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A200746 Completely multiplicative function with a(prime(k)) = prime(k)*prime(k-1), a(2) = 2.

%I A200746 #24 Jul 06 2025 21:38:06
%S A200746 1,2,6,4,15,12,35,8,36,30,77,24,143,70,90,16,221,72,323,60,210,154,
%T A200746 437,48,225,286,216,140,667,180,899,32,462,442,525,144,1147,646,858,
%U A200746 120,1517,420,1763,308,540,874,2021,96,1225,450,1326,572,2491,432,1155,280,1938
%N A200746 Completely multiplicative function with a(prime(k)) = prime(k)*prime(k-1), a(2) = 2.
%H A200746 Antti Karttunen, <a href="/A200746/b200746.txt">Table of n, a(n) for n = 1..10000</a>
%H A200746 <a href="/index/Di#divseq">Index to divisibility sequences</a>.
%F A200746 a(n) = n * A064989(n).
%F A200746 Sum_{n>=1} 1/a(n) = 2/Product_{k>=1} (1 - 1/(prime(k)*prime(k+1))) = 2.75321091... . - _Amiram Eldar_, Jan 07 2023
%t A200746 a200746[n_Integer] := Block[{f, p, a},
%t A200746   f[x_] := Transpose@FactorInteger[x];
%t A200746   p[x_] := Which[
%t A200746     x == 1, 1,
%t A200746     x == 2, 2,
%t A200746     True, x NextPrime[x, -1]];
%t A200746   a[x_] := Times @@ Power[p /@ First[f[x]], Last[f[x]]];
%t A200746 a /@ Range[n]]; a200746[57] (* _Michael De Vlieger_, Dec 19 2014 *)
%o A200746 (PARI) a(n)=local(fm=factor(n));prod(k=1,matsize(fm)[1],(fm[k,1]*if(fm[k,1]==2,1,precprime(fm[k,1]-1)))^fm[k,2])
%o A200746 (Scheme)
%o A200746 ;; With memoization-macro definec.
%o A200746 (definec (A200746 n) (if (<= n 2) n (* n (A064989 n)))) ;; _Antti Karttunen_, Dec 18 2014
%Y A200746 Cf. A064989, A108951, A191002.
%K A200746 nonn,mult,look
%O A200746 1,2
%A A200746 _Franklin T. Adams-Watters_, Nov 25 2011