A378164
Smaller of consecutive terms b < c of A076467 such that the quality q=log(rad(c))/log(rad((c-b)*b*c)) of the abc-triple c-b,b,c with gcd(c-b,b,c)=1 sets a new record.
Original entry on oeis.org
1, 81, 1296, 2187, 1419857
Offset: 1
Pairs b,c of consecutive
terms of A076467
A378165
c-b b c Quality = log(rad(c))/log(rad((c-b)*b*c))
7, 1, 8, 0.262649535...
44, 81, 125, 0.277532712...
35, 1296, 1331, 0.309605475...
10, 2187, 2197, 0.429917243...
23040, 1419857, 1442897, 0.431260235...
A378165 gives the corresponding values of c-b.
-
\\ Uses M. F. Hasler's A076467_vec from A076467
a378164_5(upto) = {my(W=A076467_vec(upto), qw=0); for(k=2, #W, my(d=W[k]-W[k-1]); if(gcd([d,W[k],W[k-1]])==1, my(C=factor(W[k])[,1], B=factor(W[k-1])[,1], A=factor(d)[,1], P=vecprod(setunion(setunion(Set(B),Set(C)),Set(A))), q=log(vecprod(C))/log(P)); if(q>qw, print([d,W[k-1],W[k],q]); qw=q)))};
a378164_5(10^16)
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.
Original entry on oeis.org
16, 64, 2744, 474552, 157529610000, 407165596771032, 1491025241529616, 173903694695292024, 661905356066769705912, 14918256451377811247508792, 19801061641727872277815512, 2718924063971620383558231552
Offset: 1
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.
A378167 gives the corresponding values of c-b.
-
\\ 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)
A378167
Differences between adjacent terms of A076467 that correspond to the locations described by A378166.
Original entry on oeis.org
8, 32, 343, 17576, 65610000, 11329982936, 26102469128, 315404039943, 152838610998696, 7327416190396311, 146668341275463896, 1097750613982270976
Offset: 1
Showing 1-3 of 3 results.
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