A125139 SENSigma: Multiplicative with a(p^e) = p*(p^e-1)/(p-1) - (-1)^e.

1, 3, 4, 5, 6, 12, 8, 15, 11, 18, 12, 20, 14, 24, 24, 29, 18, 33, 20, 30, 32, 36, 24, 60, 29, 42, 40, 40, 30, 72, 32, 63, 48, 54, 48, 55, 38, 60, 56, 90, 42, 96, 44, 60, 66, 72, 48, 116, 55, 87, 72, 70, 54, 120, 72, 120, 80, 90, 60, 120, 62, 96, 88, 125, 84, 144, 68, 90, 96

Offset

1,2

Comments

Original definition:

SENSigma(n) = (-1)^((Sum_i r_i)+Omega(n))*Sum_{d|n} (-1)^((Sum_j Max(r_j))+Omega(d))*d =Product_i (Sum_{1<=s_i<=r_i} p_i^s_i)+(-1)^(r_i+1) where n=Product_i p_i^r_i, d=Product_j p_j^r_j, p_j^max(r_j) is the largest power of p_j dividing n.

SEN stands for Signed by Exponents of prime factors and Number of prime factors.

By "Max(r_j)" I mean the following: If d|m, d = p^e*q^f, m = p^x*q^y*r^z then Max(e)=x, Max(f)=y.

Here is another version of the definition. Let n=Product_i p_i^e_i, PREX_{p}(n)=e_k, p=p_k. Example: n = 2^4*3^2*5, PREX_{2}(n)=4. Then SENSigma(m) = (-1)^((Sum_i r_i) + Omega(m))*Sum_{d|m} (-1)^((Sum_j PREX_{p_j}(m)) + Omega(d))*d = Product_i (Sum_{1 <= s_i <= r_i} p_i^s_i) + (-1)^(r_i+1) where m = Product_i p_i^r_i, d = Product_j p_j^r_j.

The function is not completely multiplicative: If p is a prime, a(p) = p+1 but a(p^2) = p + p^2 - 1. - R. J. Mathar, Nov 20 2010

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Formula

SENSigma(n) = Product_i (p_i^(r_i+1) - p_i)/(p_i-1) + (-1)^(r_i+1).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(4)/2) * Product_{p prime} (1 + 1/p^2 - 2/p^4 + 2/p^5) = 0.76715211405... . - Amiram Eldar, Nov 30 2022

Example

If n=240, d=12 then 2^max(r_j) = 2^max(2) = 2^4, 3^max(r_j) = 3^max(1) = 3^1. SENSigma(240) = (-1+2+4+8+16)*(1+3)*(1+5).

Maple

A125139 := proc(n) local ifs, i, a, r, p ; ifs := ifactors(n)[2] ; a := 1 ; if n > 1 then for i from 1 to nops(ifs) do r := op(2, op(i, ifs)) ; p := op(1, op(i, ifs)) ; a := a*(p*(1-p^r)/(1-p)-(-1)^r) ; od ; fi ; RETURN(a) ; end: for n from 1 to 80 do printf("%d, ", A125139(n)) ; od ; # R. J. Mathar, May 18 2007

Mathematica

f[p_, e_] := p*(p^e-1)/(p-1) - (-1)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 30 2022 *)

Prog

(PARI) A125139(n)={ prod( i=1, #n=factor(n)~, my( r=n[2, i], p=n[1, i]); p*(p^r-1)/(p-1)-(-1)^r ) }

Crossrefs

Cf. A013662.

Sequence in context: A188999 A186644 A337177 * A369762 A107224 A026493

Adjacent sequences: A125136 A125137 A125138 * A125140 A125141 A125142

Keyword

nonn,mult

Author

Yasutoshi Kohmoto, Jan 12 2007, Jan 29 2007

Extensions

More terms from R. J. Mathar, May 18 2007

Status

approved