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 A235342 Sum of exponents in the (unique) factorization of n as a ratio of p! terms, p prime. 2
 0, 1, 0, 2, -2, 1, -1, 3, 0, -1, -2, 2, -2, 0, -2, 4, -2, 1, -1, 0, -1, -1, 2, 3, -4, -1, 0, 1, 1, -1, 1, 5, -2, -1, -3, 2, -1, 0, -2, 1, 1, 0, 0, 0, -2, 3, -1, 4, -2, -3, -2, 0, 3, 1, -4, 2, -1, 2, 0, 0, 0, 2, -1, 6, -4, -1, -2, 0, 2, -2, 0, 3, -3, 0, -4, 1, -3, -1, 7, 2, 0, 2, -4, 1, -4, 1, 1, 1, 0, -1, -3, 4, 1, 0, -3, 5, -3, -1, -2, -2, 5, -1, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS With n>0, write n=p_1!p_2!...p_k!/(q_1!q2!...q_l!) where p_i,q_j are primes and the fraction is simplified (ie., no p_i is a q_j).  This representation is unique for positive integers (and positive rational numbers), so we let a(n):=#p_i! terms on top-#q_j! terms on bottom. LINKS Antti Karttunen, Table of n, a(n) for n = 1..5040 FORMULA a(1)=0; a(p!)=1, p prime; a(xy)=a(x)+a(y); (group homomorphism from Q^+ to Z). EXAMPLE a(1)=0 (by convention). a(2)=1 since 2=2!. a(3)=0 since 3=3!/2!. a(4)=2 since 4=2!*2!. a(5)=-2 since 5=5!/(3!*2!*2!). PROG (Sage) def plus(c, d, mult):   for elt in d:     if elt in c:       c[elt]+=mult*d[elt]     else:       c[elt]=mult*d[elt] def rep(m):   if m==1:     return {}   if m==2:     return {2:1}   f=factor(Integer(m))   #print f   if len(f)==1 and f[0][1]==1:     #print "prime", m     p=prime_range(m)[-1]     new={m:1, p:-1}     r=range(p+1, m)     #print "range", r     for k in r:       plus(new, rep(k), -1)   else:     new={}     #print "not prime", m, f     for (p, mult) in f:       #print (p, mult)       plus(new, rep(p), mult)   for elt in [elt for elt in new if new[elt]==0]:     new.pop(elt)   return new def weight(m):   w=0   r=rep(m)   for p in r:     w+=r[p]   return w A235342=[weight(m) for m in range(1, 5041)] # Above code "de-periodicized" by Antti Karttunen, Mar 28 2017 # This is just for outputting a b-file: i=0 outfp = open('b235342.txt', 'w') for an in A235342:     i = i+1     outfp.write(str(i) + " " + str(an) + "\n") outfp.close() CROSSREFS Cf. A001222, A236441. Sequence in context: A244006 A110283 A226290 * A079692 A110269 A238280 Adjacent sequences:  A235339 A235340 A235341 * A235343 A235344 A235345 KEYWORD sign AUTHOR Alexander Riasanovsky, Jan 06 2014 EXTENSIONS More terms and b-file computed with the given Sage-program by Antti Karttunen, Mar 28 2017 STATUS approved

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Last modified September 22 12:19 EDT 2019. Contains 327307 sequences. (Running on oeis4.)