A302039 Analog of A056239 for nonstandard factorization based on the sieve of Eratosthenes (A083221).

0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 6, 5, 5, 4, 7, 5, 8, 5, 6, 6, 9, 5, 6, 7, 6, 6, 10, 6, 11, 5, 7, 8, 7, 6, 12, 9, 7, 6, 13, 7, 14, 7, 8, 10, 15, 6, 8, 7, 8, 8, 16, 7, 9, 7, 8, 11, 17, 7, 18, 12, 8, 6, 8, 8, 19, 9, 9, 8, 20, 7, 21, 13, 9, 10, 9, 8, 22, 7, 9, 14, 23, 8, 10, 15, 9, 8, 24, 9, 12, 11, 10, 16, 9, 7, 25, 9, 10, 8, 26, 9, 27, 9, 10

Offset: 1

Author

Antti Karttunen, Mar 31 2018

Keywords

• nonn

Comments

Each n occurs A000041(n) times in total.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences generated by sieves

Crossrefs

Cf. A000041, A056239, A250246, A302042, A302044, A302045.

Cf. also A253557, A302041, A302050, A302051, A302052, A302055 for other similar analogs.

Programs

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, by Hasler.
A055396(n) = if(1==n,0,primepi(A020639(n)));
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A302042(n) = if((1==n)||isprime(n),1,my(c = A078898(n), 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));
A302039(n) = if(1==n,0,A055396(n) + A302039(A302042(n)));

Formula

a(1) = 0; for n > 1, a(n) = A055396(n) + a(A302042(n)).

a(1) = 0; for n > 1, a(n) = (A055396(n)*A302045(n)) + a(A302044(n)).

a(n) = A056239(A250246(n)).