A339269 a(n) is the least number that is the product of n primes (not necessarily distinct) and is the sum of n consecutive primes, or 0 if there are none.
2, 0, 425, 36, 243, 756, 29889, 4704, 207765, 8448, 4465125, 108864, 13640319, 1022976, 146146275, 3010560, 6500054871, 4259840, 60767621145, 36864000, 454444233597, 167215104, 8664659236485, 796262400, 49362406764957, 7935623168, 2310430513712625, 4160749568, 4216955409197811, 28538044416
Offset: 1
Keywords
Examples
a(3)=425 because 425 = 5^2*7 is the product of three primes and 425 = 137+139+149 is the sum of three consecutive primes, and no smaller number has this property.
Links
- Robert Israel, Table of n, a(n) for n = 1..50
Programs
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Maple
sumofconsecprimes:= proc(x, n) local P, k, p, q, t; P:= nextprime(floor(x/n)); p:= P; q:= P; for k from 1 to n-1 do if k::even or q = 2 then p:= nextprime(p); P:= P, p; else q:= prevprime(q); P:= q, P; fi od; P:= [P]; t:= convert(P, `+`); if t = x then return P fi; if t > x then while t > x do if q = 2 then return false fi; q:= prevprime(q); t:= t + q - p; P:= [q, op(P[1..-2])]; p:= P[-1]; if t = x then return P fi; od else while t < x do p:= nextprime(p); t:= t + p - q; P:= [op(P[2..-1]), p]; q:= P[1]; if t = x then return P fi; od fi; false end proc: children:= proc(r) local L, x, p, q, t, R; x:= r[1]; L:= r[2]; t:= L[-1]; p:= t[1]; q:= nextprime(p); if t[2]=1 then t:= [q, 1]; else t:= [p, t[2]-1], [q, 1] fi; R:= [x*q/p, [op(L[1..-2]), t]]; if nops(L) >= 2 then p:= L[-2][1]; q:= L[-1][1]; if L[-2][2]=1 then t:= [q, L[-1][2]+1] else t:= [p, L[-2][2]-1], [q, L[-1][2]+1] fi; R:= R, [x*q/p, [op(L[1..-3]), t]] fi; [R] end proc: f:= proc(n) local Q, t, x, v; uses priqueue; initialize(Q); if n::even then insert([-2^n, [[2, n]]], Q) else insert([-3^n, [[3, n]]], Q) fi; do t:= extract(Q); x:= -t[1]; v:= sumofconsecprimes(x, n); if v <> false then return x fi; for t in children(t) do insert(t, Q) od; od end proc: f(1):= 2: f(2):= 0: map(f, [$1..34]);
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