A297167 -id:A297167 - cp's OEIS frontend

A297168 Difference between A156552 and its Moebius transform: a(n) = A156552(n) - A297112(n).

0, 0, 0, 1, 0, 3, 0, 3, 2, 5, 0, 7, 0, 9, 6, 7, 0, 9, 0, 11, 10, 17, 0, 15, 4, 33, 6, 19, 0, 17, 0, 15, 18, 65, 12, 19, 0, 129, 34, 23, 0, 29, 0, 35, 14, 257, 0, 31, 8, 17, 66, 67, 0, 21, 20, 39, 130, 513, 0, 35, 0, 1025, 22, 31, 36, 53, 0, 131, 258, 33, 0, 39, 0, 2049, 18, 259, 24, 101, 0, 47, 14, 4097, 0, 59, 68, 8193, 514, 71, 0, 37, 40

Offset: 1

Antti Karttunen, Feb 27 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences computed from indices in prime factorization

Cf. A008683, A033265, A156552, A297112, A297112, A297113, A297167, A297169, A300827.

With[{s = Array[Total@ MapIndexed[#1 2^(First@ #2 - 1) &, Flatten@ Map[ConstantArray[2^(PrimePi@ #1 - 1), #2] & @@ # &, FactorInteger@ #]] - Boole[# == 1]/2 &, 91]}, Table[-DivisorSum[n, MoebiusMu[n/#] s[[#]] &, # < n &], {n, Length@ s}]] (* Michael De Vlieger, Mar 13 2018 *)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A297112(n) = sumdiv(n,d,moebius(n/d)*A156552(d));
A297168(n) = (A156552(n)-A297112(n));
\\ Or alternatively as:
A297168(n) = -sumdiv(n,d,(dA156552(d));

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));
A297168(n) = sumdiv(n,d,(dA297112(d)); \\ Antti Karttunen, Mar 13 2018

(define (A297168 n) (- (A156552 n) (A297112 n)))
(define (A297168 n) (if (= 1 n) 0 (- (A156552 n) (A000079 (A297167 n)))))

a(n) = -Sum_{d|n, dA008683(n/d)*A156552(d).
a(n) = Sum_{d|n, dA297112(d).
For n > 1, a(n) = Sum_{d|n, 1A033265(A156552(d)).
a(n) = A156552(n) - A297112(n).
a(1) = 0, for n > 1, a(n) = A156552(n) - 2^A297167(n).

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.

A364557 Möbius transform of A005941.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 8, 4, 4, 4, 16, 4, 32, 8, 4, 8, 64, 4, 128, 8, 8, 16, 256, 8, 8, 32, 8, 16, 512, 4, 1024, 16, 16, 64, 8, 8, 2048, 128, 32, 16, 4096, 8, 8192, 32, 8, 256, 16384, 16, 16, 8, 64, 64, 32768, 8, 16, 32, 128, 512, 65536, 8, 131072, 1024, 16, 32, 32, 16, 262144, 128, 256, 8, 524288, 16, 1048576, 2048

Offset: 1

Antti Karttunen, Jul 28 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A005941, A008683, A297112, A297113, A297167, A364558.

A364557(n) = if(1==n, 1, 2^(primepi(vecmax(factor(n)[, 1]))+(bigomega(n)-omega(n))-1));

A005941(n) = { my(f=factor(n), p, p2=1, res=0); for(i=1, #f~, p = 1 << (primepi(f[i, 1])-1); res += (p * p2 * (2^(f[i, 2])-1)); p2 <<= f[i, 2]); (1+res) }; \\ (After David A. Corneth's program for A156552)
A364557(n) = sumdiv(n,d,moebius(n/d)*A005941(d));

from sympy import factorint, primepi
def A364557(n): return 1<1 else 1 # Chai Wah Wu, Jul 29 2023

a(n) = Sum_{d|n} A008683(n/d) * A005941(d).

a(1) = 1; for n > 1, a(n) = A297112(n) = 2^(A297113(n)-1) = 2^A297167(n).

