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Michael Hardy
Michael Hardy
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On the p$p$-adic valuation of the sum of the first n$n$ factorials

This is more a curiosity than anything useful. Consider $n>3$ and define$A(n)= 1!+2!+…n!$ $A(n)= 1!+2!+\cdots+n!$

It seems that if $p$ is the largest prime divisor of $A(n)$, then the $p$-adic valuation of $A(n)$ is 1. I’ve tested several hundred cases on mathematica. This function is a translate of Kurepa’s “left factorial” function, but I haven’t found anything in the literature related to this specific question. Perhaps this has an elementary proof.

On the p-adic valuation of the sum of the first n factorials

This is more a curiosity than anything useful. Consider $n>3$ and define$A(n)= 1!+2!+…n!$

It seems that if $p$ is the largest prime divisor of $A(n)$, then the $p$-adic valuation of $A(n)$ is 1. I’ve tested several hundred cases on mathematica. This function is a translate of Kurepa’s “left factorial” function, but I haven’t found anything in the literature related to this specific question. Perhaps this has an elementary proof.

On the $p$-adic valuation of the sum of the first $n$ factorials

This is more a curiosity than anything useful. Consider $n>3$ and define $A(n)= 1!+2!+\cdots+n!$

It seems that if $p$ is the largest prime divisor of $A(n)$, then the $p$-adic valuation of $A(n)$ is 1. I’ve tested several hundred cases on mathematica. This function is a translate of Kurepa’s “left factorial” function, but I haven’t found anything in the literature related to this specific question. Perhaps this has an elementary proof.

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Marc
Marc
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On the p-adic valuation of the sum of the first n factorials

This is more a curiosity than anything useful. Consider $n>3$ and define$A(n)= 1!+2!+…n!$

It seems that if $p$ is the largest prime divisor of $A(n)$, then the $p$-adic valuation of $A(n)$ is 1. I’ve tested several hundred cases on mathematica. This function is a translate of Kurepa’s “left factorial” function, but I haven’t found anything in the literature related to this specific question. Perhaps this has an elementary proof.