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%I #16 Mar 08 2019 20:14:49
%S 1,3,4,7,6,12,8,15,13,18,12,28,14,24,24,31,18,39,20,42,40,36,24,60,31,
%T 42,32,56,30,72,32,63,78,54,48,91,38,60,48,90,42,120,44,84,121,72,48,
%U 124,57,93,124,98,54,96,156,120,104,90,60,168,62,96,56,127,72,234,68,126,240,144,72,195,74,114,72,140,96,144,80
%N An analog of sigma (A000203) for nonstandard factorization based on the sieve of Eratosthenes (A083221).
%H Antti Karttunen, <a href="/A324545/b324545.txt">Table of n, a(n) for n = 1..16384</a>
%H Antti Karttunen, <a href="/A324545/a324545.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%H <a href="/index/Si#sieve">Index entries for sequences generated by sieves</a>
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F a(n) = A000203(A250246(n)) = A324535(n) + A250246(n).
%F a(1) = 1; for n > 1, let p = A020639(n) [the smallest prime factor of n], then a(n) = (((p^(1+A302045(n)))-1) / (p-1)) * a(A302044(n)).
%F a(n) = A324054(A252754(n)).
%o (PARI)
%o up_to = 65537;
%o ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
%o A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
%o A055396(n) = if(1==n,0,primepi(A020639(n)));
%o v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
%o A078898(n) = v078898[n];
%o A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
%o A250246(n) = if(1==n,n,my(k = 2*A250246(A078898(n)), r = A055396(n)); if(1==r, k, while(r>1, k = A003961(k); r--); (k)));
%o A324545(n) = sigma(A250246(n));
%o (PARI)
%o \\ Or alternatively, using also A078898 defined above:
%o A000265(n) = (n/2^valuation(n, 2));
%o A001511(n) = 1+valuation(n,2);
%o A302045(n) = A001511(A078898(n));
%o A302044(n) = { my(c = A000265(A078898(n))); if(1==c,1,my(p = prime(-1+primepi(A020639(n))+primepi(A020639(c))), d = A078898(c), k=0); while(d, k++; if((1==k)||(A020639(k)>=p),d -= 1)); (k*p)); };
%o A324545(n) = if(1==n,n,my(p=A020639(n)); (((p^(A302045(n)+1))-1)/(p-1))*A324545(A302044(n)));
%Y Cf. A000203, A020639, A078898, A250246, A252754, A302044, A302045, A324054, A324535, A324544, A324546.
%Y Cf. also A302051, A302055, A323243.
%K nonn
%O 1,2
%A _Antti Karttunen_, Mar 06 2019
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