A353345
Numbers k such that the elements of the continued fractions of the harmonic means of the divisors of k and k+1 are anagrams of each other.
Original entry on oeis.org
688126, 29900656, 35217656, 71624168, 154979487, 527560886, 871173148, 1370592266, 2461226804, 3232529461, 3232684430, 3431178214, 3471121856, 3486231973, 3527029430, 5732671200, 6258062402, 8784477355, 9334188311, 12670993089, 12707869077, 15120804392, 16317131894
Offset: 1
688126 is a term since sequences of elements of the continued fractions of the harmonic means of the divisors of 688126 and 688127, 688126/70281 and 688127/77304, are {9, 1, 3, 1, 3, 1, 2, 9, 1, 1, 6, 8} and {8, 1, 9, 6, 3, 1, 2, 1, 3, 1, 1, 9} respectively, and they are anagrams of each other.
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h[n_] := Sort[ContinuedFraction[DivisorSigma[0, n]/DivisorSigma[-1, n]]]; seq[max_] := Module[{s = {}, n = 2, c = 0, h1 = h[1], h2}, While[n < max, h2 = h[n]; If[h1 == h2, AppendTo[s, n - 1]]; h1 = h2; n++]; s]; seq[4*10^7]
A349500
a(n) is the least number k such that A349474(k) = A349474(k+1) = n, or -1 if no such k exists.
Original entry on oeis.org
2, 7, 76, 56, 81, 63, 913, 892, 1969, 4824, 22855, 16819, 48922, 170649, 273216, 607783, 1204354, 1910608, 3433671, 10104969, 19546522, 21424744, 66961728, 103366113, 217458328, 813832568, 771821712, 2370545332, 4638470426, 7190276806, 9309810824, 35730615937
Offset: 2
a(2) = 2 since A349474(2) = A349474(3) = 2 and there is no smaller pair of consecutive numbers with this property.
a(3) = 7 since A349474(7) = A349474(8) = 3 and there is no smaller pair of consecutive numbers with this property.
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c[n_] := Length @ ContinuedFraction[DivisorSigma[0, n]/DivisorSigma[-1, n]]; seq[len_, nmax_] := Module[{s = Table[0, {len}], k = 1, n = 1, i}, s[[1]] = -1; While[n < nmax && k < len, i = c[n]; If[c[n+1] == i && i <= len && s[[i]] == 0, k++; s[[i]] = n]; n++]; Rest @ s]; seq[15, 10^6]
A349501
a(n) is the least start of a run of exactly n consecutive numbers with the same length of the continued fraction of the harmonic mean of their divisors (A349474).
Original entry on oeis.org
1, 2, 59, 280, 3539, 57575, 65, 15410548, 9286977451, 24510585369
Offset: 1
a(2) = 2 since A349474(2) = A349474(3) = 2 and there is no smaller pair of consecutive numbers with this property.
a(3) = 59 since A349474(59) = A349474(60) = A349474(61) = 3 and there is no smaller triple of consecutive numbers with this property.
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d[n_] := Length @ ContinuedFraction[DivisorSigma[0, n] / DivisorSigma[-1, n]]; seq[len_, nmax_] := Module[{s = Table[0, {len}], dprev = -1, n = 1, c = 0, k = 0}, While[k < len && n < nmax, d1 = d[n]; If[d1 == dprev, c++, If[c > 0 && c <= len && s[[c]] == 0, k++; s[[c]] = n - c]; c = 1]; n++; dprev = d1]; TakeWhile[s, # > 0 &]]; seq[7, 10^5]
Showing 1-3 of 3 results.
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