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];

A324203 Lexicographically earliest sequence such that a(i) = a(j) => A324202(i) = A324202(j) for all i, j >= 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 7, 2, 8, 2, 9, 4, 4, 2, 10, 3, 4, 5, 9, 2, 11, 2, 12, 4, 4, 4, 13, 2, 4, 4, 14, 2, 11, 2, 9, 6, 4, 2, 15, 3, 8, 4, 9, 2, 16, 4, 17, 4, 4, 2, 18, 2, 4, 9, 19, 4, 11, 2, 9, 4, 11, 2, 20, 2, 4, 8, 9, 4, 11, 2, 21, 7, 4, 2, 20, 4, 4, 4, 17, 2, 22, 4, 9, 4, 4, 4, 23, 2, 8, 9, 24, 2, 11, 2, 17, 11

Offset: 1

Antti Karttunen, Feb 19 2019

nonn

Restricted growth sequence transform of A324202.
For all i, j:
  a(i) = a(j) => A324190(i) = A324190(j),
  a(i) = a(j) => A324191(i) = A324191(j).

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

Cf. A300827, A324181, A324190, A324191, A324202.

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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A324202(n) = A046523(factorback(apply(x -> prime(1+x),apply(A297167, select(d -> d>1,divisors(n))))));
v324203 = rgs_transform(vector(up_to, n, A324202(n)));
A324203(n) = v324203[n];

A318891 Filter sequence combining the prime signature of n (A046523) with the largest prime factor of n (A006530).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 10, 15, 16, 12, 17, 18, 14, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 19, 30, 14, 31, 32, 33, 23, 34, 35, 36, 37, 38, 18, 39, 40, 41, 42, 18, 30, 43, 44, 21, 19, 45, 33, 46, 47, 48, 49, 50, 25, 51, 23, 52, 53, 54, 39, 36, 55, 56, 57, 58, 18, 59, 19, 60, 61, 62, 63, 64, 65, 66, 30, 67, 46, 68, 69, 48, 23, 70, 50

Offset: 1

Antti Karttunen, Sep 16 2018

nonn

Restricted growth sequence transform of A286356.

For all i, j: a(i) = a(j) => A297112(i) = A297112(j). (Also, equivalently, A297113 or A297167 in place of A297112.)

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A006530, A046523, A061395, A286356, A318890.

Cf. also A297112, A297113, A297167.

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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A318891aux(n) = [A046523(n), A061395(n)];
v318891 = rgs_transform(vector(up_to,n,A318891aux(n)));
A318891(n) = v318891[n];

A324180 SumXOR variant of A297168.

Original entry on oeis.org

0, 0, 0, 1, 0, 3, 0, 3, 2, 5, 0, 3, 0, 9, 6, 7, 0, 5, 0, 3, 10, 17, 0, 3, 4, 33, 6, 3, 0, 5, 0, 15, 18, 65, 12, 7, 0, 129, 34, 15, 0, 9, 0, 3, 6, 257, 0, 3, 8, 9, 66, 3, 0, 9, 20, 23, 130, 513, 0, 15, 0, 1025, 6, 31, 36, 17, 0, 3, 258, 9, 0, 3, 0, 2049, 10, 3, 24, 33, 0, 23, 14, 4097, 0, 23, 68, 8193, 514, 39, 0, 9, 40, 3

Offset: 1

Antti Karttunen, Feb 19 2019

nonn

Cf. A061395, A156552, A297106, A297112, A297167, A297168, A324181 (rgs-transform), A324120 (number of 1-bits).

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); };

a(n) = Cumulative XOR of A297112(d), where d ranges over the proper divisors d of n.

A324196 Lexicographically earliest sequence such that a(i) = a(j) => A324195(i) = A324195(j) for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 6, 7, 8, 9, 10, 7, 11, 12, 8, 13, 14, 7, 15, 13, 16, 17, 18, 13, 19, 20, 21, 22, 23, 7, 24, 25, 26, 27, 19, 13, 28, 29, 30, 25, 31, 32, 33, 34, 21, 35, 36, 25, 37, 38, 39, 40, 41, 13, 42, 43, 44, 45, 46, 13, 47, 48, 49, 43, 50, 51, 52, 53, 54, 38, 55, 25, 56, 57, 21, 58, 37, 59, 60, 43, 49, 61, 62, 25, 63, 64, 65, 66, 67, 13, 68, 69, 70, 71, 72, 43

Offset: 1

Antti Karttunen, Feb 20 2019

nonn

Restricted growth sequence transform of A324195.

