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A285893
Least number to start a run of exactly n nondecreasing values of sigma (sum of divisors, A000203).
7
45, 5, 313, 1, 356067536821, 36721681
OFFSET
1,1
COMMENTS
a(6) = 36721681, see also A028965.
The analogous sequence based on tau = A000005 instead of sigma is A284597.
EXAMPLE
We have the following values of sigma for n = 1..10:
n 1 2 3 4 5 6 7 8 9 10 ...
sigma(n) 0 1 1 2 1 2 1 3 2 2 ...
We see a run of 4 nondecreasing values starting at 1, ending at 4, therefore a(4) = 1. There is a run of 2 nondecreasing values starting at 5, ending at 6, therefore a(2) = 5.
Correspondingly, a run of length 1 corresponds to a number n such that sigma(n-1) > sigma(n) > sigma(n+1). This happens first at a(1) = 45.
MATHEMATICA
Function[s, {45}~Join~Map[Function[r, Select[s, Last@ # == r &][[1, 1]]], Range[2, Max[s[[All, -1]] ] ]]]@ Map[{#[[1, 1]], Length@ # + 1} &, DeleteCases[SplitBy[#, #[[-1]] >= 0 &], k_ /; k[[1, -1]] < 0]] &@ MapIndexed[{First@ #2, #1} &, Differences@ Array[DivisorSigma[1, #] &, 10^6]] (* Michael De Vlieger, May 06 2017 *)
PROG
(PARI) alias(A, A285893); A=vector(19); apply(scan(N, s=1, t=sigma(s))=for(k=s+1, N, t>(t=sigma(k))||next; k-s>#A||A[k-s]||printf("a(%d)=%d, ", k-s, s)||A[k-s]=s; s=k); done, [10^8]) \\ Search may be extended using scan(END, START).
CROSSREFS
Cf. A000005, A000203, A028965, A284597 (analog for sigma_0), A286287 (analog for omega = A001221), A286288 (analog for bigomega = A001222).
Sequence in context: A131994 A037940 A037218 * A070818 A129536 A108909
KEYWORD
nonn,more,hard
AUTHOR
M. F. Hasler, May 06 2017
EXTENSIONS
a(5)-a(6) from Giovanni Resta, May 07 2017
STATUS
approved