A349398 -id:A349398 - cp's OEIS frontend

A349399 a(n) = A349397(n) + A349398(n).

2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, 16, 25, 0, 0, 0, 0, 0, 0, -80, 0, 0, 0, -12, 0, 0, 0, 64, 0, -6, 0, 0, 0, 60, 0, 4, 0, 0, 0, 0, 30, 0, 0, -96, 0, 0, 0, 18, 0, 0, 0, -48, 0, -10, 0, 0, 0, 0, 0, 24, 0, -190

Offset: 1

Antti Karttunen, Nov 19 2021

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

Cf. A048673, A064216, A349397, A349398.

A349399(n) = (A349397(n) + A349398(n)); \\ Needs also code from A349397 and A349398.

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A349397(d) * A349398(n/d). [As the sequences are Dirichlet inverses of each other]

A349614 Dirichlet convolution of A064664 (the inverse permutation of EKG-permutation, A064413) with the Dirichlet inverse of A064413.

Original entry on oeis.org

1, 0, 1, -3, 7, -7, 2, 6, -8, -10, 5, 9, 14, 2, -41, -1, 17, 27, 15, -6, -38, -18, 13, 10, -32, -29, 18, 33, 18, 62, 29, -13, -31, -53, -107, 25, 48, -51, -86, 13, 30, 116, 58, 23, 88, -34, 37, -47, -30, 56, -113, 3, 45, -39, -137, -154, -73, -67, 41, 160, 84, -91, 174, 56, -154, 152, 91, 6, -113, 246, 58, -185, 56

Offset: 1

Antti Karttunen, Nov 23 2021

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Obviously, convolving this with A064413 gives its inverse permutation A064664.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to EKG sequence

Cf. A064413, A064664, A349400, A349613 (Dirichlet inverse), A349615 (sum with it), A349617.

Cf. also pairs A349376, A349377 and A349397, A349398 for similar constructions.

up_to = 32768;
v064413 = readvec("b064413_upto65539_terms_only.txt"); \\ Data prepared with Chai Wah Wu's Dec 08 2014 Python-program given in A064413.
A064413(n) = v064413[n];
\\ Then its inverse A064664 is prepared:
m064664 = Map();
for(n=1,65539,mapput(m064664,A064413(n),n));
A064664(n) = mapget(m064664,n);
memoA349400 = Map();
A349400(n) = if(1==n,1,my(v); if(mapisdefined(memoA349400,n,&v), v, v = -sumdiv(n,d,if(dA064413(n/d)*A349400(d),0)); mapput(memoA349400,n,v); (v)));
A349614(n) = sumdiv(n,d,A064664(d)*A349400(n/d));

a(n) = Sum_{d|n} A064664(d) * A349400(n/d).

A349376 Dirichlet convolution of A006368 with the Dirichlet inverse of A006369, where A006368 is the "amusical permutation", and A006369 is its inverse permutation.

Original entry on oeis.org

1, 0, 0, 1, -3, 5, -4, -2, 1, 11, -7, -7, -7, 14, 7, 4, -10, 2, -11, -22, 10, 25, -14, 16, 7, 25, 0, -26, -17, -41, -18, -8, 17, 36, 34, 7, -21, 39, 17, 52, -24, -52, -25, -48, 1, 50, -28, -36, 8, -51, 24, -48, -31, 7, 62, 60, 27, 61, -35, 136, -35, 64, 0, 16, 62, -93, -39, -70, 34, -178, -42, -26, -42, 75, -27, -74

Offset: 1

Antti Karttunen, Nov 17 2021

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Obviously, convolving this sequence with A006369 gives its inverse A006368 from n >= 1 onward.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A006368, A006369, A349368, A349377 (Dirichlet inverse), A349378 (sum with it).

Cf. also pairs A349613, A349614 and A349397, A349398 for similar constructions.

A006368(n) = ((3*n)+(n%2))\(2+((n%2)*2));
A006369(n) = if(!(n%3),(2/3)*n,(1/3)*if(1==(n%3),((4*n)-1),((4*n)+1)));
memoA349368 = Map();
A349368(n) = if(1==n,1,my(v); if(mapisdefined(memoA349368,n,&v), v, v = -sumdiv(n,d,if(dA006369(n/d)*A349368(d),0)); mapput(memoA349368,n,v); (v)));
A349376(n) = sumdiv(n,d,A006368(d)*A349368(n/d));

a(n) = Sum_{d|n} A006368(d) * A349368(n/d).

