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A356732
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Let u defined by u(1) = p and for 1 < i, u(i) = u(i-1) + primorial(i), such that all u(i) are primes for 1 <= i <= k, and u(k+1) is not prime. Let m the length of the longest run of primes obtained when u is repeatedly applied to an n-digit p. Triangle read by rows: for 1 <= n, 1 <= k <= m, T(n,k) is the least n-digit prime p begining a run of only k primes when applied u, or -1 if no such prime p exists.
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0
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2, -1, 7, 5, 19, 13, 53, 11, 37, 23, -1, -1, -1, 61, 109, 107, 131, 257, 103, 101, 331, -1, 193, 1009, 1063, 1087, 1013, 1601, 1543, 1447, 9397, 1741, 10007, 10061, 10133, 10847, 11251, 10253, 17203, 10267, 47563, 100003, 100043, 100357, 101833, 101113, 109583, 115657, 101287, 106747, 895667, 306847
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OFFSET
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1,1
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COMMENTS
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Row 8, [10000019, 10000223, 10000651, 10000763, 10013687, 10061593, 10143821, 10207621, 11390641, 11021033, 22583537, -1, -1, -1, 65987791] has length 15.
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LINKS
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EXAMPLE
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Let u(1) = 7; 7 is a 1-digit prime. u(2) = u(1) + primorial(2) = 7 + 2*3 = 13 is prime, u(3) = u(2) + primorial(3) = 13 + 30 = 43 is prime and u(4) = u(3) + primorial(4) = 43 + 210 = 253 = 11 * 23 is not prime. No lesser 1-digit p satisfies this, hence T(1,3) = 7.
Let u(1) = 13, 13 is a 2-digit prime. u(2) = u(1) + primorial(2) = 13 + 2*3 = 19 is prime and u(3) = u(2) + primorial(3) = 19 + 30 = 49 = 7^2 is not prime. No lesser 2-digit p satisfies this, hence T(2,2) = 13.
Triangle begins:
[2, -1, 7, 5]
[19, 13, 53, 11, 37, 23, -1, -1, -1, 61]
[109, 107, 131, 257, 103, 101, 331, -1, 193]
[1009, 1063, 1087, 1013, 1601, 1543, 1447, 9397, 1741]
[10007, 10061, 10133, 10847, 11251, 10253, 17203, 10267, 47563]
...
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PROG
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(PARI)
primorielle(n)=factorback(primes(n))
card(p, c0=1)=my(c=c0, r=0); while(isprime(p), c++; r=primorielle(c); p+=r); c-c0
row(chiffres, c0=1)=my(w=[], y=0, m=0, nmin=10^(chiffres-1), nmax=10^chiffres-1, va=vector(30)); forprime(n=nmin, nmax, y=card(n, c0); if(y>m, m=y); if(y>0&&va[y]==0, va[y]=n; print(); print(y" "n))); w=if(m>0, Vec(va, m), []); for(i=1, #w, if(w[i]==0, w[i]=-1)); w
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CROSSREFS
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KEYWORD
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sign,tabf,base
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AUTHOR
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STATUS
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approved
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