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

Author

Bernard Schott, Feb 14 2019

Keywords

• nonn
• easy

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.

Examples

			22 = 5 + 17; 39 = 5 + 11 + 23; 68 = 5 + 11 + 23 + 29; 139 = 11 + 17 + 23 + 29 + 59.
		

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).

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).

Programs

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 *)

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

Formula

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