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A307749
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Lengths of the hypotenuse of primitive pythagorean triples if prime, whose shorter legs sum to the hypotenuse of prime length of another primitive pythagorean triple whose shorter legs sum to a prime number.
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0
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13, 53, 97, 137, 233, 313, 421, 461, 641, 821, 877, 929, 997, 1061, 1093, 1129, 1201, 1217, 1229, 1693, 1709, 1873, 2213, 2309, 3001, 3049, 3169, 3181, 3469, 3517, 3581, 3593, 3677, 3701, 3733, 3881, 3917, 4057, 4397, 4409, 4621, 4813, 5237, 5437, 5441, 5953, 6257, 6301, 6577, 6637, 6661, 6857, 7229, 7481, 7669
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OFFSET
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1,1
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COMMENTS
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Embedded in this sequence are subsets based on the definition, for example {97,137}, and {3049,3881,5441,7481}. These arise when terms are both the length of the hypotenuse of one primitive Pythagorean triple and the sum of the two shorter legs of another.
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LINKS
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EXAMPLE
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13 is a term because 13^2 = 12^2 + 5^2 and 12 + 5 = 17 and 17^2 = 15^2 + 8^3 and 15 + 8 = 23.
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PROG
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(PARI) is(n) = {if((n%4 != 1) || !isprime(n), return(0)); my(v=thue(T, n^2), q); for(i=1, #v, if(v[i][1]>0 && v[i][2]>=v[i][1] && (q=vecsum(v[i])) && isprime(q), return(q)); ); 0; }
isok(p) = isprime(p) && (q=is(p)) && is(q);
lista(nn) = T=thueinit('x^2+1, 1); forprime(p=2, nn, if (isok(p), print1(p, ", "))); \\ Michel Marcus, May 01 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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