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A026477 a(1) = 1, a(2) = 2, a(3) = 3; and for n > 3, a(n) = smallest number > a(n-1) and not of the form a(i)*a(j)*a(k) for 1 <= i < j < k < n. 22
1, 2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 120, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 168, 169, 173, 179, 181, 191, 193, 197, 199, 210, 211, 216, 223 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

From Bob Selcoe, Aug 25 2016: (Start)

Once a term with a given prime signature S (i.e., multiset of prime exponents) appears, then all numbers with the same prime signature follow. So either all or no terms with the same prime signature appear (first conjectured by Charles R Greathouse IV).

Proof:

i. Let S = {i,j,k..,y}, i>=j>=k.. be prime signatures, i.e., numbers of the form (p^i*q^j*r^k..*z^y) where {p,q,r,..,z} are distinct primes; denote S = {} as the signature for p^0 = 1.

ii. By definition, 1 and all primes p appear in the sequence; so terms where S = {k} (i.e., one-digit signatures denoting prime powers p^k) are k=0 and k = A026474(n) = {1,2,4,8,15,22,29..}, because k cannot be the sum of any combination of 3 smaller k. So the only prime power terms are p^0 = 1 and p^A026474(n), or S = {}, {1}, {2}, {4}, {8}, {15}, {22}, {29}...

iii. By induction, once S = {k} is determined, all other terms with the same prime signature appear (or do not) depending on the various combinations of signature exponents of previous terms with smaller signature sums. So for example, terms with the same two-digit signature S = {k,k} (i.e. (pq)^k)) are constrained by the following: since p = {1} and q = {1} appear, then (pq) = {1,1} does not because in this case {1}*{1}*{} = {1,1}; since p^2 = {2} and q^2 = {2} appear, then (pq)^2 = {2,2} does not; since {4} appears but {3} does not, then {3,3} appears but {4,4} and {5,5} do not (the latter because {3,3}*{2}*{2} = {5,5} when p^2,q^2 = {2}). Note that {3,3} would not appear if {3,2} appeared because (3,2}*{1}*{} = {3,3} when q = {1}; but because p = {1} and p^2,q^2 = {2} appear, then {2}*{2}*{1} = {3,2} does not. {6,6} does not appear because {6,5} appears (by virtue of other constraints) and {6,5}*{1}*{} = {6,6} when q = {1}. Determining which signatures appear and which do not becomes increasingly complicated as the sequence increases.

Don Reble offered a proof on the Sequence Fans Mailing List which seems to be different (and certainly more formal) than mine. Perhaps mine is more of an "explanation" than a "proof"? (End)

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

MATHEMATICA

a = {1, 2, 3}; no = {1 2 3};

Do[x = SelectFirst[Range[Last[a] + 1, 1000], ! MemberQ[no, #] &]; AppendTo[a, x]; no = Union[Times @@@ Subsets[a, {3}]], 200]; a (* Robert Price, May 26 2019 *)

PROG

(PARI) list(lim)=my(v=List(), n, d, k); while(n++<=lim, d=divisors(n); for(i=1, #d-2, if(!setsearch(v, d[i]), next); for(j=i+1, #d-1, if(!setsearch(v, d[j]), next); k=n/(d[i]*d[j]); if(d[j]>=k, break); if(denominator(k)==1 && setsearch(v, k), next(3)))); listput(v, n)); Vec(v) \\ Charles R Greathouse IV, Sep 16 2015

CROSSREFS

Cf. A000028, A026416, A066724, A050376, A084400, A026474.

There are six related sequences: A026477: 1 <= i <  j <  k < n  starting 1,2,3; A026478: 1 <= i <= j <= k < n  starting 1,2,3; A026479: 1 <= i <  j <  k < n  starting 1,2,4; A026480: 1 <= i <= j <= k < n  starting 1,2,4; A026481: 1 <= i <  j <  k < n  starting 1,3,4; A026482: 1 <= i <= j <= k < n  starting 1,3,4.

Sequence in context: A066724 A089237 A009087 * A079852 A084400 A050376

Adjacent sequences:  A026474 A026475 A026476 * A026478 A026479 A026480

KEYWORD

nonn

AUTHOR

Clark Kimberling

EXTENSIONS

More terms from Christian G. Bower, Nov 15 1999

STATUS

approved

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Last modified September 27 10:35 EDT 2020. Contains 337380 sequences. (Running on oeis4.)