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%I #24 Mar 11 2023 14:55:13
%S 5,11,16,17,22,23,28,29,33,34,39,40,41,45,46,47,51,52,53,56,57,58,59,
%T 62,63,64,68,69,70,71,74,75,76,80,81,82,83,85,86,87,88,89,92,93,94,97,
%U 98,99,100,101,103,104,105,106
%N Integers which are the sum of distinct primes of the form 6n-1.
%C A theorem due to Andrzej Makowski: every natural number greater than 161 is the sum of distinct primes of the form "6n-1". (See Sierpiński and David Wells.) All the numbers < 161 and which are the sum of numbers of the form "6n-1" are here in this sequence, complement of A048264.
%D A. Mąkowski, Partitions into unequal primes, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 8 (1960), 125-126.
%D Wacław Sierpiński, Elementary Theory of Numbers, p. 144, Warsaw, 1964.
%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, Revised edition, 1997, p. 127.
%e 22 = 5 + 17; 39 = 5 + 11 + 23; 68 = 5 + 11 + 23 + 29;
%e 139 = 11 + 17 + 23 + 29 + 59.
%t Select[Range@ 60, Count[IntegerPartitions[#], _?(And[UnsameQ @@ #, AllTrue[#, And[PrimeQ@ #, Mod[#, 6] == 5] &]] &)] > 0 &] (* _Michael De Vlieger_, Feb 15 2019 *)
%t With[{prs=Select[Prime[Range[30]],Mod[#,6]==5&]},Select[Union[Rest[ Total/@ Subsets[ prs]]],#<=Max[prs]&]] (* _Harvey P. Dale_, Mar 11 2023 *)
%Y Cf. A002145, A048262 (not the sum of distinct primes of the form 4n-1)
%Y Cf. A002144, A048263 (not the sum of distinct primes of the form 4n+1).
%Y Cf. A007528, A048264 (not the sum of distinct primes of the form 6n-1).
%Y Cf. A002476, A048265 (not the sum of distinct primes of the form 6n+1).
%K nonn
%O 1,1
%A _Bernard Schott_, Feb 14 2019
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