A302051 An analog of A000005 for nonstandard factorization based on the sieve of Eratosthenes (A083221).

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 6, 6, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 5, 4, 2, 10, 3, 6, 6, 6, 2, 8, 4, 8, 6, 4, 2, 12, 2, 4, 4, 7, 4, 12, 2, 6, 8, 8, 2, 12, 2, 4, 4, 6, 4, 8, 2, 10, 6, 4, 2, 12, 6, 4, 8, 8, 2, 10, 4, 6, 6, 4, 4, 12, 2, 6, 4, 9, 2, 12, 2, 8, 9

Offset

1,2

Comments

See A302042, A302044 and A302045 for a description of the factorization process.

Links

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

Index entries for sequences generated by sieves

Formula

a(1) = 1, for n > 1, a(n) = (A302045(n)+1) * a(A302044(n)).

a(n) = A000005(A250246(n)).

a(n) = A106737(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
v078898 = ordinal_transform(vector(up_to, n, A020639(n)));
A078898(n) = v078898[n];
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)); };
A302051(n) = if(1==n, n, (A302045(n)+1)*A302051(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)));
A302051(n) = numdiv(A250246(n));

Crossrefs

Cf. A000005, A083221, A302042, A302044, A302045, A302052 (reduced modulo 2), A302053 (gives the positions of odd numbers).

Cf. also A253557, A302041, A302050, A302052, A302039, A302055 for other similar analogs.

Sequence in context: A122667 A122668 A073668 * A334080 A066800 A368195

Adjacent sequences: A302048 A302049 A302050 * A302052 A302053 A302054

Keyword

nonn

Author

Antti Karttunen, Apr 01 2018

Status

approved