cp's OEIS Frontend

Meyers (see Guy reference) conjectures that for all r >= 1, the least odd number not in the set {a(i): i < prime(r)} is prime(r+1). - N. J. A. Sloane, Jan 08 2021

Meyers' conjecture would be refuted if and only if for some r there were such a large gap between prime(r) and prime(r+1) that there existed a composite c for which prime(r) < c < a(c) < prime(r+1), in which case (by Bertrand's postulate) c would necessarily be a term of A246281. - Antti Karttunen, Mar 29 2021

a(n) is odd for all n and for each odd m there exists a k with a(k) = m (see A064216). a(n) > n for n > 1: bijection between the odd and all numbers. - Reinhard Zumkeller, Sep 26 2001

a(n) and n have the same number of distinct primes with (A001222) and without multiplicity (A001221). - Michel Marcus, Jun 13 2014

From Antti Karttunen, Nov 01 2019: (Start)

More generally, a(n) has the same prime signature as n, A046523(a(n)) = A046523(n). Also A246277(a(n)) = A246277(n) and A287170(a(n)) = A287170(n).

Many permutations and other sequences that employ prime factorization of n to encode either polynomials, partitions (via Heinz numbers) or multisets in general can be easily defined by using this sequence as one of their constituent functions. See the last line in the Crossrefs section for examples.

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The positive integers are partitioned between this sequence, A332821 and A332822, which list the integers in respective cosets of the subgroup.

As the sequence lists the integers in a multiplicative subgroup of the positive rationals, the sequence is closed under multiplication and, provided the result is an integer, under division.

It follows that for any n in this sequence, all powers n^k are present (k >= 0), as are all cubes.

If we take each odd term of this sequence and replace each prime in its factorization by the next smaller prime, the resulting numbers are a permutation of the full sequence; and if we take the square root of each square term we get the full sequence.

There are no primes in the sequence, therefore if k is present and p is a prime, k*p and k/p are absent (noting that k/p might not be an integer). This property extends from primes to all terms of A050376 (often called Fermi-Dirac primes), therefore to squares of primes, 4th powers of primes etc.

The terms are the even numbers in A332821 halved. The terms are also the numbers m such that 5m is in A332821, and so on for alternate primes: 11, 17, 23 etc. Likewise, the terms are the numbers m such that 3m is in A332822, and so on for alternate primes: 7, 13, 19 etc.

The numbers that are half of the even terms of this sequence are in A332822, which consists exactly of those numbers. The numbers that are one third of the terms that are multiples of 3 are in A332821, which consists exactly of those numbers. These properties extend in a pattern of alternating primes as described in the previous paragraph.

If k is an even number, exactly one of {k/2, k, 2k} is in the sequence (cf. A191257 / A067368 / A213258); and generally if k is a multiple of a prime p, exactly one of {k/p, k, k*p} is in the sequence.

If m and n are in this sequence then so is m*n (the definition of "multiplicative semigroup"), while if n is in this sequence, and x is in the complement A359830, then n*x is in A359830. This essentially follows from the fact that A048675 is totally additive sequence. Compare to A329609. - Antti Karttunen, Jan 17 2023

For any number n > 0, let f(n) be the polynomial in a single indeterminate x where the coefficient of x^e is the prime(1+e)-adic valuation of n (where prime(k) denotes the k-th prime); f establishes a bijection between the positive numbers and the polynomials in a single indeterminate x with nonnegative integer coefficients; let g be the inverse of f; T(n, k) = g(f(n) * f(k)).

This table has many similarities with A248601.

For any n > 0 and m > 0, f(n * m) = f(n) + f(m).

Also, f(1) = 0 and f(2) = 1.

The function f can be naturally extended to the set of positive rational numbers: if r = u/v (not necessarily in reduced form), then f(r) = f(u) - f(v); as such, f is a homomorphism from the multiplicative group of positive rational numbers to the additive group of polynomials of a single indeterminate x with integer coefficients.

See A297473 for the main diagonal of T.

As a binary operation, T(.,.) is related to A306697(.,.) and A329329(.,.). When their operands are terms of A050376 (sometimes called Fermi-Dirac primes) the three operations give the same result. However the rest of the multiplication table for T(.,.) can be derived from these results because T(.,.) distributes over integer multiplication (A003991), whereas for A306697 and A329329, the equivalent derivation uses distribution over A059896(.,.) and A059897(.,.) respectively. - Peter Munn, Mar 25 2020

From Peter Munn, Jun 16 2021: (Start)

The operation defined by this sequence can be extended to be the multiplicative operator of a ring over the positive rationals that is isomorphic to the polynomial ring Z[x]. The extended function f (described in the author's original comments) is the isomorphism we use, and it has the same relationship with the extended operation that exists between their unextended equivalents.

Denoting this extension of T(.,.) as t_Q(.,.), we get t_Q(n, 1/k) = t_Q(1/n, k) = 1/T(n, k) and t_Q(1/n, 1/k) = T(n, k) for positive integers n and k. The result for other rationals is derived from the distributive property: t_Q(q, r*s) = t_Q(q, r) * t_Q(q, s); t_Q(q*r, s) = t_Q(q, s) * t_Q(r, s). This may look unusual because standard multiplication of rational numbers takes on the role of the ring's additive group.

There are many OEIS sequences that can be shown to be a list of the integers in an ideal of this ring. See the cross-references.

There are some completely additive sequences that similarly define by extension completely additive functions on the positive rationals that can be shown to be homomorphisms from this ring onto the integer ring Z, and these functions relate to some of the ideals. For example, the extended function of A048675, denoted A048675_Q, maps i/j to A048675(i) - A048675(j) for positive integers i and j. For any positive integer k, the set {r rational > 0 : k divides A048675_Q(r)} forms an ideal of the ring; for k=2 and k=3 the integers in this ideal are listed in A003159 and A332820 respectively.

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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. (Cf. also A386430).

(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).

Numbers k for which k * A342671(k) = A000203(k) * A322361(k).

Numbers k such that gcd(A064987(k), A191002(k)) = gcd(A064987(k), A341529(k)).

Obviously, all odd terms in this sequence must be squares.

All the terms k of A005820 that satisfy A007949(k) < A007814(k) [i.e., whose 3-adic valuation is strictly less than their 2-adic valuation] are also terms of this sequence. Incidentally, the first six known terms of A005820 satisfy this condition, while on the other hand, any hypothetical odd 3-perfect number would be excluded from this sequence. Also, as a corollary, any hypothetical 3-perfect numbers of the form 4u+2 must not be multiples of 3 if they are to appear here. Similarly for any k which occurs in A349169, for 2*k to occur in this sequence, it shouldn't be a multiple of 3 and k should also be a term of A191218. See question 2 and its partial answer in A349169.

From Antti Karttunen, Feb 13-20 2022: (Start)

Question: Are all terms/2 (A351548) abundant, from n > 1 onward?

Note that of the 65 known 5-multiperfect numbers, all others except these three 1245087725796543283200, 1940351499647188992000, 4010059765937523916800 are also included in this sequence. The three exceptions are distinguished by the fact that their 3 and 5-adic valuations are equal. In 62 others the former is larger.

If k satisfying the condition were of the form 4u+2, then it should be one of the terms of A191218 doubled as only then both k and sigma(k) are of the form 4u+2, with equal 2-adic valuations for both. More precisely, one of the terms of A351538.

(End)

a(n) is the smallest positive k such that A191002(k) is a multiple of n.