A324545 An analog of sigma (A000203) for nonstandard factorization based on the sieve of Eratosthenes (A083221).
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, 42, 32, 56, 30, 72, 32, 63, 78, 54, 48, 91, 38, 60, 48, 90, 42, 120, 44, 84, 121, 72, 48, 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
Offset: 1
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PARI
up_to = 65537; 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; }; A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639 A055396(n) = if(1==n,0,primepi(A020639(n))); v078898 = ordinal_transform(vector(up_to,n,A020639(n))); A078898(n) = v078898[n]; A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961 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))); A324545(n) = sigma(A250246(n)); -
PARI
\\ Or alternatively, using also A078898 defined above: A000265(n) = (n/2^valuation(n, 2)); A001511(n) = 1+valuation(n,2); A302045(n) = A001511(A078898(n)); 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)); }; A324545(n) = if(1==n,n,my(p=A020639(n)); (((p^(A302045(n)+1))-1)/(p-1))*A324545(A302044(n)));