A349623 -id:A349623 - cp's OEIS frontend

A366295 Lexicographically earliest infinite sequence such that a(i) = a(j) => A349623(i) = A349623(j) for all i, j >= 1, where A349623 is the Dirichlet inverse of A064989(sigma(A003961(n))).

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

Offset: 1

Antti Karttunen, Oct 07 2023

nonn

Restricted growth sequence transform of A349623.

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

Cf. A000203, A003961, A064989, A326042, A349623.

Cf. also A286603, A366294.

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; };
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A326042(n) = A064989(sigma(A003961(n)));
v366295 = rgs_transform(DirInverseCorrect(vector(up_to,n,A326042(n))));
A366295(n) = v366295[n];

A349625 Dirichlet convolution of A000027 (identity function) with the Dirichlet inverse of A326042, where A326042(n) = A064989(sigma(A003961(n))).

Original entry on oeis.org

1, 1, 1, -8, 4, 1, 5, 2, -22, 4, 6, -8, 9, 5, 4, 50, 14, -22, 17, -32, 5, 6, 17, 2, -13, 9, 20, -40, 28, 4, 14, -120, 6, 14, 20, 176, 27, 17, 9, 8, 34, 5, 41, -48, -88, 17, 39, 50, -46, -13, 14, -72, 47, 20, 24, 10, 17, 28, 30, -32, 48, 14, -110, -1126, 36, 6, 63, -112, 17, 20, 40, -44, 70, 27, -13, -136, 30, 9, 69

Offset: 1

Antti Karttunen, Nov 26 2021

Multiplicative because A000027 and A349623 are.

Cf. A000203, A003961, A064989, A326042, A349623, A349624 (Dirichlet inverse).

f1[p_, e_] := NextPrime[p]^e; s1[1] = 1; s1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[2, e_] := 1; f2[p_, e_] := NextPrime[p, -1]^e; s2[1] = 1; s2[n_] := Times @@ f2 @@@ FactorInteger[n]; s[n_] := s2[DivisorSigma[1, s1[n]]]; sinv[1] = 1; sinv[n_] := sinv[n] = -DivisorSum[n, sinv[#] * s[n/#] &, # < n &]; a[n_] := DivisorSum[n, # * sinv[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 27 2021 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
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)};
A326042(n) = A064989(sigma(A003961(n)));
memoA349623 = Map();
A349623(n) = if(1==n,1,my(v); if(mapisdefined(memoA349623,n,&v), v, v = -sumdiv(n,d,if(dA326042(n/d)*A349623(d),0)); mapput(memoA349623,n,v); (v)));
A349625(n) = sumdiv(n,d,d*A349623(n/d));

a(n) = Sum_{d|n} d * A349623(n/d).

A349626 Möbius transform of A326042, where A326042(n) = A064989(sigma(A003961(n))).

Original entry on oeis.org

1, 0, 1, 10, 0, 0, 1, -8, 27, 0, 4, 10, 3, 0, 0, 46, 2, 0, 1, 0, 1, 0, 5, -8, 33, 0, -7, 10, 0, 0, 16, 6, 4, 0, 0, 270, 9, 0, 3, 0, 6, 0, 1, 40, 0, 0, 7, 46, 83, 0, 2, 30, 5, 0, 0, -8, 1, 0, 28, 0, 12, 0, 27, 1036, 0, 0, 3, 20, 5, 0, 30, -216, 2, 0, 33, 10, 4, 0, 9, 0, 447, 0, 11, 10, 0, 0, 0, -32, 24, 0, 3, 50, 16

Offset: 1

Antti Karttunen, Nov 26 2021

Dirichlet convolution of Euler phi (A000010) with A349624.

Multiplicative because A326042 is.

Cf. A000010, A003961, A008683, A064989, A326042, A349623, A349624.

f1[p_, e_] := NextPrime[p]^e; s1[1] = 1; s1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[2, e_] := 1; f2[p_, e_] := NextPrime[p, -1]^e; s2[1] = 1; s2[n_] := Times @@ f2 @@@ FactorInteger[n]; s[n_] := s2[DivisorSigma[1, s1[n]]]; a[n_] := DivisorSum[n, s[#] * MoebiusMu[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 27 2021 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
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)};
A326042(n) = A064989(sigma(A003961(n)));
A349626(n) = sumdiv(n,d,moebius(n/d)*A326042(d));

a(n) = Sum_{d|n} A008683(n/d) * A326042(d).

a(n) = Sum_{d|n} A000010(n/d) * A349624(d).

cp's OEIS Frontend

A366295 Lexicographically earliest infinite sequence such that a(i) = a(j) => A349623(i) = A349623(j) for all i, j >= 1, where A349623 is the Dirichlet inverse of A064989(sigma(A003961(n))).

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A349625 Dirichlet convolution of A000027 (identity function) with the Dirichlet inverse of A326042, where A326042(n) = A064989(sigma(A003961(n))).

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A349626 Möbius transform of A326042, where A326042(n) = A064989(sigma(A003961(n))).

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