

A303645


a(n) is the smallest number not yet seen such that sopfr(a(n)) is the least possible integer.


1



1, 2, 3, 4, 5, 6, 8, 9, 7, 10, 12, 15, 16, 18, 14, 20, 24, 27, 21, 25, 30, 32, 36, 11, 28, 40, 45, 48, 54, 35, 42, 50, 60, 64, 72, 81, 13, 22, 56, 63, 75, 80, 90, 96, 108, 33, 49, 70, 84, 100, 120, 128, 135, 144, 162, 26, 44, 105, 112, 125, 126, 150, 160, 180, 192, 216, 243, 39, 55, 66, 98, 140, 168, 189
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OFFSET

1,2


COMMENTS

The sequence is a permutation of the positive integers, listing in increasing order elements of the finite sets S(k) = {x: sopfr(x)=k}, k >= 0, where sopfr(x) = 0 iff x = 1. When a(n) = A056240(k) for some k >= 2, then sopfr(a(n)) = k and a(n) is the first of A000607(k) terms, all of which have sopfr = k. (A000607(k) is the number of partitions of k into prime parts.) Consequently the sequence follows a sawtooth profile, rising from a(n) = A056240(k) to A000792(k), the greatest number with sopfr = k, then starting over with A056240(k+1) for the next larger value of sopfr. [Edited by M. F. Hasler, Jan 19 2019]


LINKS

David A. Corneth, Table of n, a(n) for n = 1..10077 (The bfile was updated Dec 14 2018 and is correct.  N. J. A. Sloane, Jan 19 2019)


FORMULA

If a(n) = A056240(k) for some k then a(n+A000607(k)1) = A000792(k).


EXAMPLE

S(0) = {1}, S(1) = {}, S(2) = {2}, S(3) = {3}, S(4) = {4}, S(5) = {5, 6},
S(6) = {8, 9}, S(7) = {7, 10, 12}, etc. Therefore the sequence begins:
1, 2, 3, 4, 5, 6, 8, 9, 7, 10, 12, .... [Edited by M. F. Hasler, Jan 19 2019]


PROG

(PARI) lista(nn) = {nmax = A000792(nn); v = vector(nmax, k, A001414(k)); for (n=1, nn, vn = select(x>x==n, v, 1); for (k = 1, #vn, print1(vn[k], ", ")))} \\ Michel Marcus, May 01 2018
(PARI) A303645_vec(N, k=6, L=9)={vector(N, i, if(i<7, N=i, until(A001414(N+=1)==k, ); N<L, N, k++; L=3^((k2)\3)*(2+(k2)%3); N+0*N=A056240(k)1))} \\ To compute terms up to a given value of k=sopfr(n) and/or for large N >> 1000, it is more efficient to use code similar to lista() above, with "for(k...)" replaced by "a=concat(a, vn)".  M. F. Hasler, Jan 19 2019


CROSSREFS

Cf. A056240, A000607, A000792, A001414 (sopfr), A064364.
Sequence in context: A265552 A303936 A064364 * A269855 A209274 A303595
Adjacent sequences: A303642 A303643 A303644 * A303646 A303647 A303648


KEYWORD

nonn


AUTHOR

David James Sycamore, Apr 27 2018


EXTENSIONS

Prepended 1.  N. J. A. Sloane, Dec 14 2018
Edited and bfile doublechecked by M. F. Hasler, Jan 19 2019


STATUS

approved



