A346234 -id:A346234 - cp's OEIS frontend

A349382 Dirichlet convolution of A064989 with A346234 (Dirichlet inverse of A003961), where A003961 and A064989 are fully multiplicative sequences that shift the prime factorization of n one step towards larger and smaller primes respectively.

1, -2, -3, -2, -4, 6, -6, -2, -6, 8, -6, 6, -6, 12, 12, -2, -6, 12, -6, 8, 18, 12, -10, 6, -12, 12, -12, 12, -8, -24, -8, -2, 18, 12, 24, 12, -10, 12, 18, 8, -6, -36, -6, 12, 24, 20, -10, 6, -30, 24, 18, 12, -12, 24, 24, 12, 18, 16, -8, -24, -8, 16, 36, -2, 24, -36, -10, 12, 30, -48, -6, 12, -8, 20, 36, 12, 36, -36

Offset: 1

Antti Karttunen, Nov 17 2021

Multiplicative because both A064989 and A346234 are.

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

Cf. A003961, A064989, A151799, A151800, A346234, A349381 (Dirichlet inverse), A349383 (sum with it).

Cf. also A349355, A349356.

f[p_, e_] := If[p == 2, -2, NextPrime[p, -1]^e - NextPrime[p]*NextPrime[p, -1]^(e - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 17 2021 *)

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); };
A346234(n) = (moebius(n)*A003961(n));
A349382(n) = sumdiv(n,d,A064989(n/d)*A346234(d));

a(n) = Sum_{d|n} A064989(n/d) * A346234(d).
a(n) = A349383(n) - A349381(n).
Multiplicative with a(p^e) = -2 if p = 2, and prevprime(p)^e - nextprime(p) * prevprime(p)^(e-1) otherwise, where prevprime function is A151799 and nextprime function is A151800. - Amiram Eldar, Nov 17 2021

A349388 Dirichlet convolution of A000027 with A346234 (Dirichlet inverse of A003961), where A003961 is fully multiplicative with a(p) = nextprime(p).

Original entry on oeis.org

1, -1, -2, -2, -2, 2, -4, -4, -6, 2, -2, 4, -4, 4, 4, -8, -2, 6, -4, 4, 8, 2, -6, 8, -10, 4, -18, 8, -2, -4, -6, -16, 4, 2, 8, 12, -4, 4, 8, 8, -2, -8, -4, 4, 12, 6, -6, 16, -28, 10, 4, 8, -6, 18, 4, 16, 8, 2, -2, -8, -6, 6, 24, -32, 8, -4, -4, 4, 12, -8, -2, 24, -6, 4, 20, 8, 8, -8, -4, 16, -54, 2, -6, -16, 4, 4, 4

Offset: 1

Antti Karttunen, Nov 17 2021

Multiplicative because A000027 and A346234 are.

Cf. A000027, A000040, A001223, A003961, A151800, A346234, A349387 (Dirichlet inverse), A349389 (sum with it), A378607 (inverse Möbius transform).

Cf. also A347238.

f[p_, e_] := p^e - NextPrime[p] * p^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A346234(n) = (moebius(n)*A003961(n));
A349388(n) = sumdiv(n,d,d*A346234(n/d));

a(n) = Sum_{d|n} d * A346234(n/d).
For all n >= 1, a(A000040(n)) = -A001223(n).
Multiplicative with a(p^e) = p^e - nextprime(p) * p^(e-1), where nextprime function is A151800. - Amiram Eldar, Nov 18 2021

A378607 Dirichlet convolution of sigma and the Dirichlet inverse of A003961 (A346234).

Original entry on oeis.org

1, 0, -1, -2, -1, 0, -3, -6, -7, 0, -1, 2, -3, 0, 1, -14, -1, 0, -3, 2, 3, 0, -5, 6, -11, 0, -25, 6, -1, 0, -5, -30, 1, 0, 3, 14, -3, 0, 3, 6, -1, 0, -3, 2, 7, 0, -5, 14, -31, 0, 1, 6, -5, 0, 1, 18, 3, 0, -1, -2, -5, 0, 21, -62, 3, 0, -3, 2, 5, 0, -1, 42, -5, 0, 11, 6, 3, 0, -3, 14, -79, 0, -5, -6, 1, 0, 1, 6, -7, 0, 9, 10

Offset: 1

Antti Karttunen, Dec 11 2024

Cf. A000203, A003961, A016825, A151800, A346234, A378606 (Dirichlet inverse).

