117069 a(n) consists of the leading terms of the rows of differences above.
%t A117069 A117069[nmax_]:=Module[{d=Prime[Range[nmax+1]]^2},Table[First[d=Abs[Differences[d]]],nmax]];A117069[200] (* _Paolo Xausa_, May 14 2023 *)
%Y A117069 Cf. A001248 (1st row), A069482 (2nd row).
%K A117069 nonn
%O A117069 1,1
%A A117069 _Joseph L. Pe_, Apr 17 2006