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

Author

Antti Karttunen, Mar 06 2019

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
• Index entries for sequences generated by sieves
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A020639, A078898, A250246, A252754, A302044, A302045, A324054, A324535, A324544, A324546.

Cf. also A302051, A302055, A323243.

Programs

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)));

Formula

a(n) = A000203(A250246(n)) = A324535(n) + A250246(n).

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)).

a(n) = A324054(A252754(n)).