Problems & Puzzles: Puzzles

Puzzle 1021. p(k)+p(k+1)+1

Paolo Lava proposes the following nice puzzle:

Least prime p(k) such that p(k) + p(k+1) + 1 is squarefree and with n prime factors

1 5 -> 5 + 7 + 1 = 13.
2 2 -> 2 + 3 + 1 = 6 = 2*3.
3 191 -> 191 + 193 + 1 = 5*7*11.
4 1021 -> 1201 + 1213 + 1 = 2415 = 3*5*7*23.
5 20521 -> 20521 + 20533 +1 = 41055 = 3*5*7*17*23.
6 390001 -> 390001 + 390043 + 1 = 780045 = 3*5*7*17*19*23.

and

Least prime p(k) such that p(k) + p(k+1) + p(k+2) is squarefree and with n prime factors

1 5 -> 5 + 7 + 11 = 23.
2 2 -> 2 + 3 + 5 = 10 = 2*5.
3 331 -> 331 + 337 + 347 = 1015 = 5*7*29.
4 3049 -> 3049 + 3061 + 3067 = 9177 = 3*7*19*23.
5 35227 -> 35227 + 35251 + 35257 = 105735 = 3*5*7*19*53.
6 299903 -> 299903 + 299909 + 299933 = 899745 = 3*5*7*11*19*41.

Q. Find terms for n>6. Is there a demonstration that such numbers exist for any n?

 


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