# Additional (less important) items — comments, formulas, references, links, programs, etc. — related to the prime numbers, A000040

There is a unique decomposition of the primes: provided the weight A117078(n) is > 0, we have prime(n) = weight * level + gap, or A000040(n) = A117078(n) * A117563(n) + A001223(n). - _RĂ©mi Eismann_, Feb 16 2007

Equals row sums of triangle A143350. - _Gary W. Adamson_, Aug 10 2008

APSO (Alternating partial sums of sequence) a-b+c-d+e-f+g... = (a+b+c+d+e+f+g...)-2*(b+d+f...): APSO(A000040) = A008347=A007504 - 2*(A077126 repeated) (A007504-A008347)/2 = A077131 alternated with A077126. - _Eric Desbiaux_, Oct 28 2008

a(n) = A008864(n) - 1 = A052147(n) - 2 = A113395(n) - 3 = A175221(n) - 4 = A175222(n) - 5 = A139049(n) - 6 = A175223(n) - 7 = A175224(n) - 8 = A140353(n) - 9 = A175225(n) - 10. - _Jaroslav Krizek_, Mar 06 2010

For prime n, the sum of divisors of n > the product of divisors of n. Sigma(n)==1 (mod n). - _Juri-Stepan Gerasimov_, Mar 12 2011

Reading the primes (excluding 2,3,5) mod 90 divides them into 24 classes, which are described by A181732, A195993, A198382, A196000, A201804, A196007, A201734, A201739, A201819, A201816, A201817, A201818, A202104, A201820, A201822, A201101, A202113, A202105, A202110, A202112, A202129, A202114, A202115 and A202116. - _J. W. Helkenberg_, Jul 24 2013

The old definition of prime numbers was "positive integers that have no divisors other than 1 and itself", which gives A008578, not this sequence. - _Omar E. Pol_, Oct 05 2013

The primes appear as the denominators of the only fractions in the table of integers and reduced fractions for: (k!/e) * Sum_{n>=0} Sum_{j=0..n} j^k/n!, k>=0, occurring at k=p-1, where p is a prime, with p=2 occurring at both k=1 and k=3. - _Richard R. Forberg_, Dec 23 2014.

The preceding comment also applies to the z-sequence of the Sheffer matrix, when multiplied by the factorial of its index. See A130190. - _Richard R. Forberg_, Dec 28 2014

It is easily proved that (a(n+m)^j + a(n)^k)/2 and (a(n+m)^j - a(n)^k)/2 are coprime for all m, j, k > 0 and n>1. Conjecture: All coprime pairs can be so constructed, assuming repeated division by 2 of the even number in the resulting pair until it is odd. - _Richard R. Forberg_, Jun 07 2015

Prime numbers are zeros of the functions V_s(x) = Sum_{n>=1} (moebius(n) / n^s) * x^(s*omega(n)), for each s > 1. - _Dimitris Valianatos_, Jun 29 2016

Union of A030430, A030431, A030432, A030433, {2,5}. - _Muniru A Asiru_, Oct 20 2016