A302045 a(1) = 0, for n > 1, a(n) = A001511(A078898(n)); Number of instances of the smallest prime factor A020639(n) in nonstandard factorization of n that is based on the sieve of Eratosthenes (A083221).

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 4, 1, 1, 4, 2, 1, 1, 2, 1, 1, 3, 3, 2, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 3, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 2, 2, 1, 1, 3, 1, 1, 3, 2, 5, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 2

Offset

1,4

Comments

Iterating the map n -> A302044(n) until 1 is reached, and taking the smallest prime factor (A020639) of each term gives a sequence of distinct primes in ascending order, while applying this function (A302045) to those terms gives the corresponding "exponents" of those primes, that is, the count of consecutive occurrences of each prime when iterating the map n -> A302042(n), which gives the same primes with repetitions. Permutation pair A250245/A250246 maps between this non-standard prime factorization of n and the ordinary factorization of n. See also comments and examples in A302042.

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) = A001511(A078898(n)).

For n > 1, a(n) = A250245(A067029(A250246(n))).

Prog

(PARI)
A020639(n) = { if(1==n, n, vecmin(factor(n)[, 1])); };
A078898(n) = { if(n<=1, n, my(spf=A020639(n), k=1, m=n/spf); while(m>1, if(A020639(m)>=spf, k++); m--); (k)); };
A001511(n) = 1+valuation(n, 2);
A302045(n) = if(1==n, 0, A001511(A078898(n)));

Crossrefs

Cf. A001511, A067029, A078898, A250245, A250246, A268674, A302042, A302044.

Cf. also A302035, A302050, A302051.

Sequence in context: A067029 A087179 A290109 * A302035 A307907 A362333

Adjacent sequences: A302042 A302043 A302044 * A302046 A302047 A302048

Keyword

nonn

Author

Antti Karttunen, Mar 31 2018

Status

approved