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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 (list; graph; refs; listen; history; text; internal format)
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, ...
LINKS
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 *)
CROSSREFS
Sequence in context: A233865 A090552 A024520 * A341446 A015613 A135013
KEYWORD
nonn
AUTHOR
Michel Lagneau, Dec 26 2020
STATUS
approved

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)