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A324545 An analog of sigma (A000203) for nonstandard factorization based on the sieve of Eratosthenes (A083221). 6
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 (list; graph; refs; listen; history; text; internal format)
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 A084250 A090128

Adjacent sequences:  A324542 A324543 A324544 * A324546 A324547 A324548

KEYWORD

nonn

AUTHOR

Antti Karttunen, Mar 06 2019

STATUS

approved

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Last modified January 28 06:00 EST 2021. Contains 340490 sequences. (Running on oeis4.)