cp's OEIS Frontend

A349169 Numbers k such that k * gcd(sigma(k), A003961(k)) is equal to the odd part of {sigma(k) * gcd(k, A003961(k))}, where A003961 shifts the prime factorization one step towards larger primes, and sigma is the sum of divisors function.

%I A349169 #81 Aug 28 2025 17:24:29
%S A349169 1,15,105,3003,3465,13923,45045,264537,459459,745875,1541475,5221125,
%T A349169 8729721,10790325,14171625,29288025,34563375,57034575,71430975,
%U A349169 99201375,109643625,144729585,205016175,255835125,295708875,356080725,399242025,419159475,449323875,928602675,939495375,1083656925,1941623775,1962350685,2083228875
%N A349169 Numbers k such that k * gcd(sigma(k), A003961(k)) is equal to the odd part of {sigma(k) * gcd(k, A003961(k))}, where A003961 shifts the prime factorization one step towards larger primes, and sigma is the sum of divisors function.
%C A349169 Numbers k such that A348990(k) [= k/gcd(k, A003961(k))] is equal to A348992(k), which is the odd part of A349162(k), thus all terms must be odd, as A348990 preserves the parity of its argument.
%C A349169 Equally, numbers k for which gcd(A064987(k), A191002(k)) is equal to A000265(gcd(A064987(k), A341529(k))).
%C A349169 Also odd numbers k for which A348993(k) = A319627(k).
%C A349169 Odd terms of A336702 are given by the intersection of this sequence and A349174.
%C A349169 Conjectures:
%C A349169   (1) After 1, all terms are multiples of 3. (Why?)
%C A349169   (2) After 1, all terms are in A104210, in other words, for all n > 1, gcd(a(n), A003961(a(n))) > 1. Note that if we encountered a term k with gcd(k, A003961(k)) = 1, then we would have discovered an odd multiperfect number.
%C A349169   (3) Apart from 1, 15, 105, 3003, 13923, 264537, all other terms are abundant. [These apparently are also the only terms that are not Zumkeller, A083207. Note added Dec 05 2024]
%C A349169   (4) After 1, all terms are in A248150. (Cf. also A386430).
%C A349169   (5) After 1, all terms are in A348748.
%C A349169   (6) Apart from 1, there are no common terms with A349753.
%C A349169 Note: If any of the last four conjectures could be proved, it would refute the existence of odd perfect numbers at once. Note that it seems that gcd(sigma(k), A003961(k)) < k, for all k except these four: 1, 2, 20, 160.
%C A349169 Questions:
%C A349169   (1) For any term x here, can 2*x be in A349745? (Partial answer: at least x should be in A191218 and should not be a multiple of 3). Would this then imply that x is an odd perfect number? (Which could explain the points (1) and (4) in above, assuming the nonexistence of opn's).
%H A349169 Amiram Eldar, <a href="/A349169/b349169.txt">Table of n, a(n) for n = 1..63</a> (terms below 10^11)
%H A349169 <a href="/index/O#opnseqs">Index entries for sequences where odd perfect numbers must occur, if they exist at all</a>
%H A349169 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A349169 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A349169 For all n >= 1, A007949(A000203(a(n))) = A007949(a(n)). [sigma preserves the 3-adic valuation of the terms of this sequence] - _Antti Karttunen_, Nov 29 2021
%t A349169 Select[Range[10^6], #1/GCD[#1, #3] == #2/(2^IntegerExponent[#2, 2]*GCD[#2, #3]) & @@ {#, DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &] (* _Michael De Vlieger_, Nov 11 2021 *)
%o A349169 (PARI)
%o A349169 A000265(n) = (n >> valuation(n, 2));
%o A349169 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A349169 isA349169(n) = { my(s=sigma(n),u=A003961(n)); (n*gcd(s,u) == A000265(s)*gcd(n,u)); }; \\ (Program simplified Nov 30 2021)
%Y A349169 Cf. A000203, A003961, A007949, A064989, A083207, A104210, A161942, A191002, A191218, A228058, A319627, A322361, A326134, A329963, A342671, A348748, A348990, A348992, A348993, A349162, A349164, A349174.
%Y A349169 Cf. also A349745, A349746, A349753.
%Y A349169 Subsequence of A349749, and of A386430.
%K A349169 nonn,changed
%O A349169 1,2
%A A349169 _Antti Karttunen_, Nov 10 2021
%E A349169 Name changed and comment section rewritten by _Antti Karttunen_, Nov 29 2021