A346248 -id:A346248 - cp's OEIS frontend

A378755 Dirichlet convolution of A033879 and A346248.

1, 0, 1, 2, 1, 2, 3, 10, 11, 2, 1, 14, 3, 6, 7, 42, 1, 26, 3, 14, 17, 2, 5, 70, 15, 6, 81, 34, 1, 22, 5, 170, 7, 2, 17, 146, 3, 6, 17, 70, 1, 54, 3, 14, 69, 10, 5, 326, 55, 34, 7, 34, 5, 234, 7, 162, 17, 2, 1, 138, 5, 10, 155, 682, 17, 22, 3, 14, 27, 54, 1, 730, 5, 6, 89, 34, 17, 54, 3, 326, 571, 2, 5, 318, 7, 6, 7

Offset: 1

Antti Karttunen, Dec 11 2024

nonn

Cf. A033879, A346248, A378754 (Dirichlet inverse).

A033879(n) = (n+n-sigma(n));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A252748(n) = (A003961(n) - (2*n));
memoA346248 = Map();
A346248(n) = if(1==n,1,my(v); if(mapisdefined(memoA346248,n,&v), v, v = -sumdiv(n,d,if(dA252748(n/d)*A346248(d),0)); mapput(memoA346248,n,v); (v)));
A378755(n) = sumdiv(n,d,A033879(d)*A346248(n/d));

a(n) = Sum_{d|n} A033879(d)*A346248(n/d).

A346246 Dirichlet inverse of A344587, 2*A003961(n) - sigma(A003961(n)).

Original entry on oeis.org

1, -2, -4, -1, -6, 10, -10, -2, -3, 14, -12, 4, -16, 22, 26, -4, -18, 2, -22, 6, 42, 26, -28, 6, -5, 34, -6, 10, -30, -66, -36, -8, 50, 38, 62, 7, -40, 46, 66, 10, -42, -106, -46, 12, 14, 58, -52, 8, -9, 2, 74, 16, -58, -2, 74, 18, 90, 62, -60, -18, -66, 74, 26, -16, 98, -126, -70, 18, 114, -150, -72, 18, -78, 82, 12, 22

Offset: 1

Antti Karttunen, Jul 19 2021

Dirichlet inverse of the deficiency of prime shifted n.

Cf. A000203, A003961, A003973, A323910, A344587, A346247, A346251 (positions of zeros).

Cf. also A346235, A346248, A346254.

up_to = 16384;
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); }; \\ From A003961
A344587(n) = { my(u=A003961(n)); (u+u - sigma(u)); };
v346246 = DirInverseCorrect(vector(up_to,n,A344587(n)));
A346246(n) = v346246[n];

a(n) = A323910(A003961(n)).

a(n) = A346247(n) - A344587(n).

A346250 Sum of -A252748 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 1, 0, 2, 0, -3, 1, 6, 0, -11, 0, 6, 6, -17, 0, -23, 0, -17, 6, 18, 0, -39, 9, 18, -15, -25, 0, -48, 0, -51, 18, 30, 18, -49, 0, 30, 18, -77, 0, -72, 0, -35, -61, 34, 0, -85, 9, -31, 30, -43, 0, -123, 54, -97, 30, 54, 0, -117, 0, 50, -77, -89, 54, -96, 0, -53, 34, -104, 0, -19, 0, 66, -55, -61, 54, -120, 0

Offset: 1

Antti Karttunen, Jul 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. A003961, A252748, A346248.

Cf. also A323911, A346236, A346247, A346255.

up_to = 16384;
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); }; \\ From A003961
A252748(n) = (A003961(n) - (2*n));
v346248 = DirInverseCorrect(vector(up_to,n,-A252748(n)));
A346248(n) = v346248[n];
A346250(n) = (A346248(n)-A252748(n));

a(n) = A346248(n) - A252748(n).

A346235 Dirichlet inverse of A341530, gcd(nsigma(A003961(n)), sigma(n)A003961(n)).

Original entry on oeis.org

1, -1, -2, 0, -2, -32, -4, -4, 3, 2, -2, 34, -2, -16, -112, 8, -2, 125, -4, 0, 8, 0, -6, -128, 3, -14, -8, -124, -2, 8, -2, -10, -4, 2, -320, 920, -2, -4, 4, 8, -2, 64, -4, -358, 430, -12, -6, 528, -3, -5, -352, 16, -6, -368, -48, 224, 0, 2, -2, 104, -2, -12, -12, 28, -4, 16, -4, 0, -36, 400, -2, -440, -2, -2, 450, 8, -248, 128

Offset: 1

Antti Karttunen, Jul 11 2021

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Cf. A000203, A003961, A341530, A346236.

Cf. also A346246, A346248, A346254.

up_to = 65537;
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); }; \\ From A003961
A341530(n) = { my(t=A003961(n), s=sigma(t)); gcd((n*s), sigma(n)*t); };
v346235 = DirInverseCorrect(vector(up_to,n,A341530(n)));
A346235(n) = v346235[n];

A346254 Dirichlet inverse of A336849.

Original entry on oeis.org

1, -3, -5, 0, -7, 25, -11, 0, 0, 21, -13, -30, -17, 55, 35, 0, -19, -100, -23, 0, 55, 39, -29, 36, 0, 85, 0, -66, -31, -175, -37, 0, 65, 57, 77, 400, -41, 115, 85, 0, -43, -495, -47, 108, 0, 145, -53, -216, 0, 98, 171, -68, -59, 500, 169, 0, 115, 93, -61, 210, -67, 111, 0, 0, 119, -325, -71, 0, 261, -385, -73, -120, -79, 205, 0, -138

Offset: 1

Antti Karttunen, Jul 19 2021

sign

Cf. A000203, A003961, A003973, A336849, A346255, A346256 (positions of zeros).

