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A341529 a(n) = sigma(n) * A003961(n), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of the divisors of n.

%I A341529 #43 Jan 28 2023 10:35:51
%S A341529 1,9,20,63,42,180,88,405,325,378,156,1260,238,792,840,2511,342,2925,
%T A341529 460,2646,1760,1404,696,8100,1519,2142,5000,5544,930,7560,1184,15309,
%U A341529 3120,3078,3696,20475,1558,4140,4760,17010,1806,15840,2068,9828,13650,6264,2544,50220,6897,13671,6840,14994,3186,45000,6552,35640
%N A341529 a(n) = sigma(n) * A003961(n), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of the divisors of n.
%C A341529 Question: Does the maximum value of ratio A341529(n)/A341528(n) stay below 2?
%C A341529 From _Amiram Eldar_ and _Antti Karttunen_, Jan 28 2023: (Start)
%C A341529 Answer to the above question is yes: Sup_{n>=1} A341529(n)/A341528(n) = 2.
%C A341529 Proof:
%C A341529   f(n) = A341529(n)/A341528(n) is a multiplicative function with f(p^e) = (1 + 1/p + ... + 1/p^e)/(1 + 1/q + ... + 1/q^e), where q = nextprime(p).
%C A341529   First we prove a lemma which states that f(p^(1+e)) / f(p^e) > 1, for any prime p, and exponent e.
%C A341529   We note that (sigma(p^(1+e))/(p^(1+e))) / (sigma(p^e)/(p^e)) = (sigma(p^(1+e))/(p*sigma(p^e))) = sigma(p^(1+e)) / (sigma(p^(1+e)) - 1), so setting q = nextprime(p), we can write the ratio f(p^(1+e)) / f(p^e) as (sigma(p^(1+e))/(sigma(p^(1+e))-1)) / (sigma(q^(1+e))/(sigma(q^(1+e))-1)), and to prove this to be > 1, we just note that the denominator is less than the numerator, because sigma(p^e) is monotonically growing with respect to the increasing prime p.
%C A341529   Since q > p, we have f(p^e) > 1 for all p and all e>=1, and together with the above lemma this shows that f(n) <= f(n*m) for all m>=1.
%C A341529   Suppose n = Product_i p_i^e_i, and let pmax = max(p_i), emax = max(e_i), so n is a divisor of m = (pmax#)^emax, and f(n) < f(m), where p# = 2 * 3 * ... * p is the primorial of p, A034386(p).
%C A341529   Then f(m) = f(2^emax) * f(3^emax) * ... * f(pmax^emax) = (1 + 1/2 + ... + 1/2^emax)/(1 + 1/3 + ... + 1/3^emax)) * (1 + 1/3 + ... + 1/3^emax)/(1 + 1/5 + ... + 1/5^emax)) * ... * (1 + 1/p + ... + 1/p^emax)/(1 + 1/q + ... + 1/q^emax))[telescoping product] = (1 + 1/2 + ... + 1/2^emax)/(1 + 1/qmax + ... + 1/qmax^emax) <= (1 + 1/2 + ... + 1/2^emax) < 2, where qmax = nextprime(pmax).
%C A341529   So we have f(n) < 2 for all n.
%C A341529   To prove that 2 is the supremum, we have lim_{e,k -> oo) f(prime(k)#^e) = 2.
%C A341529 (End)
%H A341529 Antti Karttunen, <a href="/A341529/b341529.txt">Table of n, a(n) for n = 1..8191</a>
%H A341529 <a href="https://oeis.org/plot2a?name1=A341529&amp;name2=A341528&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=ratio&amp;drawpoints=true">Ratio A341529(n)/A341528(n) plotted with OEIS Plot tool</a>
%H A341529 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.
%H A341529 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A341529 Multiplicative with a(p^e) = q^e * (p^(e+1)-1)/(p-1), where q = nextPrime(p).
%F A341529 a(n) = A000203(n) * A003961(n).
%F A341529 For all n > 1, a(n) > A341528(n).
%F A341529 For all n >= 1, A072861(n) <= a(n) <= A003961(n)^2. [See A286385].
%F A341529 a(n) = A341528(n) + A341512(n) = A342671(n) * A342672(n) = A342661(A003961(n)). - _Antti Karttunen_, Mar 22 2021
%F A341529 Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} p^4*(p-1)/((p^3-nextprime(p))*(p^2-nextprime(p))) = 3.0664809..., where nextprime is A151800. - _Amiram Eldar_, Dec 08 2022
%t A341529 Array[DivisorSigma[1, #]*Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] &, 56] (* _Michael De Vlieger_, Feb 22 2021 *)
%o A341529 (PARI)
%o A341529 A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
%o A341529 A341529(n) = (sigma(n)*A003961(n));
%Y A341529 Cf. A000203, A003961, A003973, A028982 (positions of odd terms), A072861, A151800, A286385, A341512, A341526, A341527, A341528, A341530, A342661, A342667, A342671, A342672.
%Y A341529 Cf. also arrays A341605/A341606.
%K A341529 nonn,mult
%O A341529 1,2
%A A341529 _Antti Karttunen_, Feb 16 2021