login
A340001
Number of ways prime(n) is a sum of five distinct primes.
1
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 5, 6, 11, 14, 16, 25, 29, 39, 57, 68, 75, 88, 92, 109, 169, 198, 235, 240, 322, 331, 379, 437, 497, 565, 635, 634, 803, 798, 896, 888, 1091, 1328, 1477, 1444, 1616, 1753, 1730, 2080, 2262, 2452, 2627, 2588, 2790, 3043, 3004, 3535
OFFSET
1,16
COMMENTS
Conjecture: all primes >= 43 are the sum of five distinct primes.
The sequence of the prime numbers that are the sum of five distinct prime numbers begins with 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, ...
The primes in the sequence are 2, 5, 11, 29, 109, 331, 379, 1091, 1753, ...
The squares in the sequence are 0, 1, 16, 25, 169, 1444, ...
FORMULA
a(n) = A219199(A000040(n)).
a(n) = [x^prime(n)*y^5] Product_{i>=1} (1+x^prime(i)*y). - Alois P. Heinz, Dec 30 2020
EXAMPLE
a(14) = 1 because prime(14) = 43 = 3 + 5 + 7 + 11 + 17.
a(17) = 5 because prime(17) = 59 = 3 + 5 + 7 + 13 + 31 = 3 + 5 + 11 + 17 + 23 = 3 + 7 + 13 + 17 + 19 = 5 + 7 + 11 + 13 + 23 = 5 + 7 + 11 + 17 + 19.
MAPLE
b:= proc(n, i) option remember; series(`if`(n=0, 1,
`if`(i<1, 0, b(n, i-1)+(p-> `if`(p>n, 0,
x*b(n-p, i-1)))(ithprime(i)))), x, 6)
end:
a:= n-> coeff(b(ithprime(n), n), x, 5):
seq(a(n), n=1..100); # Alois P. Heinz, Dec 30 2020
MATHEMATICA
b[n_, i_] := b[n, i] = Series[If[n == 0, 1,
If[i < 1, 0, b[n, i - 1] + Function[p, If[p > n, 0,
x*b[n - p, i - 1]]][Prime[i]]]], {x, 0, 6}];
a[n_] := SeriesCoefficient[b[Prime[n], n], {x, 0, 5}];
Array[a, 100] (* Jean-François Alcover, Apr 26 2021, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[p, {5}], AllTrue[#, PrimeQ]&&Length[Union[#]]==5&]], {p, Prime[Range[70]]}] (* Harvey P. Dale, Jul 07 2024 *)
KEYWORD
nonn
AUTHOR
Michel Lagneau, Dec 26 2020
STATUS
approved