A349385 -id:A349385 - cp's OEIS frontend

A349386 a(n) = A349384(n) + A349385(n).

2, 0, 0, 1, 0, 4, 0, 3, 4, 6, 0, 2, 0, 10, 12, 7, 0, 0, 0, 3, 20, 12, 0, -4, 9, 16, 16, 5, 0, -36, 0, 15, 24, 18, 30, -10, 0, 22, 32, -6, 0, -60, 0, 6, 0, 28, 0, -20, 25, -3, 36, 8, 0, -20, 36, -10, 44, 30, 0, -48, 0, 36, 0, 31, 48, -72, 0, 9, 56, -90, 0, -32, 0, 40, -6, 11, 60, -96, 0, -30, 52, 42, 0, -80, 54, 46, 60

Offset: 1

Antti Karttunen, Nov 17 2021

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

Cf. A003961, A048673, A349384, A349385.

Cf. also A349383.

A349386(n) = (A349384(n)+A349385(n)); \\ Needs also code from A349384 and A349385.

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A349384(d) * A349385(n/d). [As the sequences are Dirichlet inverses of each other]

A336840 Inverse Möbius transform of A048673.

Original entry on oeis.org

1, 3, 4, 8, 5, 14, 7, 22, 17, 18, 8, 42, 10, 26, 26, 63, 11, 65, 13, 55, 38, 30, 16, 124, 30, 38, 80, 81, 17, 100, 20, 185, 44, 42, 50, 206, 22, 50, 56, 164, 23, 148, 25, 94, 127, 62, 28, 368, 68, 117, 62, 120, 31, 316, 58, 244, 74, 66, 32, 318, 35, 78, 189, 550, 74, 172, 37, 133, 92, 196, 38, 626, 41, 86, 174, 159

Offset: 1

Antti Karttunen, Aug 07 2020

nonn

Arithmetic mean of the number of divisors (A000005) and prime-shifted sigma (A003973), thus a(n) is the average between the number of and the sum of divisors of A003961(n).

The local minima occur on primes p, where p/2 < a(p) <= (p+1).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000005, A003961, A003973, A007503, A048673, A113415, A336838, A336839, A336841, A349371, A349384, A349385, A349910 [= a(n)-A048673(n)], A364063, A378520 (Dirichlet inverse).

A048673(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)+1)/2; };
A336840(n) = sumdiv(n,d,A048673(d));

A336840(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(numdiv(n)+sigma(factorback(f))); };

a(n) = Sum_{d|n} A048673(d).
a(n) = (1/2) * (A000005(n) + A003973(n)).
a(n) = A113415(A003961(n)). - Antti Karttunen, Jun 01 2022
a(n) = A349371(A003961(n)) = A364063(A048673(n)). - Antti Karttunen, Nov 30 2024

A349398 Dirichlet convolution of A048673 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, -7, -50, 55, 8, -24, 6, -41, -15, 58, 20, -17, -31, 108, 27, 70, -37, -24, 0, -20, -49, -98, 6, 26, -13, 21, -15, 62, 158, 84, -22, 9, -49, 130, -67, 12, -49, 62, -29, 112, 4, -60, 103, 16

Offset: 1

Antti Karttunen, Nov 19 2021

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Dirichlet convolution of A048673 with A349358, which is the Dirichlet inverse of A064216 (inverse permutation of A048673). Therefore, convolving A064216 with this sequence gives A048673.

Note how for n = 1 .. 35, a(n) = -A349397(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, A349397 (Dirichlet inverse), A349399 (sum with it).

Cf. also A349376, A349377, A349385.

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); };
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)));
A349398(n) = sumdiv(n,d,A048673(n/d)*A349358(d));

a(n) = Sum_{d|n} A048673(n/d) * A349358(d).

A349384 Dirichlet convolution of A003961 with the Dirichlet inverse of A048673, where A003961 is fully multiplicative with a(p) = nextprime(p), and A048673(n) = (1+A003961(n))/2.

Original entry on oeis.org

1, 1, 2, 2, 3, 0, 5, 4, 6, 0, 6, -2, 8, 0, 0, 8, 9, -4, 11, -3, 0, 0, 14, -8, 12, 0, 18, -5, 15, -12, 18, 16, 0, 0, 0, -14, 20, 0, 0, -12, 21, -20, 23, -6, -12, 0, 26, -24, 30, -9, 0, -8, 29, -24, 0, -20, 0, 0, 30, -24, 33, 0, -20, 32, 0, -24, 35, -9, 0, -30, 36, -36, 39, 0, -18, -11, 0, -32, 41, -36, 54, 0, 44

Offset: 1

Antti Karttunen, Nov 17 2021

sign

Convolving this with A336840 gives A003973.

