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%I #43 Jan 27 2017 13:20:19
%S 3050208,27150208,712250208,4198150208,9887150208,29407950208,
%T 186613550208,254756450208,412941550208,496967350208,553174550208,
%U 1710112750208,8023681250208,9908919150208,20053008750208,20931113950208,22635692110208,24734957450208,39291663950208
%N Refactorable numbers n such that 3*n + 1 is also a refactorable number.
%C Corresponding first four values of 3*n + 1 are 5^4 * 11^4, 5^4 * 19^4, 5^4 * 43^4, 5^4 * 67^4.
%C Primes p such that both (5*p)^4 and ((5*p)^4 - 1)/3 are refactorable numbers begin 11, 19, 43, 67, 83, 109, 173, 211, 227, 443, 467, 557, 563, 587, 659, 739, 787, 821, 829, 853, 1123, 1187, 1229, 1277, 1453, 1523, 1571, 1709, 1901, 1973, 2083, 2099, 2237, 2467, 2531, 2621, 2909, 3347, 3517, 3877, 3923, 4099, 4243, 4253, 4259, 4483, 4547, ...; for each, p == 3 or 5 (mod 8). - _Jon E. Schoenfield_, Jan 21 2017
%C From _Altug Alkan_, Jan 25 2017: (Start)
%C Although numbers of the form ((5*p)^4 - 1)/3 appear in the beginning of sequence, note that not all terms are of the form ((5*p)^4 - 1)/3, i.e., (6239^16-1)/3.
%C However we can show that all terms are of the form 8 * A001318(m).
%C Proof: If an odd number n is in this sequence, then n must be a square and 3*n + 1 = 3 * (2*k + 1)^2 + 1 = 12 * k * (k + 1) + 4 = 24 * A000217(k) + 4 is a refactorable number. 4 = 2^2 is the highest power of 2 that divides 24 * A000217(k) + 4 because 6 * A000217(k) + 1 is an odd number. Since 24 * A000217(k) + 4 is not divisible by 3, 3*n + 1 cannot be a refactorable number when n is an odd refactorable number.
%C Since we proved that n is an even number, 3 * n + 1 is odd and it must be a square. If 3 * n + 1 = (2 * t + 1)^2, then n = ((2 * t + 1)^2 - 1) / 3 = 4 * t * (t + 1) / 3 = 8 * A001318(m). (End)
%H S. Colton, <a href="https://cs.uwaterloo.ca/journals/JIS/colton/joisol.html">Refactorable Numbers - A Machine Invention</a>, J. Integer Sequences, Vol. 2, 1999.
%H Joshua Zelinsky, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL5/Zelinsky/zelinsky9.html">Tau Numbers: A Partial Proof of a Conjecture and Other Results </a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8
%e 3050208 is a term because d(3050208) = 144 divides 3050208 and 3050208*3 + 1 = 5^4 * 11^4 is divisible by d(55^4) = 25.
%o (PARI) isA033950(n) = n % numdiv(n) == 0;
%o is(n) = isA033950(n) && isA033950(3*n+1);
%Y Cf. A001318, A033950, A281294.
%K nonn
%O 1,1
%A _Altug Alkan_, Jan 21 2017