A324179 -id:A324179 - cp's OEIS frontend

A324190 Number of distinct values A297167 obtains over the divisors > 1 of n; a(1) = 0.

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 4, 2, 2, 1, 4, 2, 2, 3, 4, 1, 3, 1, 5, 2, 2, 2, 4, 1, 2, 2, 5, 1, 3, 1, 4, 3, 2, 1, 5, 2, 3, 2, 4, 1, 4, 2, 6, 2, 2, 1, 4, 1, 2, 4, 6, 2, 3, 1, 4, 2, 3, 1, 5, 1, 2, 3, 4, 2, 3, 1, 6, 4, 2, 1, 5, 2, 2, 2, 6, 1, 4, 2, 4, 2, 2, 2, 6, 1, 3, 4, 5, 1, 3, 1, 6, 3

Offset: 1

Antti Karttunen, Feb 19 2019

nonn

Number of distinct values of the sum {excess of d} + {the index of the largest prime factor of d} (that is, A046660(d) + A061395(d)) that occurs over all divisors d > 1 of n.

Number of distinct values A297112 obtains over the divisors > 1 of n; a(1) = 0.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000005, A001221, A001222, A046660, A061395, A324120, A324179, A324191, A324192, A324202, A324203.

Cf. also A156552, A297112.

A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A324190(n) = #Set(apply(A297167, select(d -> d>1,divisors(n))));

a(n) = A001221(A324202(n)).
a(n) >= A324120(n).
a(n) >= A001222(n) >= A001221(n). [See A324179 and A324192 for differences]
a(n) <= A000005(n)-1. [See A324191 for differences]
For all primes p, a(p^k) = k.

A324120 Binary weight of SumXOR variant of A297168: a(n) = A000120(A324180(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 2, 0, 2, 2, 3, 0, 2, 0, 2, 2, 2, 0, 2, 1, 2, 2, 2, 0, 2, 0, 4, 2, 2, 2, 3, 0, 2, 2, 4, 0, 2, 0, 2, 2, 2, 0, 2, 1, 2, 2, 2, 0, 2, 2, 4, 2, 2, 0, 4, 0, 2, 2, 5, 2, 2, 0, 2, 2, 2, 0, 2, 0, 2, 2, 2, 2, 2, 0, 4, 3, 2, 0, 4, 2, 2, 2, 4, 0, 2, 2, 2, 2, 2, 2, 2, 0, 2, 2, 3, 0, 2, 0, 4, 2

Offset: 1

Antti Karttunen, Feb 19 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000043, A324179, A324180, A324181, A324190, A324191, A324192, A324201.

A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A297112(n) = if(1==n, 0, 2^A297167(n));
A324180(n) = { my(v=0); fordiv(n, d, if(dA297112(d)))); (v); };
A324120(n) = hammingweight(A324180(n));

a(n) = A000120(A324180(n)).
a(n) <= A324190(n).
a(p^k) = k-1 for all primes p and exponents k >= 1.

A324191 Number of divisors of n minus the number of distinct values that A297167 obtains over the divisors > 1 of n: a(n) = A000005(n) - A324190(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 1, 2, 1, 5, 1, 1, 2, 2, 2, 5, 1, 2, 2, 3, 1, 5, 1, 2, 3, 2, 1, 5, 1, 3, 2, 2, 1, 4, 2, 2, 2, 2, 1, 8, 1, 2, 2, 1, 2, 5, 1, 2, 2, 5, 1, 7, 1, 2, 3, 2, 2, 5, 1, 4, 1, 2, 1, 7, 2, 2, 2, 2, 1, 8, 2, 2, 2, 2, 2, 6, 1, 3, 2, 4, 1, 5, 1, 2, 5

Offset: 1

Antti Karttunen, Feb 19 2019

nonn

a(p^k) = 1 for all primes p and all exponents k >= 0, because with prime powers there are k divisors larger than 1 and A297167 obtains a distinct value for each one of them.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000005, A001221, A001222, A046660, A061395, A297167, A324120, A324179, A324120, A324190, A324192.

Cf. A000961 (positions of ones).

A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A324190(n) = #Set(apply(A297167, select(d -> d>1,divisors(n))));
A324191(n) = (numdiv(n)-A324190(n));

a(n) = A000005(n) - A324190(n).

A324192 Number of distinct values A297167 obtains over divisors > 1 of n, minus number of distinct prime factors of n: a(n) = A324190(n) - A001221(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 2, 0, 0, 0, 2, 1, 0, 2, 2, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 2, 1, 0, 0, 3, 1, 1, 0, 2, 0, 2, 0, 4, 0, 0, 0, 1, 0, 0, 2, 5, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 1, 2, 0, 0, 0, 4, 3, 0, 0, 2, 0, 0, 0, 4, 0, 1, 0, 2, 0, 0, 0, 4, 0, 1, 2, 3, 0, 0, 0, 4, 0

Offset: 1

Antti Karttunen, Feb 19 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A001222, A324179.

A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A324190(n) = #Set(apply(A297167, select(d -> d>1,divisors(n))));
A324192(n) = (A324190(n)-omega(n));

a(n) = A324190(n) - A001221(n).

a(n) >= A324179(n).

cp's OEIS Frontend

A324190 Number of distinct values A297167 obtains over the divisors > 1 of n; a(1) = 0.

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A324120 Binary weight of SumXOR variant of A297168: a(n) = A000120(A324180(n)).

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A324191 Number of divisors of n minus the number of distinct values that A297167 obtains over the divisors > 1 of n: a(n) = A000005(n) - A324190(n).

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A324192 Number of distinct values A297167 obtains over divisors > 1 of n, minus number of distinct prime factors of n: a(n) = A324190(n) - A001221(n).

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