A256416 -id:A256416 - cp's OEIS frontend

A256417 The EKG sequence (A064413) smoothed by replacing each prime p with 2p and each thrice-prime 3p also with 2p.

1, 4, 4, 4, 6, 6, 12, 8, 10, 10, 10, 18, 14, 14, 14, 24, 16, 20, 22, 22, 22, 27, 30, 25, 35, 28, 26, 26, 26, 36, 32, 34, 34, 34, 42, 38, 38, 38, 45, 40, 44, 46, 46, 46, 48, 50, 52, 54, 56, 49, 63, 60, 55, 65, 70, 58, 58, 58, 66, 62, 62

Offset: 1

N. J. A. Sloane, Apr 05 2015

nonn

This smoothing of A064413 is discussed in Lagarias et al. (2002).

a(n) = A256415(A064413(n)). - Reinhard Zumkeller, Apr 06 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The EKG sequence, arXiv:math/0204011 [math.NT], 2002.
J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The EKG Sequence, Exper. Math. 11 (2002), 437-446.

Cf. A064413, A256415, A256416.

a256417 = a256415 . a064413  -- Reinhard Zumkeller, Apr 06 2015

ekg[s_] := Block[{m = s[[-1]], k = 3}, While[MemberQ[s, k] || GCD[m, k] == 1, k++]; Append[s, k]];
Nest[ekg, {1, 2}, 100] /. {n_ /; PrimeQ[n] -> 2n, n_ /; PrimeQ[n/3] -> 2n/3 } (* Jean-François Alcover, Aug 04 2018, after Robert G. Wilson v *)

A256415 Positive integers with primes p replaced by 2p and also 3p replaced by 2p.

Original entry on oeis.org

1, 4, 6, 4, 10, 4, 14, 8, 6, 10, 22, 12, 26, 14, 10, 16, 34, 18, 38, 20, 14, 22, 46, 24, 25, 26, 27, 28, 58, 30, 62, 32, 22, 34, 35, 36, 74, 38, 26, 40, 82, 42, 86, 44, 45, 46, 94, 48, 49, 50, 34, 52, 106, 54, 55, 56, 38, 58, 118, 60, 122, 62, 63, 64, 65, 66, 134, 68, 46, 70

Offset: 1

N. J. A. Sloane, Apr 05 2015

nonn

The smoothed version of A064413 (A256417) is a rearrangement of these terms.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A010051, A064413, A256416 (the sorted sequence), A256417.

a256415 n | a010051 n == 1 = 2 * n
          | r == 0 && a010051 n' == 1 = 2 * n'
          | otherwise = n
          where (n', r) = divMod n 3
-- Reinhard Zumkeller, Apr 05 2015

# Apply p->2p, 3p->2p to a sequence t1
SMOOTH:=proc(t1) local M,t2,n; t2:=[]: M:=nops(t1):
for n from 1 to M do
if isprime(t1[n]) then t2:=[op(t2),2*t1[n]];
elif (t1[n] mod 3 = 0 ) and isprime(t1[n]/3) then t2:=[op(t2),2*t1[n]/3];
else t2:=[op(t2),t1[n]]; fi; od: t2; end;
SMOOTH([seq(n,n=1..200)]);

Range[70] /. {n_ /; PrimeQ[n] -> 2n, n_ /; PrimeQ[n/3] -> 2n/3} (* Jean-François Alcover, Aug 04 2018 *)
Table[Which[PrimeQ[n],2n,PrimeQ[n/3],2 n/3,True,n],{n,120}] (* Harvey P. Dale, Jun 08 2019 *)

cp's OEIS Frontend

A256417 The EKG sequence (A064413) smoothed by replacing each prime p with 2p and each thrice-prime 3p also with 2p.

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