A054994 Numbers of the form q1^b1 * q2^b2 * q3^b3 * q4^b4 * q5^b5 * ... where q1=5, q2=13, q3=17, q4=29, q5=37, ... (A002144) and b1 >= b2 >= b3 >= b4 >= b5 >= ....
1, 5, 25, 65, 125, 325, 625, 1105, 1625, 3125, 4225, 5525, 8125, 15625, 21125, 27625, 32045, 40625, 71825, 78125, 105625, 138125, 160225, 203125, 274625, 359125, 390625, 528125, 690625, 801125, 1015625, 1185665, 1221025, 1373125, 1795625
Offset: 1
Examples
1=5^0, 5=5^1, 25=5^2, 65=5*13, 125=5^3, 325=5^2*13, 625=5^4, etc.
Links
- Ray Chandler, Table of n, a(n) for n = 1..10000
- Eric Weisstein's World of Mathematics, Pythagorean Triple
Crossrefs
Programs
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Mathematica
maxTerm = 10^15;(* this limit gives ~ 500 terms *) maxNumberOfExponents = 9;(* this limit has to be increased until the number of reaped terms no longer changes *) bmax = Ceiling[ Log[ maxTerm]/Log[q]]; q = Reap[For[k = 0 ; cnt = 0, cnt <= maxNumberOfExponents, k++, If[PrimeQ[4*k + 1], Sow[4*k + 1]; cnt++]]][[2, 1]]; Clear[b]; b[maxNumberOfExponents + 1] = 0; iter = Sequence @@ Table[{b[k], b[k + 1], bmax[[k]]}, {k, maxNumberOfExponents, 1, -1}]; Reap[ Do[an = Product[q[[k]]^b[k], {k, 1, maxNumberOfExponents}]; If[an <= maxTerm, Print[an]; Sow[an]], Evaluate[iter]]][[2, 1]] // Flatten // Union (* Jean-François Alcover, Jan 18 2013 *) -
PARI
list(lim)= { my(u=[1], v=List(), w=v, pr, t=1); forprime(p=5,, if(p%4>1, next); t*=p; if(t>lim, break); listput(w,t) ); for(i=1,#w, pr=1; for(e=1,logint(lim\=1,w[i]), pr*=w[i]; for(j=1,#u, t=pr*u[j]; if(t>lim, break); listput(v,t) ) ); if(w[i]^2 -
Python
def generate_A054994(): """generate arbitrarily many elements of the sequence. TO_DO is a list of pairs (radius, exponents) where "exponents" is a weakly decreasing sequence, and radius == prod(prime_4k_plus_1(i)**j for i,j in enumerate(exponents)) An example entry is (5525, (2, 1, 1)) because 5525 = 5**2 * 13 * 17. """ TO_DO = {(1,())} while True: radius, exponents = min(TO_DO) yield radius #, exponents TO_DO.remove((radius, exponents)) TO_DO.update(successors(radius,exponents)) def successors(radius,exponents): # try to increase each exponent by 1 if possible for i,e in enumerate(exponents): if i==0 or exponents[i-1]>e: # can add 1 in position i without violating monotonicity yield (radius*prime_4k_plus_1(i), exponents[:i]+(e+1,)+exponents[i+1:]) if exponents==() or exponents[-1]>0: # add new exponent 1 at the end: yield (radius*prime_4k_plus_1(len(exponents)), exponents+(1,)) from sympy import isprime primes_congruent_1_mod_4 = [5] # will be filled with 5,13,17,29,37,... def prime_4k_plus_1(i): # the i-th prime that is congruent to 1 mod 4 while i>=len(primes_congruent_1_mod_4): # generate primes on demand n = primes_congruent_1_mod_4[-1]+4 while not isprime(n): n += 4 primes_congruent_1_mod_4.append(n) return primes_congruent_1_mod_4[i] for n,radius in enumerate(generate_A054994()): if n==34: print(radius) break # print the first 35 elements print(radius, end=", ") # Günter Rote, Sep 12 2023
Formula
Sum_{n>=1} 1/a(n) = Product_{n>=1} 1/(1 - 1/A006278(n)) = 1.2707219403... - Amiram Eldar, Oct 20 2020
Extensions
More terms from Henry Bottomley, Mar 14 2001
Comments