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

Cf. A003961, A048673, A323893, A349385 (Dirichlet inverse), A349386 (sum with it).

Cf. also A003973, A336840, A349572.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A048673(n) = (A003961(n)+1)/2;
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)));
A349384(n) = sumdiv(n,d,A003961(n/d)*A323893(d));

a(n) = Sum_{d|n} A003961(n/d) * A323893(d).

a(n) = A349386(n) - A349385(n).

A349381 Dirichlet convolution of A003961 with A349125 (Dirichlet inverse of A064989), where A003961 and A064989 are fully multiplicative sequences that shift the prime factorization of n one step towards larger and smaller primes respectively.

Original entry on oeis.org

1, 2, 3, 6, 4, 6, 6, 18, 15, 8, 6, 18, 6, 12, 12, 54, 6, 30, 6, 24, 18, 12, 10, 54, 28, 12, 75, 36, 8, 24, 8, 162, 18, 12, 24, 90, 10, 12, 18, 72, 6, 36, 6, 36, 60, 20, 10, 162, 66, 56, 18, 36, 12, 150, 24, 108, 18, 16, 8, 72, 8, 16, 90, 486, 24, 36, 10, 36, 30, 48, 6, 270, 8, 20, 84, 36, 36, 36, 10, 216, 375, 12

Offset: 1

Antti Karttunen, Nov 17 2021

Multiplicative because both A003961 and A349125 are.

Convolving this with A349127 gives A003972.

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

Cf. A003961, A064989, A349125, A349382 (Dirichlet inverse), A349383 (sum with it).

Cf. also A003972, A349127, and A349355, A349356 and A349384, A349385, and A349387.

A003961(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)); 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); };
A349125(n) = (moebius(n)*A064989(n));
A349381(n) = sumdiv(n,d,A003961(n/d)*A349125(d));

a(n) = Sum_{d|n} A003961(n/d) * A349125(d).

a(n) = A349383(n) - A349382(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

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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).

A378220 Dirichlet inverse of phi(A003961(n)), where A003961 is fully multiplicative function with a(prime(i)) = prime(i+1).

Original entry on oeis.org

1, -2, -4, -2, -6, 8, -10, -2, -4, 12, -12, 8, -16, 20, 24, -2, -18, 8, -22, 12, 40, 24, -28, 8, -6, 32, -4, 20, -30, -48, -36, -2, 48, 36, 60, 8, -40, 44, 64, 12, -42, -80, -46, 24, 24, 56, -52, 8, -10, 12, 72, 32, -58, 8, 72, 20, 88, 60, -60, -48, -66, 72, 40, -2, 96, -96, -70, 36, 112, -120, -72, 8, -78, 80, 24

Offset: 1

Antti Karttunen, Nov 23 2024

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

Cf. A023900, A003961, A048673, A151800, A349385, A378216.
Dirichlet inverse of A003972.
Inverse Möbius transform of A346234.
After the initial term, A349385 doubled.
Cf. A346246, A378216.

f[p_, e_] := 1 - NextPrime[p]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 13 2025 *)

A378220(n) = factorback(apply(p -> 1-nextprime(1+p), factor(n)[, 1]));

from math import prod
from sympy import nextprime, primefactors
def A378220(n): return prod(1-nextprime(p) for p in primefactors(n)) # Chai Wah Wu, Nov 23 2024

Multiplicative with a(p^e) = (1-q), where q = A151800(p), i.e., the least prime > p.
a(n) = A023900(A003961(n)).
For n > 1, a(n) = 2*A349385(n).
a(n) = Sum_{d|n} A346234(d).
a(n) = Sum_{d|n} A346246(d)*A378216(n/d).

cp's OEIS Frontend

A349386 a(n) = A349384(n) + A349385(n).

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A336840 Inverse Möbius transform of A048673.

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A349398 Dirichlet convolution of A048673 with the Dirichlet inverse of its inverse permutation.

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A349384 Dirichlet convolution of A003961 with the Dirichlet inverse of A048673, where A003961 is fully multiplicative with a(p) = nextprime(p), and A048673(n) = (1+A003961(n))/2.

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A349381 Dirichlet convolution of A003961 with A349125 (Dirichlet inverse of A064989), where A003961 and A064989 are fully multiplicative sequences that shift the prime factorization of n one step towards larger and smaller primes respectively.

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

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A378220 Dirichlet inverse of phi(A003961(n)), where A003961 is fully multiplicative function with a(prime(i)) = prime(i+1).

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