A302055 An arithmetic derivative analog for nonstandard factorization process based on the sieve of Eratosthenes (A083221).

0, 0, 1, 1, 4, 1, 5, 1, 12, 6, 7, 1, 16, 1, 9, 8, 32, 1, 21, 1, 24, 27, 13, 1, 44, 10, 15, 10, 32, 1, 31, 1, 80, 39, 19, 12, 60, 1, 21, 14, 68, 1, 75, 1, 48, 102, 25, 1, 112, 14, 45, 55, 56, 1, 47, 75, 92, 57, 31, 1, 92, 1, 33, 16, 192, 16, 111, 1, 72, 150, 59, 1, 156, 1, 39, 20, 80, 18, 67, 1, 176, 81, 43, 1, 192, 95, 45, 71, 140, 1, 249, 147, 96

Offset: 0

Author

Antti Karttunen, Mar 31 2018

Keywords

• nonn

Comments

The formula is analogous to Reinhard Zumkeller's May 09 2011 formula in A003415, with A032742 replaced by A302042. See the comments in the latter sequence.

Note that this cannot be computed just as f(n) = A003415(A250246(n)), in contrast to many other such analogs, like A253557, A302039, A302041, A302050, A302051 and A302052.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65537
• Index entries for sequences generated by sieves

Crossrefs

Cf. A003415, A020639, A078898, A083221, A302042.

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.
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));
A302055(n) = if(n<2,0,my(k=A302042(n)); (A020639(n)*A302055(k))+k);

Formula

a(0) = a(1) = 0; for n > 1, a(n) = (A020639(n)*a(A302042(n))) + A302042(n).