For all i, j: a(i) = a(j) => A324197(i) = A324197(j) => A324190(i) = A324190(j).

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

Cf. A297112, A297167, A297168, A061395, A324190, A324195, A324197.

Cf. also A324181.

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));
A324195(n) = { my(v=0); fordiv(n, d, v = bitor(v,A297112(d))); (v); };
v324196 = rgs_transform(vector(up_to, n, A324195(n)));
A324196(n) = v324196[n];

A324197 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A324195(n) for all other numbers except f(2) = -1 and f(n) = -2 when n is an odd prime.

Original entry on oeis.org

1, 2, 3, 4, 3, 4, 3, 5, 6, 7, 3, 5, 3, 8, 6, 9, 3, 5, 3, 9, 10, 11, 3, 9, 12, 13, 14, 15, 3, 5, 3, 16, 17, 18, 12, 9, 3, 19, 20, 16, 3, 21, 3, 22, 14, 23, 3, 16, 24, 25, 26, 27, 3, 9, 28, 29, 30, 31, 3, 9, 3, 32, 33, 29, 34, 35, 3, 36, 37, 25, 3, 16, 3, 38, 14, 39, 24, 40, 3, 29, 33, 41, 3, 16, 42, 43, 44, 45, 3, 9, 46, 47, 48, 49, 50, 29, 3, 51, 52, 16, 3, 53, 3, 54, 14

Offset: 1

Antti Karttunen, Feb 20 2019

nonn

For all i, j: a(i) = a(j) => A324190(i) = A324190(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A297112, A297167, A297168, A061395, A324190, A324195, A324196.

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));
A324195(n) = { my(v=0); fordiv(n, d, v = bitor(v,A297112(d))); (v); };
Aux324197(n) = if(isprime(n),-(n%2)-1,A324195(n));
v324197 = rgs_transform(vector(up_to, n, Aux324197(n)));
A324197(n) = v324197[n];

A329372 Dirichlet convolution of the identity function with A156552.

Original entry on oeis.org

0, 1, 2, 5, 4, 12, 8, 17, 12, 22, 16, 44, 32, 40, 32, 49, 64, 61, 128, 78, 56, 76, 256, 132, 32, 142, 50, 136, 512, 152, 1024, 129, 104, 274, 88, 209, 2048, 532, 188, 230, 4096, 256, 8192, 252, 148, 1048, 16384, 356, 80, 159, 356, 454, 32768, 240, 160, 392, 680, 2078, 65536, 504, 131072, 4128, 248, 321, 280, 464, 262144, 858, 1328, 400

Offset: 1

Antti Karttunen, Nov 12 2019

nonn

Equally, Dirichlet convolution of sigma (A000203) with A297112 (Möbius transform of A156552).

Antti Karttunen, Table of n, a(n) for n = 1..1024
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A000203, A061395, A156552, A297112, A297167, A329374.

Cf. also A329371, A329373.

A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A329372(n) = sumdiv(n,d,(n/d)*A156552(d));

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));
A329372(n) = sumdiv(n,d,sigma(n/d)*A297112(d));

a(n) = Sum_{d|n} d * A156552(n/d).
a(n) = Sum_{d|n} A000203(n/d) * A297112(d).
A000265(a(n)) = A329374(n).

cp's OEIS Frontend

A297168 Difference between A156552 and its Moebius transform: a(n) = A156552(n) - A297112(n).

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

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A364557 Möbius transform of A005941.

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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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A324203 Lexicographically earliest sequence such that a(i) = a(j) => A324202(i) = A324202(j) for all i, j >= 1.

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A318891 Filter sequence combining the prime signature of n (A046523) with the largest prime factor of n (A006530).

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A324180 SumXOR variant of A297168.

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A324196 Lexicographically earliest sequence such that a(i) = a(j) => A324195(i) = A324195(j) for all i, j >= 1.

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A324197 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A324195(n) for all other numbers except f(2) = -1 and f(n) = -2 when n is an odd prime.

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