A160135 Sum of non-exponential divisors of n.

1, 1, 1, 1, 1, 6, 1, 5, 1, 8, 1, 10, 1, 10, 9, 9, 1, 15, 1, 12, 11, 14, 1, 30, 1, 16, 10, 14, 1, 42, 1, 29, 15, 20, 13, 19, 1, 22, 17, 40, 1, 54, 1, 18, 18, 26, 1, 58, 1, 33, 21, 20, 1, 60, 17, 50, 23, 32, 1, 78, 1, 34, 20, 49, 19, 78, 1, 24, 27, 74, 1, 75, 1, 40, 34, 26, 19, 90, 1, 76, 28

Offset: 1

Author

Jaroslav Krizek, May 02 2009

Keywords

• nonn

Comments

The non-exponential divisors d|n of a number n = p(i)^e(i) are divisors d not of the form p(i)^s(i), s(i)|e(i) for all i.

Examples

			a(8) = A000203(8) - A051377(8) = 15 - 10 = 5. a(8) = a(2^3) = (2^4-1)/(2-1) - (2^1+2^3) = 5.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A000203, A051377, A000040, A006881, A120944, A000961, A053810.

Programs

Maple

lpowp := proc(n,p) local e; for e from 0 do if n mod p^(e+1) <> 0 then RETURN(e) ; fi; od: end:
expdvs := proc(n) local a,d,nfcts,b,f,iseDiv ; a := {} ; nfcts := ifactors(n)[2] ; for d in ( numtheory[divisors](n) minus {1} ) do iseDiv := true; for f in nfcts do b := lpowp(d,op(1,f) ) ; if b = 0 or op(2,f) mod b <> 0 then iseDiv := false; fi; od: if iseDiv then a := a union {d} ; fi; od: a ; end proc:
A051377 := proc(n) local k ; add( k, k = expdvs(n)) ; end: A160135 := proc(n) if n = 1 then 1; else numtheory[sigma][1](n)-A051377(n) ; fi; end: seq(A160135(n),n=1..120) ; # R. J. Mathar, May 08 2009

Mathematica

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; a[1] = 1; a[n_] := DivisorSigma[1, n] - esigma[n]; Array[a, 100] (* Amiram Eldar, Oct 26 2021 after Jean-François Alcover at A051377 *)

PARI

A051377(n) = { my(f=factor(n)); prod(i=1, #f[, 1], sumdiv(f[i, 2], d, f[i, 1]^d)); }; \\ From A051377
A160135(n) = if(1==n,n, sigma(n) - A051377(n)); \\ Antti Karttunen, Mar 04 2018

Formula

a(n) = A000203(n) - A051377(n) for n >= 2.

a(1) = 1, a(p) = 1, a(p*q) = 1 + p + q, a(p*q*...*z) = (p + 1)*(q + 1)*...*(z + 1) - p*q*...*z, for p, q,..,z = primes (A000040), p*q = product of two distinct primes (A006881), p*q*...*z = product of k (k > 0) distinct primes (A120944).

a(p^k) = (p^(k+1)-1)/(p-1)- Sum_{d|k} p^d for p = primes (A000040), p^k = prime powers A000961(n>1), k = natural numbers (A000027)>

a(p^q) = 1+(p^1-p^1)+p^2+p^3+...+p^(q-1), for p, q = primes (A000040), p^q = prime powers of primes (A053810).

Extensions

Edited by R. J. Mathar, May 08 2009