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A324076 Integers which are the sum of distinct primes of the form 6*n - 1.

%I A324076 #42 Mar 02 2025 09:58:44
%S A324076 5,11,16,17,22,23,28,29,33,34,39,40,41,45,46,47,51,52,53,56,57,58,59,
%T A324076 62,63,64,68,69,70,71,74,75,76,80,81,82,83,85,86,87,88,89,92,93,94,97,
%U A324076 98,99,100,101,103,104,105,106
%N A324076 Integers which are the sum of distinct primes of the form 6*n - 1.
%C A324076 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 A324076 A. Mąkowski, Partitions into unequal primes, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 8 (1960), 125-126.
%D A324076 Wacław Sierpiński, Elementary Theory of Numbers, p. 144, Warsaw, 1964.
%D A324076 David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, Revised edition, 1997, p. 127.
%H A324076 Ray Chandler, <a href="/A324076/b324076.txt">Table of n, a(n) for n = 1..101</a>
%H A324076 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A324076 a(n) = n + 60 for n > 101. - _Stefano Spezia_, Mar 01 2025
%e A324076 22 = 5 + 17; 39 = 5 + 11 + 23; 68 = 5 + 11 + 23 + 29; 139 = 11 + 17 + 23 + 29 + 59.
%t A324076 Select[Range@ 60, Count[IntegerPartitions[#], _?(And[UnsameQ @@ #, AllTrue[#, And[PrimeQ@ #, Mod[#, 6] == 5] &]] &)] > 0 &] (* _Michael De Vlieger_, Feb 15 2019 *)
%t A324076 With[{prs=Select[Prime[Range[30]],Mod[#,6]==5&]},Select[Union[Rest[ Total/@ Subsets[ prs]]],#<=Max[prs]&]] (* _Harvey P. Dale_, Mar 11 2023 *)
%o A324076 (Python)
%o A324076 def A324076(n): return int('050b101116171c1d21222728292d2e2f33343538393a3b3e3f40444546474a4b4c5051525355565758595c5d5e61626364656768696a6b6d6e6f707173747576797a7b7c7e7f808182838485868788898a8b8c8d8e909192939495969798999a9c9d9e9fa0'[n-1<<1:n<<1],16) if n<102 else n+60 # _Chai Wah Wu_, Feb 26 2025
%o A324076 (Python)
%o A324076 from itertools import combinations
%o A324076 from sympy import primerange
%o A324076 def A324076(n):
%o A324076     if n>101: return n+60
%o A324076     plist = list(p for p in primerange(161) if p%6==5)
%o A324076     xlist = sorted(set(sum(d) for i in range(1,len(plist)+1) for d in combinations(plist,i) if sum(d) < 162))
%o A324076     return xlist[n-1] # _Chai Wah Wu_, Feb 28 2025
%Y A324076 Cf. A002145, A048262 (not the sum of distinct primes of the form 4n-1)
%Y A324076 Cf. A002144, A048263 (not the sum of distinct primes of the form 4n+1).
%Y A324076 Cf. A007528, A048264 (not the sum of distinct primes of the form 6n-1).
%Y A324076 Cf. A002476, A048265 (not the sum of distinct primes of the form 6n+1).
%K A324076 nonn,easy
%O A324076 1,1
%A A324076 _Bernard Schott_, Feb 14 2019