

A061898


Swap each prime in factorization of n with "neighbor" prime.


8



1, 3, 2, 9, 7, 6, 5, 27, 4, 21, 13, 18, 11, 15, 14, 81, 19, 12, 17, 63, 10, 39, 29, 54, 49, 33, 8, 45, 23, 42, 37, 243, 26, 57, 35, 36, 31, 51, 22, 189, 43, 30, 41, 117, 28, 87, 53, 162, 25, 147, 38, 99, 47, 24, 91, 135, 34, 69, 61, 126, 59, 111, 20, 729, 77, 78, 71, 171, 58
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OFFSET

1,2


COMMENTS

Here "neighbor" primes are just paired in order: 2<>3, 5<>7, 11<>13, etc. Selfinverse permutation of the integers. Multiplicative.


LINKS



FORMULA

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2p)/(p^2q(p)) = 0.9229142333..., where q(p) is the "neighbor" of p.  Amiram Eldar, Nov 29 2022


EXAMPLE

a(60) = 126 since 60 = 2^2*3*5, swapping 2<>3 and 5<>7 gives 3^2*2*7 = 126 (and of course then a(126) = 60).


MAPLE

p:= proc(n) option remember; `if`(numtheory[pi](n)::odd,
nextprime(n), prevprime(n))
end:
a:= n> mul(p(i[1])^i[2], i=ifactors(n)[2]):


MATHEMATICA

p[n_] := p[n] = If[OddQ[PrimePi[n]], NextPrime[n], NextPrime[n, 1]];
a[1] = 1; a[n_] := Product[p[i[[1]]]^i[[2]], {i, FactorInteger[n]}];


PROG

(PARI) a(n) = my(f=factor(n)); for (i=1, #f~, ip = primepi(f[i, 1]); if (ip % 2, f[i, 1] = prime(ip+1), f[i, 1] = prime(ip1))); factorback(f); \\ Michel Marcus, Jun 09 2014


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



