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

0,5

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

Formula

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

Crossrefs

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

Sequence in context: A328385 A328099 A003415 * A086300 A028271 A168066

Adjacent sequences: A302052 A302053 A302054 * A302056 A302057 A302058

Keyword

nonn

Author

Antti Karttunen, Mar 31 2018

Status

approved