A141447 a(0)=0; for n >= 1, a(n) is the least positive integer that cannot be written in the form Sum_{k>=1} s(k)*k!, with s(k) in {a(0), a(1), ..., a(n-1)}.

0, 1, 4, 5, 22, 23, 82, 83, 466, 467, 478, 479, 1090486, 1090487, 1090774, 1090775, 1090846, 1090847, 1199566, 1199567

Offset

0,3

Comments

This sequence was suggested by Kevin O'Bryant, who provided terms a(0)-a(11); a(12)-a(17) were from David S. Newman.

The numbers always come in pairs, the first even, then the following odd number.

a(20) > 10^10. - Andrew Howroyd, Mar 02 2023

Links

Sidney Cadot, Table of n, a(n) for n = 0..19

Example

When a(0) = 0 and a(1) = 1, the numbers from 1 to 3 may be written 1 = 1*1!, 2 = 1*2!, 3 = 1*1! + 1*2!, but there is no way to write 4 as a sum having the required form, so a(2) = 4.

Prog

(Python)
def make_reachable(coefficients, factorials, max_k):
    reachable = set([0])
    for factorial in factorials:
        reachable_extra = set(r + coefficient * factorial for r in reachable for coefficient in coefficients if r + coefficient * factorial <= max_k)
        reachable.update(reachable_extra)
    return reachable
def generate_sequence_elements(num_elements):
    m = [0]
    factorials = [1]
    k = 1
    max_k = 1
    reachable = make_reachable(m, factorials, max_k)
    while len(m) < num_elements:
        if k > max_k:
            max_k *= 2
            reachable = make_reachable(m, factorials, max_k)
        if factorials[-1] * (len(factorials) + 1) <= k:
            factorials.append(factorials[-1] * (len(factorials) + 1))
            reachable = make_reachable(m, factorials, max_k)
        if k not in reachable:
            m.append(k)
            reachable = make_reachable(m, factorials, max_k)
        k += 1
    return m
print(generate_sequence_elements(20)) # Sidney Cadot, Mar 01 2023
(PARI)
Fact(lim)={my(k=1); while(k!<=lim, k++); vector(k-1, i, i!)}
Scan(lim, F, S)=my(b=2^lim); for(i=1, #F, my(f=F[i], t=0); for(j=1, #S, my(s=f*S[j]); if(s && s<=lim, t=bitor(t, b>>s))); b=bitor(b, t)); b=2^(lim+1)-1-b; if(!b, 0, lim-logint(b, 2))
upto(lim) = {my(S=[0], t=0, b=0); while(t || b<lim, if(t, S=concat(S, t), b=min(lim, 2*b+1)); t=Scan(b, Fact(b), S)); S}
{ upto(1200000) } \\ Andrew Howroyd, Mar 02 2023

Crossrefs

Sequence in context: A350827 A284911 A091130 * A129346 A291670 A176957

Adjacent sequences: A141444 A141445 A141446 * A141448 A141449 A141450

Keyword

nonn,more

Author

David S. Newman, Dec 16 2010

Extensions

a(18)-a(19) from Sidney Cadot, Mar 01 2023

Status

approved