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

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

Ray Chandler, Table of n, a(n) for n = 1..101

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Formula

a(n) = n + 60 for n > 101. - Stefano Spezia, Mar 01 2025

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 *)

Prog

(Python)
def A324076(n): return int('050b101116171c1d21222728292d2e2f33343538393a3b3e3f40444546474a4b4c5051525355565758595c5d5e61626364656768696a6b6d6e6f707173747576797a7b7c7e7f808182838485868788898a8b8c8d8e909192939495969798999a9c9d9e9fa0'[n-1<<1:n<<1], 16) if n<102 else n+60 # Chai Wah Wu, Feb 26 2025
(Python)
from itertools import combinations
from sympy import primerange
def A324076(n):
    if n>101: return n+60
    plist = list(p for p in primerange(161) if p%6==5)
    xlist = sorted(set(sum(d) for i in range(1, len(plist)+1) for d in combinations(plist, i) if sum(d) < 162))
    return xlist[n-1] # Chai Wah Wu, Feb 28 2025

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 A379036 A314082

Adjacent sequences: A324073 A324074 A324075 * A324077 A324078 A324079

Keyword

nonn,easy,changed

Author

Bernard Schott, Feb 14 2019

Status

approved