A324180 -id:A324180 - cp's OEIS frontend

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

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.

A324181 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A324180(n) for n > 1 and f(1) = -1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 5, 6, 2, 4, 2, 7, 8, 9, 2, 6, 2, 4, 10, 11, 2, 4, 12, 13, 8, 4, 2, 6, 2, 14, 15, 16, 17, 9, 2, 18, 19, 14, 2, 7, 2, 4, 8, 20, 2, 4, 21, 7, 22, 4, 2, 7, 23, 24, 25, 26, 2, 14, 2, 27, 8, 28, 29, 11, 2, 4, 30, 7, 2, 4, 2, 31, 10, 4, 32, 13, 2, 24, 33, 34, 2, 24, 35, 36, 37, 38, 2, 7, 39, 4, 40, 41, 42, 4, 2, 11, 8, 43, 2, 16, 2, 44, 10

Offset: 1

Antti Karttunen, Feb 19 2019

nonn

For all i, j: a(i) = a(j) => A324120(i) = A324120(j).

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

Cf. A000040 (positions of 2's), A156552, A297112, A324120, A324180.

Cf. also A300827, A323914, A324203.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
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); };
Aux324181(n) = if((1==n),-n,A324180(n));
v324181 = rgs_transform(vector(up_to, n, Aux324181(n)));
A324181(n) = v324181[n];

A324193 a(1) = 0; for n > 1, a(n) = Product_{d|n, d>1, dA297167(d)).

Original entry on oeis.org

0, 1, 1, 2, 1, 6, 1, 6, 3, 10, 1, 54, 1, 14, 15, 30, 1, 90, 1, 150, 21, 22, 1, 1350, 5, 26, 15, 294, 1, 2250, 1, 210, 33, 34, 35, 6750, 1, 38, 39, 5250, 1, 6174, 1, 726, 375, 46, 1, 66150, 7, 350, 51, 1014, 1, 3150, 55, 16170, 57, 58, 1, 1181250, 1, 62, 735, 2310, 65, 23958, 1, 1734, 69, 17150, 1, 1653750, 1, 74, 525, 2166, 77, 39546, 1, 404250, 105

Offset: 1

Antti Karttunen, Feb 20 2019

nonn

An auxiliary sequence for defining A300827, which is the restricted growth sequence transform of this sequence. A324202 is a similar sequence, but is not limited to the proper divisors of n, and in contrast to this, also finds the least prime signature representative (A046523) of the product formed.

Antti Karttunen, Table of n, a(n) for n = 1..4096
Index entries for sequences computed from indices in prime factorization

Cf. A001221, A001222, A007913, A061395, A087207, A297167, A300827 (rgs-transform), A324120, A324180.

Cf. also A324202.

A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A324193(n) = { my(m=1); if(n<=2, n-1, fordiv(n, d, if((d>1)&(dA297167(d)))); (m)); };

a(1) = 0; for n > 1, a(n) = Product_{d|n, d>1, dA297167(d)).
For all n > 0:
A001222(a(n)) = A000005(n)-2.
A001221(A007913(a(n))) = A324120(n).
A087207(A007913(a(n))) = A324180(n).

A324195 Cumulative bitwise-OR of A297112(d), where d ranges over the divisors d of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 3, 8, 7, 6, 5, 16, 7, 32, 9, 6, 15, 64, 7, 128, 15, 10, 17, 256, 15, 12, 33, 14, 27, 512, 7, 1024, 31, 18, 65, 12, 15, 2048, 129, 34, 31, 4096, 11, 8192, 51, 14, 257, 16384, 31, 24, 13, 66, 99, 32768, 15, 20, 63, 130, 513, 65536, 15, 131072, 1025, 30, 63, 36, 19, 262144, 195, 258, 13, 524288, 31, 1048576, 2049, 14, 387, 24, 35, 2097152, 63, 30

Offset: 1

Antti Karttunen, Feb 20 2019

nonn

A324180 differs from this one in that it uses XOR instead of OR, and uses only the proper divisors of n.

Antti Karttunen, Table of n, a(n) for n = 1..4096

Cf. A000120, A003986, A297112, A297167, A297168, A061395, A324190, A324196, A324197.

Cf. also A324180.

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));
A324195(n) = { my(v=0); fordiv(n, d, v = bitor(v,A297112(d))); (v); };

A000120(a(n)) = A324190(n).

cp's OEIS Frontend

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

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A324181 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A324180(n) for n > 1 and f(1) = -1.

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A324193 a(1) = 0; for n > 1, a(n) = Product_{d|n, d>1, dA297167(d)).

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A324195 Cumulative bitwise-OR of A297112(d), where d ranges over the divisors d of n.

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