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Let G = C n 1 ⊕ … ⊕ C n r with 1 < n 1 | … | n r be a finite abelian group. The Davenport constant D ( G ) is the smallest integer t such that every sequence S over G of length | S | ≥ t has a non-empty zero-sum subsequence. It is a starting point of zero-sum theory. It has a trivial lower bound D ⁎ ( G ) = n 1 + … + n r − r + 1 , which equals D ( G ) over p -groups. We investigate the non-dispersive sequences over groups C n r , thereby revealing the growth of D ( G ) − D ⁎ ( G ) over non- p -groups G = C n r ⊕ C k n with n , k ≠ 1 . We give a general lower bound of D ( G ) over non- p -groups and show that if G is an abelian group with exp ( G ) = m and rank r , fix m > 0 a non-prime-power, then for each N > 0 there exists an ε > 0 such that if | G | / m r < ε , then D ( G ) − D ⁎ ( G ) > N .
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