A324076 - OEIS

A324076

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

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

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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.

LINKS

Table of n, a(n) for n=1..54.

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

Adjacent sequences: A324073 A324074 A324075 * A324077 A324078 A324079

KEYWORD

nonn

AUTHOR

Bernard Schott, Feb 14 2019

STATUS

approved