A378166 Terms c = A076467(k) such that the distinct prime factors of b = A076467(k-1) and of c-b are subsets of the prime factors of c, i.e., rad(c)/rad((c-b)*b*c) = 1.

16, 64, 2744, 474552, 157529610000, 407165596771032, 1491025241529616, 173903694695292024, 661905356066769705912, 14918256451377811247508792, 19801061641727872277815512, 2718924063971620383558231552

Offset: 1

Author

Hugo Pfoertner, Nov 20 2024

Keywords

• nonn
• hard
• more

Comments

a(13) > 5*10^27.

Examples

			                      Pairs b,c of consecutive terms of A076467
          A378167
              c-b                      b               c = a(n)
                8,                     8,                    16,
               32,                    32,                    64,
              343,                  2401,                  2744,
            17576,                456976,                474552,
         65610000,          157464000000,          157529610000,
      11329982936,       407154266788096,       407165596771032,
      26102469128,      1490999139060488,      1491025241529616,
     315404039943,    173903379291252081,    173903694695292024,
  152838610998696, 661905203228158707216, 661905356066769705912.
		

Crossrefs

A378167 gives the corresponding values of c-b.

Cf. A007947 (rad), A076467, A378164, A378165.

Programs

PARI

\\ Uses M. F. Hasler's A076467_vec from A076467
rad(x) = vecprod(factor(x)[,1]);
a378166_7(upto) = {my(W=A076467_vec(upto)); for(k=2, #W, my(d=W[k]-W[k-1], q=rad(W[k])/rad(W[k]*W[k-1]*d)); if(q==1, print([d, W[k-1], W[k]])))};
\\ Alternative program not using rad, more efficient
a378166_7(upto) = {my(W=A076467_vec(upto)); for(k=2, #W, my(C=Set(factor(W[k])[,1]), d=W[k]-W[k-1]); if(#setminus(Set(factor(d)[,1]), C)>0, , if(#setminus(Set(factor(W[k-1])[,1]), C)==0, print([d, W[k-1], W[k]]))))};
a378166_7(10^18)