A007444 -id:A007444 - cp's OEIS frontend

A130071 Triangle, A007444(k) in each column interspersed with k zeros.

2, 2, 1, 2, 0, 3, 2, 1, 0, 4, 2, 0, 0, 0, 9, 2, 1, 3, 0, 0, 7, 2, 0, 0, 0, 0, 0, 15, 2, 1, 0, 4, 0, 0, 0, 12, 2, 0, 3, 0, 0, 0, 0, 0, 18, 2, 1, 0, 0, 9, 0, 0, 0, 0, 17

Offset: 1

Gary W. Adamson, May 05 2007

Row sums = the primes. T(n,k) = 0 if k does not divide n. If k divides n, extract A007444(k) which become the nonzero terms of row n, sum = n-th prime. Example: The factors of 6 are (1, 2, 3 and 6) = k's for A007444(k) = (2 + 1 + 3 + 7) = p(6) = 13. A007444 = the Moebius transform of the primes, (2, 1, 3, 4, 9, 7, 15, 12, ...), as the right diagonal of A130071.

			First few rows of the triangle:
  2;
  2,  1;
  2,  0,  3;
  2,  1,  0,  4;
  2,  0,  0,  0,  9;
  2,  1,  3,  0,  0,  7;
  2,  0,  0,  0,  0,  0, 15;
  2,  1,  0,  4,  0,  0,  0, 12;
  2,  0,  3,  0,  0,  0,  0,  0, 18;
  2,  1,  0,  0,  9,  0,  0,  0,  0, 17;
  ...

Cf. A130070, A007444, A054525, A000040.

Given the Moebius transform of the primes, A007444: (2, 1, 3, 4, 9, 7, 15, ...), the k-th term (k= 1,2,3,...) of this sequence generates the k-th column of A130071, interspersed with (k-1) zeros.

A030013 Moebius transform of {1, primes}.

Original entry on oeis.org

1, 1, 2, 3, 6, 7, 12, 12, 16, 15, 28, 17, 36, 27, 34, 30, 52, 32, 60, 41, 56, 43, 78, 40, 82, 59, 82, 59, 106, 43, 112, 80, 100, 83, 120, 70, 150, 95, 124, 88, 172, 73, 180, 115, 134, 117, 198, 98, 210, 122, 174, 133, 238, 100, 216, 142, 200, 161, 270, 107, 280, 169, 206, 180

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane, Transforms

Cf. A008578 ({1, primes}), A007444 (Moebius transform of primes), A008683.

a(n) = sumdiv(n, d, if (d==1, 1, prime(d-1))*moebius(n/d)); \\ Michel Marcus, Nov 04 2018

Offset 1 and more terms from Michel Marcus, Nov 04 2018

A062774 Inverse Moebius transform of PrimePi function.

Original entry on oeis.org

0, 1, 2, 3, 3, 6, 4, 7, 6, 8, 5, 13, 6, 11, 11, 13, 7, 17, 8, 18, 14, 14, 9, 26, 12, 16, 15, 22, 10, 29, 11, 24, 18, 19, 18, 35, 12, 21, 20, 34, 13, 37, 14, 30, 29, 24, 15, 47, 19, 32, 24, 33, 16, 42, 24, 42, 26, 27, 17, 61, 18, 30, 36, 42, 27, 48, 19, 40, 30, 48, 20, 68, 21, 34

Offset: 1

Labos Elemer, Jul 18 2001

nonn

			n = 12: divisors = D(12) = {1,2,3,4,6,12}, pi(D(12)) = {0,1,2,2,3,5} of which the sum is 0+1+2+2+3+5 = 13 so a(12) = 13; a(p(n)) = 0+n = n, for n-th prime p(n).

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000720, A007444, A007445.

