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

1,2

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)

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)).

Prog

(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)));

Crossrefs

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

Cf. also A302051, A302055, A323243.

Sequence in context: A097012 A143348 A000203 * A003979 A378648 A084250

Adjacent sequences: A324542 A324543 A324544 * A324546 A324547 A324548

Keyword

nonn

Author

Antti Karttunen, Mar 06 2019

Status

approved