Inverse Möbius transform of A349388.

f[p_, e_] := (p^(e + 1) - NextPrime[p]*(p^e - 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 12 2025 *)

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A346234(n) = (moebius(n)*A003961(n));
A378607(n) = sumdiv(n,d,sigma(d)*A346234(n/d));

a(n) = Sum_{d|n} A000203(d)*A346234(n/d).
a(n) = Sum_{d|n} A349388(d).
Multiplicative with a(p^e) = (p^(e+1) - nextprime(p)*(p^e-1) - 1)/(p-1), where nextprime(p) = A151800(p). - Amiram Eldar, Jan 12 2025

A349632 Dirichlet convolution of A250469 with A346234, which is Dirichlet inverse of A003961.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, -6, 0, 6, 0, -12, 0, 6, 0, -18, 0, -24, 0, -24, 0, 24, 0, 0, 0, 24, -60, -36, 0, -48, 0, -42, 20, 42, 0, 12, 0, 42, 10, -12, 0, -72, 0, -60, -60, 48, 0, 24, 0, -42, 30, -72, 0, 84, 0, -12, 30, 78, 0, 120, 0, 72, -120, -90, 0, -180, 0, -96, 30, -132, 0, 48, 0, 96, -60, -108, 0, -174, 0, 12, -120

Offset: 1

Antti Karttunen, Nov 27 2021

sign

Note that for n = 2..36, a(n) = -A349631(n).

Dirichlet convolution of this sequence with A003972 is A347376.

Cf. A003961, A250469, A346234, A349631 (Dirichlet inverse).
Cf. also A003972, A347376, A349382.
Cf. also arrays A083221, A246278, A249821, A249822 and permutations A250245, A250246.

up_to = 20000;
A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A250469(n) = if(1==n,n,my(spn = nextprime(1+A020639(n)), c = A078898(n), k = 0); while(c, k++; if((1==k)||(A020639(k)>=spn),c -= 1)); (k*spn));
A346234(n) = (moebius(n)*A003961(n));
A349632(n) = sumdiv(n,d,A250469(n/d)*A346234(d));

a(n) = Sum_{d|n} A250469(d) * A346234(n/d).

A153881 1 followed by -1, -1, -1, ... .

Original entry on oeis.org

1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1

Offset: 1

Mats Granvik, Jan 03 2009

Dirichlet inverse of A074206.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1).

If prefixed by initial 0, we get A134824.
Cf. A074206 (Dirichlet inverse).
Cf. A033879, A055615, A057427, A063524, A344587, A346234.

[1 - 2*Sign(n-1) : n in [1..100]]; // Wesley Ivan Hurt, Jun 20 2014

A153881:=n->`if`(n=1, 1, -1); seq(A153881(n), n=1..100); # Wesley Ivan Hurt, Jun 20 2014

Table[1 - 2 Sign[n - 1], {n, 100}] (* Wesley Ivan Hurt, Jun 20 2014 *)

a(n)=if(n>1,-1,1) \\ Charles R Greathouse IV, Sep 24 2015

G.f: x*(1-2*x)/(1-x). - Mats Granvik, Mar 09 2009, rewritten R. J. Mathar, Mar 31 2010
a(n) = (-1)^A000040(n). - Juri-Stepan Gerasimov, Sep 10 2009
G.f.: x / (1 + x / (1 - 2*x)). - Michael Somos, Apr 02 2012
From Wesley Ivan Hurt, Jun 20 2014: (Start)
a(1) = 1; a(n) = -1, n > 1.
a(n) = 1 - 2*sign(n-1) = 1 - 2*A057427(n-1).
a(n) = (-1)^sign(1-n) = (-1)^A057427(1-n).
a(n) = 2*floor(1/n)-1 = 2*A063524(n)-1. (End)
Dirichlet g.f.: 2 - zeta(s). - Álvar Ibeas, Dec 30 2018
a(n) = Sum_{d|n} A033879(d)*A055615(n/d) = Sum_{d|n} A344587(d)*A346234(n/d). - Antti Karttunen, Nov 22 2024

Edited by Charles R Greathouse IV, Mar 18 2010

More terms from Antti Karttunen, Nov 22 2024

A349125 Dirichlet inverse of A064989, where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p.

Original entry on oeis.org

1, -1, -2, 0, -3, 2, -5, 0, 0, 3, -7, 0, -11, 5, 6, 0, -13, 0, -17, 0, 10, 7, -19, 0, 0, 11, 0, 0, -23, -6, -29, 0, 14, 13, 15, 0, -31, 17, 22, 0, -37, -10, -41, 0, 0, 19, -43, 0, 0, 0, 26, 0, -47, 0, 21, 0, 34, 23, -53, 0, -59, 29, 0, 0, 33, -14, -61, 0, 38, -15, -67, 0, -71, 31, 0, 0, 35, -22, -73, 0, 0, 37, -79

Offset: 1

Antti Karttunen, Nov 13 2021

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

Cf. A008683, A064989, A151800, A349126, A349127.

