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A225522 Levy's conjecture and generalization. Irregular triangle read by rows: row n gives all odd numbers T(n,k) > 2^(n+1) + 1 such that T(n,k) = p + q*2^n is impossible, where p and q are prime numbers and k = 1..r.

%I A225522 #23 Jun 01 2024 12:03:42
%S A225522 0,77,25,49,41,47,57,81,113,167,203,233,257,473,73,79,85,89,91,97,121,
%T A225522 129,141,145,151,159,181,187,211,217,225,251,279,281,301,343,349,385,
%U A225522 511,601,751,913,1433,1681,1951,137,143,149,153,155,161,163,167,173,177
%N A225522 Levy's conjecture and generalization. Irregular triangle read by rows: row n gives all odd  numbers T(n,k) > 2^(n+1) + 1 such that T(n,k) = p + q*2^n is impossible, where p and q are prime numbers and k = 1..r.
%C A225522 Levy (1963) conjectured that all odd numbers > 5 can be expressed as the sum of a prime plus twice a prime. This conjecture is a stronger version of the weak Goldbach conjecture and has been verified up to n <= 10^9 (according to MathWorld, Corbit 1999). The conjecture was posed by Emile Lemoine in 1895, but in more recent years came to be attributed to Hyman Levy, who pondered it in 1963.
%C A225522 But, according to the sequences A219604 and A223174, it is possible to suggest that this conjecture is generalizable with other powers of 2: the conjecture 2n+1 = p+2q becomes rightfully the conjecture 2n+1 = p + q*2^n.
%C A225522 Conjecture: for each integer n > 1, all odd numbers > 2^(n+1) + 1 are of the form p + q*2^n, except for a finite set of integers {n_1, n_2,..., n_r}, where p and q are prime numbers.
%C A225522 It is impossible to find odd numbers m < 2^(n+1)+1 such that m = p + q*2^n where p and q are primes because with the smallest values p = 3 and q = 2 then m = 2^(n+1) + 3, hence m > 2^(n+1)+1. But there exists, for each value n > 1 a finite set of odd numbers that are not decomposable into the form p + q*2^n:
%C A225522 For n=1 (Levy's conjecture), all odd number > 2^2+1 = 5 are of the form p+2q, p and q primes;
%C A225522 For n=2 (see A219604 and A219252), except the number 77, all numbers > 2^3+1 = 9 are of the form p+4q, p and q primes;
%C A225522 For n=3 (see A223174 and A223175), except the numbers 25 and 49, all numbers > 2^4+1 = 17 are of the form p+8q, p and q primes.
%C A225522 ...............
%C A225522 Rows of the triangle:
%C A225522 n=1 => {0}
%C A225522 n=2 => {77},
%C A225522 n=3 => {25, 49},
%C A225522 n=4 => {41, 47, 57, 81, 113, 167, 203, 233, 257, 473},
%C A225522 n=5 => {73, 79, 85, 89, 91, 97, 121, 129, 141, 145, 151, 159, 181, 187, 211, 217, 225, 251, 279, 281, 301, 343, 349, 385, 511, 601, 751, 913, 1433, 1681, 1951}.
%D A225522 K. Guy, Unsolved Problems in Number Theory New York: Springer-Verlag 2004: C1.
%e A225522 Row 1 = {0} because there is no odd numbers m > 2^2+1 such that 2n+1 = p+2q is impossible.
%e A225522 Row 2 = {77} because, except 77, there is no other odd numbers m > 2^3+1 such that m = p+4q is impossible.
%e A225522 Row 3 = {25, 49} because, except 25 and 49, there is no other odd numbers m > 2^3+1 such that 2n+1 = p+8q is impossible.
%p A225522 for m from 1 to 6 do: lst:={}:for n from 1 by 2 to 10000 do:jj:=0:for j from 1 to 5000 while (jj=0) do:q:=ithprime(j):p:=n-(2^m)*q:if p> 0 and type(p, prime)=true  then jj:=1: else fi:od:if jj=0 and n > 2^(m+1)+1 then lst:=lst union {n}:else fi:od: print(lst):od:
%Y A225522 Cf. A046927, A219604, A219252, A223174, A223175.
%K A225522 nonn,tabf
%O A225522 1,2
%A A225522 _Michel Lagneau_, May 09 2013