{ for (n=1, 1000, d=divisors(n); write("b062774.txt", n, " ", sum(k=1, length(d), primepi(d[k]))) ) } \\ Harry J. Smith, Aug 10 2009

a(n) = Sum_{d|n} pi(d).
G.f.: Sum_{k>=1} pi(k)*x^k/(1 - x^k), where pi(k) is the number of primes <= k (A000720). - Ilya Gutkovskiy, Jan 16 2017
a(n) = Sum_{d|n} omega(d!). - Wesley Ivan Hurt, May 23 2021

Offset changed from 0 to 1 by Harry J. Smith, Aug 10 2009

A127638 A054525 * A127640, where A127640 = infinite lower triangular matrix with the sequence of primes in the main diagonal and the rest zeros.

Original entry on oeis.org

2, -2, 3, -2, 0, 5, 0, -3, 0, 7, -2, 0, 0, 0, 11, 2, -3, -5, 0, 0, 13, -2, 0, 0, 0, 0, 0, 17, 0, 0, 0, -7, 0, 0, 0, 19, 0, 0, -5, 0, 0, 0, 0, 0, 23, 2, -3, 0, 0, -11, 0, 0, 0, 0, 29, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 31, 0, 3, 0, -7, 0, -13, 0, 0, 0, 0, 0, 37, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 41, 2, -3, 0, 0, 0, 0, -17, 0, 0, 0, 0, 0, 0, 43, 2, 0, -5, 0, -11, 0, 0

Offset: 1

Gary W. Adamson, Jan 21 2007

Right diagonal = primes: (2, 3, 5, 7, ...). Row sums = the Mobius transform of primes, A007444: (2, 1, 3, 4, 9, 7, ...).

			First few rows of the triangle:
   2;
  -2,  3;
  -2,  0,  5;
   0, -3,  0, 7;
  -2,  0,  0, 0, 11;
   2, -3, -5, 0,  0, 13;
  ...

Cf. A007444, A054525, A127639.

A054525 := proc(n,k) if n mod k = 0 then numtheory[mobius](n/k) ; else 0 ; fi ; end: A127648 := proc(n,k) A054525(n,k)*ithprime(k) ; end: for n from 1 to 16 do for k from 1 to n do printf("%d,", A127648(n,k)) ; od ; od ; # R. J. Mathar, Mar 14 2007

More terms from R. J. Mathar, Mar 14 2007

A333450 a(n) = Sum_{k=1..n} mu(k) * prime(floor(n/k)).

Original entry on oeis.org

2, 1, 1, 2, 4, 5, 7, 7, 9, 12, 12, 15, 17, 16, 16, 20, 24, 22, 26, 23, 21, 26, 28, 28, 32, 33, 31, 32, 32, 29, 41, 39, 43, 40, 44, 40, 44, 45, 45, 47, 51, 52, 60, 55, 53, 52, 62, 64, 64, 56, 54, 55, 55, 65, 67, 69, 69, 70, 74, 73, 73, 70, 80, 80, 76, 69, 81, 84, 90, 81, 83, 87, 93, 94

Offset: 1

Ilya Gutkovskiy, Mar 21 2020

nonn

Cf. A000040, A001223, A007444, A007504, A008683, A333449.

Table[Sum[MoebiusMu[k] Prime[Floor[n/k]], {k, 1, n}], {n, 1, 74}]
g[1] = 2; g[n_] := Prime[n] - Prime[n - 1]; a[n_] := Sum[Sum[MoebiusMu[k/d] g[d], {d, Divisors[k]}], {k, 1, n}]; Table[a[n], {n, 1, 74}]

a(n) = sum(k=1, n, moebius(k)*prime(n\k)); \\ Michel Marcus, Mar 22 2020

from functools import lru_cache
from sympy import prime
@lru_cache(maxsize=None)
def A333450(n):
    c, j = 2*(n+1)-prime(n), 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A333450(k1)
        j, k1 = j2, n//j2
    return 2*j-c # Chai Wah Wu, Mar 31 2021

Sum_{k=1..n} a(floor(n/k)) = prime(n).

