A112418 Primes which have a prime number of partitions into five distinct primes.

53, 59, 67, 83, 113, 151, 157, 211, 239, 601, 809, 821, 881, 971, 1237, 1297, 1427, 1669, 1759, 1973, 2069, 2129, 2243, 2333, 2659, 2677, 2719, 2789, 2803, 2999, 3329, 3613, 3623, 3769, 3797, 4001, 4451, 4759, 4817, 4951, 5273, 5711, 5867, 5879, 6053, 6079, 6131, 6317, 6343, 6653, 6917, 6967, 7211, 7559, 7607, 7867, 7919, 8171, 8669, 8753, 8863, 9013, 9059, 9239, 9293, 9311, 9511, 9629, 9769, 9811

Offset

1,1

Comments

The corresponding numbers of partitions are 2,5,11,29,109,331,379,1091...

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..102

Example

53 is there because there are 2 partitions of 53 (3+7+11+13+19, 5+7+11+13+17) and 2 is prime.

Maple

part5_prime:=proc(N) local s, n, cont, i, j, k, l, m, a; s:=1; for n from 2 to N do cont:=0; for i from 1 to n-5 do for j from i+1 to n-4 do for k from j+1 to n-3 do for l from k+1 to n-2 do for m from l+1 to n-1 do if ithprime(n)= ithprime(i)+ithprime(j)+ithprime(k)+ithprime(l)+ithprime(m) then cont:=cont+1; fi; od; od; od; od; od; if (isprime(cont)=true) then a[s]:=ithprime(n); s:=s+1; fi; od; seq(a[i], i=1..s-1); end: # Neil Sloane, Jan 24 2006
N := 500; # Finds all values up to the N-th prime
p := Array(1..N, ithprime):
S := proc(s, k, i) option remember; local j;
       if k=1 then return ifelse(s=p[i], 1, 0); end if;
       if s <= p[i] then return 0; end if;
       add(S(s-p[i], k-1, j), j=1..i-1);
end proc:
A := proc(s) local j, ans; j, ans := 1, 0;
   while p[j] < s do ans := ans + S(s, 5, j); j := j+1; end do;
   ans;
end proc:
select(X->isprime(A(X)), convert(p, list));
# Brendan McKay, Mar 01 2026

Prog

(PARI) has(n)=my(t, Q, R, S); forprime(p=n\5+1, n-26, Q=n-p; forprime(q=Q\4+1, min(p-1, Q-15), R=Q-q; forprime(r=R\3+1, min(q-1, R-8), S=R-r; forprime(s=S-r+1, (S-1)\2, isprime(S-s) && t++)))); isprime(t)
select(has, primes(100)) \\ Charles R Greathouse IV, Apr 22 2015
(PARI) list(lim)=my(v=vectorsmall(precprime(lim)), u=List(), Q, R, S); forprime(p=13, #v-26, Q=#v-p; forprime(q=11, min(p-1, Q-15), R=Q-q; forprime(r=7, min(q-1, R-8), S=R-r; forprime(s=5, min(S-2, r-1), forprime(t=3, min(S-s, s-1), v[p+q+r+s+t]++))))); forprime(p=2, lim, if(isprime(v[p]), listput(u, p))); Set(u) \\ Charles R Greathouse IV, Apr 22 2015

Crossrefs

Cf. A000009, A000041, A007963, A051034, A219199.

Sequence in context: A095509 A268913 A166385 * A059497 A059472 A165455

Adjacent sequences: A112415 A112416 A112417 * A112419 A112420 A112421

Keyword

nonn

Author

Giorgio Balzarotti and Paolo P. Lava, Dec 09 2005

Extensions

Edited by Don Reble, Jan 26 2006

a(31)-a(37) from Charles R Greathouse IV, Apr 22 2015

Status

approved