A226319
a(n) is the smallest odd k > 1 such that k is a Niven number at least in all the bases from 1 to n.
Original entry on oeis.org
3, 21, 21, 21, 675, 4725, 4725, 98175, 140175, 543375, 543375, 23186625, 35026425, 139264125, 139264125, 608679225, 11553990525, 87662479905, 87662479905, 343947649815, 2383529269275, 4005262262925, 4005262262925
Offset: 1
a(4) = 21 because 21 is the smallest odd k > 1 which is Niven in bases 1 (trivial), 2, 3, and 4, being equal to (10101)_2, (210)_3 and (111)_4.
-
a[n_] := Block[{k=3}, n > 1 && While[Max@ Mod[k, Total /@ IntegerDigits[k, Range[2, n]]] > 0, k += 2]; k]; Array[a,9]
A344512
a(n) is the least number larger than 1 which is a self number in all the bases 2 <= b <= n.
Original entry on oeis.org
4, 13, 13, 13, 287, 287, 2971, 2971, 27163, 27163, 90163, 90163, 5940609, 5940609, 6069129, 6069129, 276404649, 276404649
Offset: 2
a(2) = 4 since the least binary self number after 1 is A010061(2) = 4.
a(3) = 13 since the least binary self number after 1 which is also a self number in base 3 is A010061(4) = 13.
- Vijayshankar Shivshankar Joshi, Contributions to the theory of power-free integers and self-numbers, Ph.D. dissertation, Gujarat University, Ahmedabad (India), October, 1973.
- József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384-386.
Cf.
A003052,
A010061,
A010064,
A010067,
A010070,
A339211,
A339212,
A339213,
A339214,
A339215,
A342729,
A344513.
-
s[n_, b_] := n + Plus @@ IntegerDigits[n, b]; selfQ[n_, b_] := AllTrue[Range[n, n - (b - 1) * Ceiling @ Log[b, n], -1], s[#, b] != n &]; a[2] = 4; a[b_] := a[b] = Module[{n = a[b - 1]}, While[! AllTrue[Range[2, b], selfQ[n, #] &], n++]; n]; Array[a, 10, 2]
A344513
a(n) is the least number larger than 1 which is a self number in all the even bases b = 2*k for 1 <= k <= n.
Original entry on oeis.org
4, 13, 287, 294, 6564, 90163, 1136828, 3301262, 276404649, 5643189146
Offset: 1
a(1) = 4 since the least binary self number after 1 is A010061(2) = 4.
a(2) = 13 since the least binary self number after 1 which is also a self number in base 2*2 = 4 is A010061(4) = A010064(4) = 13.
- Vijayshankar Shivshankar Joshi, Contributions to the theory of power-free integers and self-numbers, Ph.D. dissertation, Gujarat University, Ahmedabad (India), October, 1973.
- József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384-386.
Cf.
A003052,
A010061,
A010064,
A010067,
A010070,
A339211,
A339212,
A339213,
A339214,
A339215,
A342729,
A344512.
-
s[n_, b_] := n + Plus @@ IntegerDigits[n, b]; selfQ[n_, b_] := AllTrue[Range[n, n - (b - 1) * Ceiling @ Log[b, n], -1], s[#, b] != n &]; a[1] = 4; a[n_] := a[n] = Module[{k = a[n - 1]}, While[! AllTrue[Range[1, n], selfQ[k, 2*#] &], k++]; k]; Array[a, 7]
Showing 1-3 of 3 results.
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