A339185 a(n) is the least prime p such that the sum of n consecutive primes starting with p has exactly n prime factors, counted with multiplicity, or 0 if no such p exists.
2, 0, 137, 5, 41, 109, 4253, 569, 23057, 821, 405863, 9013, 1049173, 73009, 9742969, 188017, 382355863, 236527, 3198295691, 1843111, 21640201361, 7600499, 376724314301, 33177461, 1974496270177, 305216017, 85571500507397, 148597987, 145412255489161, 951267841, 2609815945304401, 1140850357, 24575914221842531
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Keywords
Examples
a(3)=137 because the sum of 3 consecutive primes starting with 137 is 137+139+149=425=5^2*7 is the product of 3 primes counting multiplicity, and 137 is the least prime with this property.
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 v[1] 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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