A349377 Dirichlet convolution of A006369 with the Dirichlet inverse of A006368, where A006368 is the "amusical permutation", and A006369 is its inverse permutation.

Original entry on oeis.org

1, 0, 0, -1, 3, -5, 4, 2, -1, -11, 7, 7, 7, -14, -7, -3, 10, -2, 11, 16, -10, -25, 14, -6, 2, -25, 0, 18, 17, 11, 18, 4, -17, -36, -10, 20, 21, -39, -17, -18, 24, 12, 25, 34, -7, -50, 28, 2, 8, -15, -24, 34, 31, 3, -20, -16, -27, -61, 35, 30, 35, -64, -8, -5, -20, 23, 39, 50, -34, 6, 42, -44, 42, -75, -15, 52, -22, 23

Offset: 1

Antti Karttunen, Nov 17 2021

sign

Obviously, convolving this sequence with A006368 gives its inverse A006369 from n >= 1 onward.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A006368, A006369, A349351, A349376 (Dirichlet inverse), A349378 (sum with it).

Cf. also pairs A349613, A349614 and A349397, A349398 for similar constructions.

A006368(n) = ((3*n)+(n%2))\(2+((n%2)*2));
A006369(n) = if(!(n%3),(2/3)*n,(1/3)*if(1==(n%3),((4*n)-1),((4*n)+1)));
memoA349351 = Map();
A349351(n) = if(1==n,1,my(v); if(mapisdefined(memoA349351,n,&v), v, v = -sumdiv(n,d,if(dA006368(n/d)*A349351(d),0)); mapput(memoA349351,n,v); (v)));
A349377(n) = sumdiv(n,d,A006369(d)*A349351(n/d));

a(n) = Sum_{d|n} A006369(d) * A349351(n/d).

a(n) = A349378(n) - A349376(n).

A349397 Dirichlet convolution of A064216 with the Dirichlet inverse of its inverse permutation.

Original entry on oeis.org

1, 0, 0, 0, 0, -1, 5, -8, 0, 6, 3, -2, 0, -19, 5, 4, 4, -20, 19, -22, -6, 15, -3, 8, 0, 0, -16, -16, 18, -24, 40, -70, -9, 24, -21, 8, 50, -55, -8, 24, -6, 31, 15, -58, -20, 17, 31, -92, -2, -70, 37, 24, 0, 20, 49, 18, -6, -26, 13, -33, 15, -62, -158, -20, 22, -15, 49, -130, 67, 48, 49, -58, 29, -112, -4, 60, -73, -16

Offset: 1

Antti Karttunen, Nov 19 2021

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Dirichlet convolution of A064216 with A323893, which is the Dirichlet inverse of A048673. Therefore, convolving A048673 with this sequence gives A064216.

Note how for n = 1 .. 35, a(n) = -A349398(n).

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

Cf. A003961, A048673, A064216, A064989, A323893, A349398 (Dirichlet inverse), A349399 (sum with it), A349384.

Cf. also pairs A349376, A349377 and A349613, A349614 for similar constructions.

A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(1+factorback(f)); };
A064216(n) = { my(f = factor(n+n-1)); 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); };
memoA323893 = Map();
A323893(n) = if(1==n,1,my(v); if(mapisdefined(memoA323893,n,&v), v, v = -sumdiv(n,d,if(dA048673(n/d)*A323893(d),0)); mapput(memoA323893,n,v); (v)));
A349397(n) = sumdiv(n,d,A064216(n/d)*A323893(d));

a(n) = Sum_{d|n} A064216(n/d) * A323893(d).

A349613 Dirichlet convolution of A064413 (EKG-permutation) with the Dirichlet inverse of its inverse permutation.

Original entry on oeis.org

1, 0, -1, 3, -7, 7, -2, -6, 9, 10, -5, -15, -14, -2, 55, 10, -17, -41, -15, -36, 42, 18, -13, 44, 81, 29, -35, -45, -18, -180, -29, -23, 41, 53, 135, 99, -48, 51, 114, 131, -30, -140, -58, -53, -303, 34, -37, -120, 34, -196, 147, -87, -45, 226, 207, 166, 103, 67, -41, 466, -84, 91, -288, 13, 350, -258, -91, -108

Offset: 1

Antti Karttunen, Nov 23 2021

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Obviously, convolving this with A064664 gives A064413 back.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to EKG sequence

Cf. A064413, A064664, A323411, A349614 (Dirichlet inverse), A349615 (sum with it), A349616.

