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A324076
Integers which are the sum of distinct primes of the form 6n-1.
2
5, 11, 16, 17, 22, 23, 28, 29, 33, 34, 39, 40, 41, 45, 46, 47, 51, 52, 53, 56, 57, 58, 59, 62, 63, 64, 68, 69, 70, 71, 74, 75, 76, 80, 81, 82, 83, 85, 86, 87, 88, 89, 92, 93, 94, 97, 98, 99, 100, 101, 103, 104, 105, 106
OFFSET
1,1
COMMENTS
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.
REFERENCES
A. Mąkowski, Partitions into unequal primes, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 8 (1960), 125-126.
Wacław Sierpiński, Elementary Theory of Numbers, p. 144, Warsaw, 1964.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, Revised edition, 1997, p. 127.
EXAMPLE
22 = 5 + 17; 39 = 5 + 11 + 23; 68 = 5 + 11 + 23 + 29;
139 = 11 + 17 + 23 + 29 + 59.
MATHEMATICA
Select[Range@ 60, Count[IntegerPartitions[#], _?(And[UnsameQ @@ #, AllTrue[#, And[PrimeQ@ #, Mod[#, 6] == 5] &]] &)] > 0 &] (* Michael De Vlieger, Feb 15 2019 *)
With[{prs=Select[Prime[Range[30]], Mod[#, 6]==5&]}, Select[Union[Rest[ Total/@ Subsets[ prs]]], #<=Max[prs]&]] (* Harvey P. Dale, Mar 11 2023 *)
CROSSREFS
Cf. A002145, A048262 (not the sum of distinct primes of the form 4n-1)
Cf. A002144, A048263 (not the sum of distinct primes of the form 4n+1).
Cf. A007528, A048264 (not the sum of distinct primes of the form 6n-1).
Cf. A002476, A048265 (not the sum of distinct primes of the form 6n+1).
Sequence in context: A314080 A337947 A072557 * A314081 A314082 A314083
KEYWORD
nonn
AUTHOR
Bernard Schott, Feb 14 2019
STATUS
approved