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.

1, 15, 105, 3003, 3465, 13923, 45045, 264537, 459459, 745875, 1541475, 5221125, 8729721, 10790325, 14171625, 29288025, 34563375, 57034575, 71430975, 99201375, 109643625, 144729585, 205016175, 255835125, 295708875, 356080725, 399242025, 419159475, 449323875, 928602675, 939495375, 1083656925, 1941623775, 1962350685, 2083228875

Offset

1,2

Comments

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.

Equally, numbers k for which gcd(A064987(k), A191002(k)) is equal to A000265(gcd(A064987(k), A341529(k))).

Also odd numbers k for which A348993(k) = A319627(k).

Odd terms of A336702 are given by the intersection of this sequence and A349174.

Conjectures:

(1) After 1, all terms are multiples of 3. (Why?)

(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.

(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]

(4) After 1, all terms are in A248150.

(5) After 1, all terms are in A348748.

(6) Apart from 1, there are no common terms with A349753.

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.

Questions:

(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).

Links

Amiram Eldar, Table of n, a(n) for n = 1..63 (terms below 10^11)

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to sigma(n)

Formula

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

Mathematica

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 *)

Prog

(PARI)
A000265(n) = (n >> valuation(n, 2));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA349169(n) = { my(s=sigma(n), u=A003961(n)); (n*gcd(s, u) == A000265(s)*gcd(n, u)); }; \\ (Program simplified Nov 30 2021)

Crossrefs

Cf. A000203, A003961, A007949, A064989, A083207, A104210, A161942, A191002, A191218, A228058, A319627, A322361, A326134, A329963, A342671, A348748, A348990, A348992, A348993, A349162, A349164, A349174.

Cf. also A349745, A349746, A349753.

Subsequence of A349749.

Sequence in context: A319588 A344886 A165892 * A077261 A012507 A143727

Adjacent sequences: A349166 A349167 A349168 * A349170 A349171 A349172

Keyword

nonn

Author

Antti Karttunen, Nov 10 2021

Extensions

Name changed and comment section rewritten by Antti Karttunen, Nov 29 2021

Status

approved