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A306196 Irregular triangle read by rows where row n lists the primes 2n - k, with 1 < k < 2n-1, and if k is composite also 2n - p has to be prime for some prime divisor p of k.

%I A306196 #28 Dec 04 2024 20:54:56
%S A306196 2,3,2,3,5,3,5,7,2,5,7,2,3,5,7,11,3,5,7,11,13,3,5,7,11,13,2,3,5,7,11,
%T A306196 13,17,2,3,5,7,11,13,17,19,2,3,5,7,11,13,17,19,2,3,5,7,11,13,17,19,23,
%U A306196 3,5,11,13,17,23,2,7,11,13,17,19,23,2,3,5,11,13,17,19,23,29
%N A306196 Irregular triangle read by rows where row n lists the primes 2n - k, with 1 < k < 2n-1, and if k is composite also 2n - p has to be prime for some prime divisor p of k.
%C A306196 Conjectures:
%C A306196 (i) 1 <= A035026(n) <= (n-th row length of this triangle) for n >= 2;
%C A306196 (ii) a(n,1) < A171637(n,1) for n >= 4.
%C A306196 Numbers m such that m-th row length of this triangle is equal to A000720(m): 1, 2, 11, 13, 25, 56, 60, ...
%e A306196 Row 2 = [2] because 2*2 = 2 + 2;
%e A306196 Row 3 = [3] because 2*3 = 3 + 3;
%e A306196 Row 4 = [2,3,5] because 2*4 - 2 = 6 = 2*3 and 2*4 = 3 + 5;
%e A306196 Row 5 = [3,5,7] because 2*5 = 3 + 7 = 5 + 5.
%e A306196 The table starts:
%e A306196   2;
%e A306196   3;
%e A306196   2,  3,  5;
%e A306196   3,  5,  7;
%e A306196   2,  5,  7;
%e A306196   2,  3,  5,  7, 11;
%e A306196   3,  5,  7, 11, 13;
%e A306196   3,  5,  7, 11, 13;
%e A306196   2,  3,  5,  7, 11, 13, 17;
%e A306196   2,  3,  5,  7, 11, 13, 17, 19;
%e A306196   2,  3,  5,  7, 11, 13, 17, 19;
%e A306196   2,  3,  5,  7, 11, 13, 17, 19, 23;
%e A306196   3,  5, 11, 13, 17, 23;
%e A306196   2,  7, 11, 13, 17, 19, 23;
%e A306196   2,  3,  5, 11, 13, 17, 19, 23, 29;
%o A306196 (PARI) isok(k,n) = {if (isprime(2*n-k), pf = factor(k)[,1]; for (j=1, #pf, if (isprime(2*n-pf[j]), return (1));););}
%o A306196 row(n) = {my(v = []); for (k=1, 2*n, if (isok(k,n), v = concat(v, 2*n-k))); vecsort(v);} \\ _Michel Marcus_, Mar 02 2019
%Y A306196 Supersequence of A171637.
%Y A306196 Cf. A000720, A020481, A035026, A306247, A306261.
%K A306196 nonn,easy,tabf
%O A306196 2,1
%A A306196 _Juri-Stepan Gerasimov_, Jan 28 2019