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

1,6

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.

Links

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

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).

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A367563 A016534 A230821 * A178645 A010137 A245967

Adjacent sequences: A160132 A160133 A160134 * A160136 A160137 A160138

Keyword

nonn

Author

Jaroslav Krizek, May 02 2009

Extensions

Edited by R. J. Mathar, May 08 2009

Status

approved