By comparing the formula giving odd powers sums of integers from Bernoulli numbers and the Faulhaber conjecture form of them, we obtain two recurrence relations for calculating the Faulhaber coefficients. Parallelly we search for and obtain the differential operator which transform a powers sum into a Bernoulli polynomial. From this and by changing arguments from z,n into Z=z(z-1), λ=zn+n(n-1/2) we obtain a formula giving powers sums on arithmetic progressions directly from the powers sums on integers.

**Author (s) Details **

**Do Tan Si
**HoChiMinh-city Physical Association, Vietnam and Université libre de Bruxelles and UEM, Belgium.

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