A302041 An omega analog for a nonstandard factorization based on the sieve of Eratosthenes (A083221).

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 2, 3, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2

Offset

1,6

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences generated by sieves

Formula

a(1) = 0; for n > 1, a(n) = 1 + a(A302044(n)).

a(n) = A001221(A250246(n)).

a(n) = A069010(A252754(n)).

Prog

(PARI)
\\ Assuming A250469 and its inverse A268674 have been precomputed, then the following is reasonably fast:
A302044(n) = if(1==n, n, my(k=0); while((n%2), n = A268674(n); k++); n = (n/2^valuation(n, 2)); while(k>0, n = A250469(n); k--); (n));
A302041(n) = if(1==n, 0, 1+A302041(A302044(n)));
(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
v078898 = ordinal_transform(vector(up_to, n, A020639(n)));
A078898(n) = v078898[n];
A000265(n) = (n/2^valuation(n, 2));
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)); };
A302041(n) = if(1==n, 0, 1+A302041(A302044(n)));
(PARI)
\\ Or, using also some of the code from above:
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A055396(n) = if(1==n, 0, primepi(A020639(n)));
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)));
A302041(n) = omega(A250246(n));

Crossrefs

Cf. A001221, A069010, A250246, A252754, A302044.

Cf. A302040 (positions of terms < 2).

Cf. A253557 (a similar analog for bigomega), A302050, A302051, A302052, A302039, A302055 (other analogs).

Differs from A302031 for the first time at n=59, where a(59) = 1, while A302031(59) = 2.

Sequence in context: A339839 A062893 A328512 * A302031 A371883 A237353

Adjacent sequences: A302038 A302039 A302040 * A302042 A302043 A302044

Keyword

nonn

Author

Antti Karttunen, Mar 31 2018

Status

approved