A130070 Moebius transform of A130069.

Original entry on oeis.org

2, 1, 2, 3, 0, 2, 4, 1, 0, 2, 9, 0, 0, 0, 2, 7, 3, 1, 0, 0, 2, 15, 0, 0, 0, 0, 0, 2, 12, 4, 0, 1, 0, 0, 0, 2, 18, 0, 3, 0, 0, 0, 0, 0, 2, 17, 9, 0, 0, 1, 0, 0, 0, 0, 2

Offset: 1

Gary W. Adamson, May 05 2007

Row sums = the primes, A000040: (2, 3, 5, 7, ...). Left column = A007444, the Moebius transform of the primes: (2, 1, 3, 4, 9, 7, 15, ...). A130071 = triangle with reversal of nonzero terms.

			First few rows of the triangle:
   2;
   1, 2;
   3, 0, 2;
   4, 1, 0, 2;
   9, 0, 0, 0, 2;
   7, 3, 1, 0, 0, 2;
  15, 0, 0, 0, 0, 0, 2;
  12, 4, 0, 1, 0, 0, 0, 2;
  18, 0, 3, 0, 0, 0, 0, 0, 2;
  17, 9, 0, 0, 1, 0, 0, 0, 0, 2;
  ...

Cf. A054525, A130069, A130071.

A054525 * A130069.

A333177 a(n) = Sum_{d|n, gcd(d, n/d) = 1} (-1)^omega(n/d) * prime(d).

Original entry on oeis.org

2, 1, 3, 5, 9, 7, 15, 17, 21, 17, 29, 27, 39, 25, 33, 51, 57, 37, 65, 55, 53, 47, 81, 67, 95, 59, 101, 85, 107, 41, 125, 129, 103, 79, 123, 123, 155, 95, 123, 145, 177, 75, 189, 157, 165, 115, 209, 167, 225, 131, 171, 193, 239, 147, 217, 229, 199, 161, 275, 147

Offset: 1

Ilya Gutkovskiy, Mar 10 2020

nonn

Cf. A000040, A001221, A007444, A076479, A333178.

a[n_] := Sum[If[GCD[n/d, d] == 1, (-1)^PrimeNu[n/d] Prime[d], 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 60}]

a(n) = sumdiv(n, d, if (gcd(d, n/d) ==1, (-1)^omega(n/d) * prime(d))); \\ Michel Marcus, Mar 10 2020

Sum_{d|n, gcd(d, n/d) = 1} a(d) = prime(n).

A361707 Moebius transform applied twice to primes.

Original entry on oeis.org

2, -1, 1, 3, 7, 5, 13, 8, 15, 9, 27, 10, 37, 11, 23, 22, 55, 8, 63, 18, 37, 19, 79, 12, 77, 21, 62, 32, 105, -5, 123, 44, 73, 23, 101, 23, 153, 31, 83, 44, 175, 7, 187, 60, 84, 35, 207, 38, 195, 20, 113, 72, 237, 18, 181, 76, 133, 55, 273, 34, 279, 41, 148, 102, 217

Offset: 1

Ilya Gutkovskiy, Mar 21 2023

Winston de Greef, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms

Cf. A000040, A007427, A007431, A007444, A361706.

a:= (proc(p) proc(n) uses numtheory;
       add(p(d)*mobius(n/d), d=divisors(n))
     end end@@2)(ithprime):
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 23 2023

A007427[n_] := Sum[MoebiusMu[n/d] MoebiusMu[d], {d, Divisors[n]}]; a[n_] := Sum[A007427[n/d] Prime[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 65}]

A007427(n) = if( n<1, 0, direuler(p=2, n, (1 - 'x)^2)[n]) \\ from Michael Somos, Nov 15 2002
A361707(n)=sumdiv(n, d, A007427(n/d) * prime(d)) \\ Winston de Greef, Mar 23 2023

a(n) = Sum_{d|n} A007427(n/d) * prime(d).

