cp's OEIS Frontend

A324535 An analog of sigma(n)-n (A001065) for nonstandard factorization based on the sieve of Eratosthenes (A083221).

%I A324535 #7 Mar 08 2019 20:14:32
%S A324535 0,1,1,3,1,6,1,7,4,8,1,16,1,10,9,15,1,21,1,22,13,14,1,36,6,16,11,28,1,
%T A324535 42,1,31,33,20,13,55,1,22,15,50,1,66,1,40,40,26,1,76,8,43,49,46,1,54,
%U A324535 31,64,41,32,1,108,1,34,17,63,17,144,1,58,105,74,1,123,1,40,21,64,19,78,1,106,57,44,1,172,73,46,87,92,1,201,57,76,121
%N A324535 An analog of sigma(n)-n (A001065) for nonstandard factorization based on the sieve of Eratosthenes (A083221).
%H A324535 Antti Karttunen, <a href="/A324535/b324535.txt">Table of n, a(n) for n = 1..16384</a>
%H A324535 Antti Karttunen, <a href="/A324535/a324535.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%H A324535 <a href="/index/Si#sieve">Index entries for sequences generated by sieves</a>
%H A324535 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A324535 a(n) = A001065(A250246(n)) = A324545(n) - A250246(n).
%F A324535 a(n) = A250246(n) - A324546(n).
%o A324535 (PARI)
%o A324535 up_to = 65537;
%o A324535 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 A324535 A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
%o A324535 A055396(n) = if(1==n,0,primepi(A020639(n)));
%o A324535 v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
%o A324535 A078898(n) = v078898[n];
%o A324535 A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
%o A324535 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 A324535 A001065(n) = (sigma(n)-n);
%o A324535 A324535(n) = A001065(A250246(n));
%Y A324535 Cf. A001065, A250246, A324545, A324546.
%K A324535 nonn
%O A324535 1,4
%A A324535 _Antti Karttunen_, Mar 08 2019