A302050 An analog of Möbius function (A008683) for nonstandard factorization based on the sieve of Eratosthenes (A083221).

1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, -1, 1, 1, 0, -1, 0, -1, 0, 0, 1, -1, 0, 0, 1, 1, 0, -1, -1, -1, 0, 0, 1, 1, 0, -1, 1, 1, 0, -1, 0, -1, 0, 0, 1, -1, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 1, -1, 0, -1, 1, 1, 0, 1, 0, -1, 0, 0, -1, -1, 0, -1, 1, 1, 0, 1, -1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, -1, 0, 0, 0, 0, 1, 1, 0, -1, 0, 1, 0, -1, 0, -1, 0, 0

Offset

1

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, if A302045(n) > 1, then a(n) = 0, otherwise a(n) = -1*a(A302044(n)).

a(n) = A008683(A250246(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)); };
A302050(n) = if(1==n, n, if(A302045(n)>1, 0, -1*A302050(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)));
A302050(n) = moebius(A250246(n));

Crossrefs

Cf. A008683, A083221, A250246, A302044, A302045.

Cf. also A253557, A302041, A302051, A302052, A302039, A302055 for other similar analogs.

Sequence in context: A075437 A130047 A293233 * A008683 A008966 A080323

Adjacent sequences: A302047 A302048 A302049 * A302051 A302052 A302053

Keyword

sign

Author

Antti Karttunen, Apr 01 2018

Status

approved