A316190 a(n) is the distance between n and the nearest prime power (in the sense of A246655) other than n.
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 3, 2, 1, 2, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3
Offset: 1
Keywords
Examples
Note that 1369, 1373, 1381 and 1399 are prime powers. This leads to the mapping: 1373 -> 4, 1374 -> 1, 1375 -> 2, 1376 -> 3, 1377 -> 4, 1378 -> 3, 1379 -> 2, 1380 -> 1, 1381 -> 8.
References
- S. M. Gonek, A formula of Landau and mean values of Zeta(s), Topics in Analytic Number Theory, ed. by S. W. Graham and J. D. Vaaler, 92-97, Univ. Texas Press 1985.
- S. M. Gonek, An explicit formula of Landau and its applications to the theory of the zeta-function, Contemporary Math. 143 (1993), 395-413.
Links
- Peter Luschny, Table of n, a(n) for n = 1..10000
- E. Landau, Über die Nullstellen der Zetafunktion, Math. Annalen 71, 548-564, (1911).
Programs
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Maple
A316190_list := proc(N) local a, b, d, m, k, P, R; R := NULL; m := 1; P := select(t -> nops(numtheory:-factorset(t)) = 1 or t = 0, [$0..N]); for k from 1 to nops(P)-1 do a := P[k]; b := P[k+1]; if m = a then R := R, min(m - P[k-1] , b - m); m := m + 1; fi; while m < b do R := R, min(m - a , b - m); m := m + 1; od; od; [R] end: A316190_list(100);
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Mathematica
a[n_] := Module[{k = 1}, While[!PrimePowerQ[n+k] && !PrimePowerQ[n-k], k++]; k]; Array[a, 100] (* Jean-François Alcover, Jul 25 2019 *)