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%I #28 Mar 11 2023 08:07:50
%S 0,1,3,4,7,6,15,8,12,13,31,12,63,18,18,24,127,14,255,20,39,48,511,24,
%T 28,84,24,48,1023,32,2047,32,54,176,42,40,4095,258,144,56,8191,38,
%U 16383,68,36,800,32767,48,60,31,252,132,65535,30,91,72,528,1302,131071,44,262143,2736,60,104,126,96,524287,304,774,42,1048575,72,2097151,4356,42
%N a(1) = 0; for n > 1, a(n) = A000203(A156552(n)).
%H Antti Karttunen, <a href="/A323243/b323243.txt">Table of n, a(n) for n = 1..10000</a> (based on Hans Havermann's factorization of A156552)
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F a(1) = 0; for n > 1, a(n) = A000203(A156552(n)).
%F a(n) = 2*A156552(n) - A323244(n).
%F a(n) = A323247(n) - A323248(n).
%F From _Antti Karttunen_, Mar 12 2019: (Start)
%F a(A000040(n)) = A000225(n).
%F a(n) = Sum_{d|n} A324543(d).
%F For n > 1, a(2*A246277(n)) = A324118(n).
%F gcd(a(n), A156552(n)) = A324396(n).
%F A000035(a(n)) = A324823(n).
%F (End)
%t Array[If[# == 0, 0, DivisorSigma[1, #]] &@ Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ #]] &, 75] (* _Michael De Vlieger_, Apr 21 2019 *)
%o (PARI)
%o A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
%o A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
%o A323243(n) = if(1==n, 0, sigma(A156552(n)));
%o (PARI)
%o \\ For computing terms a(n), with n > ~4000 use Hans Havermann's factorization file https://oeis.org/A156552/a156552.txt
%o v156552sigs = readvec("a156552.txt"); \\ First read it in as a PARI-vector.
%o A323243(n) = if(n<=2,n-1,my(prsig=v156552sigs[n],ps=prsig[1],es=prsig[2]); prod(i=1,#ps,((ps[i]^(1+es[i]))-1)/(ps[i]-1))); \\ Then play sigma
%o \\ _Antti Karttunen_, Mar 15 2019
%o (Python)
%o from sympy import divisor_sigma, primepi, factorint
%o def A323243(n): return divisor_sigma(sum((1<<primepi(p)-1)<<i for i, p in enumerate(factorint(n,multiple=True)))) if n > 1 else 0 # _Chai Wah Wu_, Mar 10 2023
%Y Cf. A000203, A156552, A323244, A323247, A323248, A324118, A324543 (Möbius transform), A324396, A324823.
%Y Cf. A323173, A324054, A324184, A324545 for other permutations of sigma, and also A324573, A324653.
%K nonn
%O 1,3
%A _Antti Karttunen_, Jan 10 2019
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