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