Cf. also pairs A349376, A349377 and A349397, A349398 for similar constructions.

up_to = 32768;
v064413 = readvec("b064413_upto65539_terms_only.txt"); \\ Data prepared with Chai Wah Wu's Dec 08 2014 Python-program given in A064413.
A064413(n) = v064413[n];
\\ Then its inverse A064664 is prepared:
m064664 = Map();
for(n=1,65539,mapput(m064664,A064413(n),n));
A064664(n) = mapget(m064664,n);
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA064664(n)));
A323411(n) = v323411[n];
A349613(n) = sumdiv(n,d,A064413(d)*A323411(n/d));

a(n) = Sum_{d|n} A064413(d) * A323411(n/d).

A349358 Dirichlet inverse of A064216, which is A064989(2n-1), where A064989 is fully multiplicative with a(2) = 1 and a(p) = prevprime(p) for odd primes p.

Original entry on oeis.org

1, -2, -3, -1, -4, 5, -11, 6, -4, -1, -10, 3, -9, 36, 1, -24, -14, 25, -31, 38, 29, -1, -12, -29, -9, 10, 4, -11, -34, 53, -59, 62, 27, -5, 50, -41, -71, 106, 19, -83, -16, -125, -39, 98, 51, -7, -58, 184, 32, 112, -13, -15, -30, -84, -27, -170, 77, 79, -44, -109, -49, 162, 184, -84, -10, 31, -85, 192, -59, -75, -86

Offset: 1

Antti Karttunen, Nov 17 2021

sign

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

Cf. A064216, A064989, A349359, A349398.

Cf. also A323893, A349125.

A064989(n) = { my(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); };
A064216(n) = A064989((2*n)-1);
memoA349358 = Map();
A349358(n) = if(1==n,1,my(v); if(mapisdefined(memoA349358,n,&v), v, v = -sumdiv(n,d,if(dA064216(n/d)*A349358(d),0)); mapput(memoA349358,n,v); (v)));

a(1) = 1; a(n) = -Sum_{d|n, d < n} A064216(n/d) * a(d).

a(n) = A349359(n) - A064216(n).

A349571 Dirichlet convolution of A048673 with A055615 (Dirichlet inverse of n).

Original entry on oeis.org

1, 0, 0, 1, -1, 2, -1, 4, 4, 3, -4, 4, -4, 5, 6, 13, -7, 6, -7, 5, 10, 6, -8, 10, 5, 8, 24, 9, -13, -2, -12, 40, 12, 9, 16, 16, -16, 11, 16, 11, -19, -2, -19, 8, 14, 14, -20, 28, 19, 9, 18, 12, -23, 26, 22, 21, 22, 15, -28, 2, -27, 18, 26, 121, 28, -8, -31, 11, 28, -8, -34, 46, -33, 20, 18, 15, 34, -8, -37, 29, 124

Offset: 1

Antti Karttunen, Nov 23 2021

sign

Also Dirichlet convolution of A349385 with A349387.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A048673, A055615, A349385, A349387, A349572 (Dirichlet inverse).

Cf. also A349398, A349573.

f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := (1 + Times @@ f @@@ FactorInteger[n])/2; a[n_] := DivisorSum[n, # * MoebiusMu[#] * s[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 23 2021 *)

A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(1+factorback(f)); };
A055615(n) = (n*moebius(n));
A349571(n) = sumdiv(n,d,A048673(n/d)*A055615(d));

a(n) = Sum_{d|n} A048673(n/d) * A055615(d).

a(n) = Sum_{d|n} A349385(n/d) * A349387(d).

cp's OEIS Frontend

A349399 a(n) = A349397(n) + A349398(n).

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A349614 Dirichlet convolution of A064664 (the inverse permutation of EKG-permutation, A064413) with the Dirichlet inverse of A064413.

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A349376 Dirichlet convolution of A006368 with the Dirichlet inverse of A006369, where A006368 is the "amusical permutation", and A006369 is its inverse permutation.

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A349377 Dirichlet convolution of A006369 with the Dirichlet inverse of A006368, where A006368 is the "amusical permutation", and A006369 is its inverse permutation.

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A349397 Dirichlet convolution of A064216 with the Dirichlet inverse of its inverse permutation.

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A349613 Dirichlet convolution of A064413 (EKG-permutation) with the Dirichlet inverse of its inverse permutation.

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A349358 Dirichlet inverse of A064216, which is A064989(2n-1), where A064989 is fully multiplicative with a(2) = 1 and a(p) = prevprime(p) for odd primes p.

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A349571 Dirichlet convolution of A048673 with A055615 (Dirichlet inverse of n).

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