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A302041 An omega analog for a nonstandard factorization based on the sieve of Eratosthenes (A083221). 14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,6
LINKS
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. 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 A237353 A293460
KEYWORD
nonn
AUTHOR
Antti Karttunen, Mar 31 2018
STATUS
approved

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Last modified March 28 04:13 EDT 2024. Contains 371235 sequences. (Running on oeis4.)