Cf. also A055615, A346234, A349134.

f[p_, e_] := If[e == 1, If[p == 2, -1, -NextPrime[p, -1]], 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 13 2021 *)

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));

A349125(n) = { my(f = factor(n)); prod(i=1, #f~, if(1

from sympy import prevprime, factorint, prod
def f(p, e):
    return 0 if e > 1 else -1 if p == 2 else -prevprime(p)
def a(n):
    return prod(f(p, e) for p, e in factorint(n).items()) # Sebastian Karlsson, Nov 13 2021

a(1) = 1; a(n) = -Sum_{d|n, d < n} A064989(n/d) * a(d).
a(n) = A349126(n) - A064989(n).
Multiplicative with a(p^e) = 0 if e > 1, -1 if p = 2 and -prevprime(p) otherwise. - Sebastian Karlsson, Nov 13 2021
a(n) = A008683(n) * A064989(n). [Because A064989 is fully multiplicative. See "Properties" section in the Wikipedia article]

A349385 Dirichlet convolution of A048673 with the Dirichlet inverse of A003961, 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, -1, -3, 4, -5, -1, -2, 6, -6, 4, -8, 10, 12, -1, -9, 4, -11, 6, 20, 12, -14, 4, -3, 16, -2, 10, -15, -24, -18, -1, 24, 18, 30, 4, -20, 22, 32, 6, -21, -40, -23, 12, 12, 28, -26, 4, -5, 6, 36, 16, -29, 4, 36, 10, 44, 30, -30, -24, -33, 36, 20, -1, 48, -48, -35, 18, 56, -60, -36, 4, -39, 40, 12, 22, 60

Offset: 1

Antti Karttunen, Nov 17 2021

sign

Convolving this with A003973 gives A336840.

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

Cf. A003961, A048673, A346234, A349384 (Dirichlet inverse), A349386 (sum with it).

Cf. also A003973, A336840.

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;
A346234(n) = (moebius(n)*A003961(n));
A349385(n) = sumdiv(n,d,A048673(n/d)*A346234(d));

a(n) = Sum_{d|n} A048673(n/d) * A346234(d).

a(n) = A349386(n) - A349384(n).

A346479 Dirichlet inverse of A250469.

Original entry on oeis.org

1, -3, -5, 0, -7, 15, -11, 6, 0, 15, -13, 12, -17, 27, 35, 0, -19, 24, -23, 42, 55, 15, -29, -66, 0, 27, 60, 54, -31, -27, -37, -12, 45, 15, 77, -144, -41, 27, 75, -102, -43, -63, -47, 132, 60, 39, -53, -24, 0, 84, 65, 144, -59, -384, 91, -162, 85, 15, -61, -558, -67, 39, 120, 0, 119, 165, -71, 222, 115, 9, -73, 168

Offset: 1

Antti Karttunen, Jul 30 2021

sign

Not all zeros occur on squares. For example, a(1445) = a(5 * 17^2) = 0.

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

Cf. A250469, A346480.

Cf. also A346234, A346477.

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(dA250469(n)));
A346479(n) = v346479[n];

a(1) = 1; and for n > 2, a(n) = -Sum_{d|n, dA250469(n/d).

a(n) = A346480(n) - A250469(n).

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).

A378221 Dirichlet inverse of sigma(A003961(n)), where A003961 is fully multiplicative function with a(prime(i)) = prime(i+1).

Original entry on oeis.org

1, -4, -6, 3, -8, 24, -12, 0, 5, 32, -14, -18, -18, 48, 48, 0, -20, -20, -24, -24, 72, 56, -30, 0, 7, 72, 0, -36, -32, -192, -38, 0, 84, 80, 96, 15, -42, 96, 108, 0, -44, -288, -48, -42, -40, 120, -54, 0, 11, -28, 120, -54, -60, 0, 112, 0, 144, 128, -62, 144, -68, 152, -60, 0, 144, -336, -72, -60, 180, -384, -74, 0

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.

Dirichlet inverse of A003973.

Cf. A003961, A046692, A151800, A346234, A378220.

f[p_, e_] := Which[e == 1, -NextPrime[p]-1, e == 2, NextPrime[p], e >= 3, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 12 2025 *)

A378221(n) = { my(f=factor(n)~); prod(i=1, #f, if(1==f[2,i], -(nextprime(1+f[1,i])+1), if(2==f[2,i], nextprime(1+f[1,i]), 0))); };

from math import prod
from sympy import nextprime, factorint
def A378221(n): return 0 if any(map(lambda x:x>2,(f:=factorint(n)).values())) else prod(-nextprime(p)-1 if e&1 else nextprime(p) for p,e in f.items()) # Chai Wah Wu, Nov 23 2024

Multiplicative with a(p) = -q-1, a(p^2) = q, a(p^k) = 0 for k > 2, where q = A151800(p), the least prime > p.

a(n) = A046692(A003961(n)).

cp's OEIS Frontend

A349382 Dirichlet convolution of A064989 with A346234 (Dirichlet inverse of A003961), 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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A349388 Dirichlet convolution of A000027 with A346234 (Dirichlet inverse of A003961), where A003961 is fully multiplicative with a(p) = nextprime(p).

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A378607 Dirichlet convolution of sigma and the Dirichlet inverse of A003961 (A346234).

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A349632 Dirichlet convolution of A250469 with A346234, which is Dirichlet inverse of A003961.

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A153881 1 followed by -1, -1, -1, ... .

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A349125 Dirichlet inverse of A064989, where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p.

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A349385 Dirichlet convolution of A048673 with the Dirichlet inverse of A003961, where A003961 is fully multiplicative with a(p) = nextprime(p), and A048673(n) = (1+A003961(n))/2.

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A346479 Dirichlet inverse of A250469.