A361709 Moebius transform of nonprimes.

Original entry on oeis.org

1, 3, 5, 4, 8, 1, 11, 6, 9, 4, 17, 6, 20, 7, 10, 11, 25, 8, 27, 10, 15, 12, 33, 9, 27, 14, 24, 14, 41, 12, 44, 21, 25, 20, 30, 14, 51, 23, 29, 20, 56, 15, 59, 25, 30, 27, 64, 20, 56, 26, 39, 30, 73, 24, 50, 31, 45, 35, 80, 18, 83, 37, 45, 41, 59, 26, 90, 39, 54, 30

Offset: 1

Ilya Gutkovskiy, Mar 21 2023

nonn

Paolo Xausa, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms

Cf. A007444, A018252, A361708.

With[{np = Select[Range[100], !PrimeQ[#] &]}, Table[DivisorSum[n, MoebiusMu[n/#]*np[[#]] &], {n, Length[np]}]] (* Paolo Xausa, Aug 21 2025 *)

a(n) = Sum_{d|n} mu(n/d) * A018252(d).

A062778 Values of Moebius-transform of PrimePi function.

Original entry on oeis.org

0, 1, 2, 1, 3, 0, 4, 2, 2, 0, 5, 1, 6, 1, 1, 2, 7, 2, 8, 3, 2, 2, 9, 2, 6, 2, 5, 2, 10, 3, 11, 5, 4, 3, 4, 2, 12, 3, 4, 2, 13, 3, 14, 5, 6, 4, 15, 4, 11, 5, 6, 5, 16, 4, 8, 5, 6, 5, 17, 2, 18, 6, 8, 7, 9, 4, 19, 7, 8, 6, 20, 5, 21, 8, 9, 8, 12, 6, 22, 8, 13, 8, 23, 6, 13, 8, 11, 7, 24, 4, 14, 9, 11, 8

Offset: 1

Labos Elemer, Jul 18 2001

nonn

			n=12, divisors = D(12) = {1,2,3,4,6,12}, pi(12/divisors) = {5,3,2,2,1,0}, mu(divisors) = {1,-1,-1,0,1,0}, Sum = 5*1 - 3*1 - 2*1 + 0 + 1*1 + 0 = 1, thus a(12)=1; for p=prime(n), pi(p/divisor) = {n,0}, mu({1,p})={1,-1}, Sum = 1*n + 0 = n, so a(prime(n)) = n.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A007444, A007445.

f[n_] := Block[{d = Divisors@n}, Plus @@ (MoebiusMu /@ (n/d)*PrimePi /@ d)]; Array[f, 94] (* Robert G. Wilson v, Dec 07 2005 *)

{ for (n=1, 1000, d=divisors(n); write("b062778.txt", n, " ", sum(k=1, length(d), primepi(n/d[k]) * moebius(d[k]))) ) } \\ Harry J. Smith, Aug 10 2009

a(n) = sumdiv(n, d, primepi(d)*moebius(n/d)); \\ Michel Marcus, Nov 05 2018

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

cp's OEIS Frontend

A130071 Triangle, A007444(k) in each column interspersed with k zeros.

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A030013 Moebius transform of {1, primes}.

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A062774 Inverse Moebius transform of PrimePi function.

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A127638 A054525 * A127640, where A127640 = infinite lower triangular matrix with the sequence of primes in the main diagonal and the rest zeros.

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A333450 a(n) = Sum_{k=1..n} mu(k) * prime(floor(n/k)).

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A130070 Moebius transform of A130069.

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A333177 a(n) = Sum_{d|n, gcd(d, n/d) = 1} (-1)^omega(n/d) * prime(d).

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A361707 Moebius transform applied twice to primes.

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Mathematica

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A361709 Moebius transform of nonprimes.

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