Cf. also A346235, A346246, A346248.

up_to = 16384;
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); }; \\ From A003961
A336849(n) = { my(u=A003961(n)); (u/gcd(u, sigma(u))); };
v346254 = DirInverseCorrect(vector(up_to,n,A336849(n)));
A346254(n) = v346254[n];

a(n) = A346255(n) - A336849(n).

A346477 Dirichlet inverse of A346476.

Original entry on oeis.org

1, -1, -1, 2, -3, 5, -3, 2, 8, 13, -9, -2, -9, 17, 11, 8, -15, -8, -15, -12, 19, 37, -17, 18, 8, 41, -4, -12, -27, -33, -25, 20, 37, 61, 25, 56, -33, 65, 35, 38, -39, -45, -39, -42, -36, 77, -41, 32, 32, -20, 53, -42, -47, 96, 35, 58, 61, 109, -57, 132, -55, 109, -48, 56, 43, -121, -63, -72, 71, -109, -69, 56

Offset: 1

Antti Karttunen, Jul 29 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences generated by sieves

Cf. A000040, A000720, A062234, A250469, A252748, A346476, A346478.

Cf. also A323910, A346248, A346479.

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA346476(n) = (n+n-A250469(n));
v346477 = DirInverseCorrect(vector(up_to,n,A346476(n)));
A346477(n) = v346477[n];

a(1) = 1; and for n > 2, a(n) = -Sum_{d|n, dA346476(n/d).
a(n) = A346478(n) - A346476(n).
a(p) = A252748(p) = A346248(p) = -A346476(p) = -A062234(A000720(p)), for any prime p.

A347098 a(1) = 1; a(n) = -Sum_{d|n, d < n} A336853(n/d) * a(d), where A336853(n) = A003961(n) - n.

Original entry on oeis.org

1, -1, -2, -4, -2, -5, -4, -10, -12, -7, -2, -1, -4, -11, -12, -16, -2, -1, -4, -7, -18, -13, -6, 42, -20, -17, -42, -5, -2, 21, -6, -4, -24, -19, -26, 106, -4, -23, -30, 38, -2, 45, -4, -25, -10, -29, -6, 196, -56, -17, -36, -23, -6, 123, -28, 82, -42, -31, -2, 225, -6, -37, 4, 80, -38, 15, -4, -43, -52, 39, -2, 413

Offset: 1

Antti Karttunen, Aug 19 2021

sign

Dirichlet inverse of the pointwise sum of A336853 and A063524 (1, 0, 0, 0, ...).

Cf. A000040, A001223, A003961, A336853, A347099, A347100.

Cf. also A346248, A347096.

f[p_, e_] := NextPrime[p]^e; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#] * s[n/#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Nov 27 2021 *)

up_to = 16384;
A336853(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)-n); };
Aux347098(n) = if(1==n,n,A336853(n));
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA347098(n) = v347098[n];

a(1) = 1; and for n > 1, a(n) = -Sum_{d|n, d < n} A336853(n/d) * a(d).

For all n >= 1, a(A000040(n)) = -A001223(n).

A376404 Dirichlet inverse of 2*phi(n) - phi(A003961(n)), where phi is Euler totient function and A003961(n) is fully multiplicative function with a(prime(i)) = prime(i+1).

Original entry on oeis.org

1, 0, 0, 2, -2, 4, -2, 10, 8, 4, -8, 16, -8, 8, 8, 42, -14, 28, -14, 12, 16, 4, -16, 72, 6, 8, 64, 28, -26, 16, -24, 170, 8, 4, 20, 144, -32, 8, 16, 52, -38, 40, -38, 0, 40, 12, -40, 328, 30, 28, 8, 16, -46, 228, 24, 124, 16, 4, -56, 112, -54, 12, 96, 682, 32, -8, -62, -12, 24, 24, -68, 712, -66, 8, 56, 4, 32, 16

Offset: 1

Antti Karttunen, Nov 15 2024

sign

Cf. A000010, A003961, A003972.
Dirichlet inverse of -A349754, inverse Möbius transform of A346248.
Cf. also A323912.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A349754(n) = (eulerphi(A003961(n))-2*eulerphi(n));
memoA376404 = Map();
A376404(n) = if(1==n,1,my(v); if(mapisdefined(memoA376404,n,&v), v, v = -sumdiv(n,d,if(dA349754(n/d)*A376404(d),0)); mapput(memoA376404,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA349754(n/d) * a(d).

a(n) = Sum_{d|n} A346248(d).

cp's OEIS Frontend

A378755 Dirichlet convolution of A033879 and A346248.

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A346246 Dirichlet inverse of A344587, 2*A003961(n) - sigma(A003961(n)).

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A346250 Sum of -A252748 and its Dirichlet inverse.

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A346235 Dirichlet inverse of A341530, gcd(n*sigma(A003961(n)), sigma(n)*A003961(n)).

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A346254 Dirichlet inverse of A336849.

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A346477 Dirichlet inverse of A346476.

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A347098 a(1) = 1; a(n) = -Sum_{d|n, d < n} A336853(n/d) * a(d), where A336853(n) = A003961(n) - n.

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Formula

A376404 Dirichlet inverse of 2*phi(n) - phi(A003961(n)), where phi is Euler totient function and A003961(n) is fully multiplicative function with a(prime(i)) = prime(i+1).

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A346235 Dirichlet inverse of A341530, gcd(nsigma(A003961(n)), sigma(n)A003961(n)).