Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #36 May 23 2021 03:04:52
%S 1,2,3,7,47,181,1307,2503,40973,46833,109177,2885373,11744311,
%T 192968969,899988745
%N Numbers that start a product crescendo of record length.
%C A product crescendo is a sequence of successive integers that can be written as products j * k where the j's form a strictly increasing sequence and the k's form a strictly decreasing sequence.
%C From _Jon E. Schoenfield_, May 22 2021: (Start)
%C a(16) <= 13399626241.
%C Numbers that start long product crescendos, but are not necessarily of record length, are easy to find by testing numbers of the form 1 + m*lcm(1..k) for sufficiently large m and k. E.g., the ones that start at 13399626241 = 1 + 18592*lcm(1..16), 442452890881 = 1 + 36112*lcm(1..17), and 521688126961 = 1 + 2241*lcm(1..19) have lengths 37, 39, and 41 respectively. (End)
%C The sequence is infinite as for any n >= 0, A038507(n) starts a product crescendo of length >= n. - _Rémy Sigrist_, May 22 2021
%e 181 is in the list because it begins a product crescendo that is longer than any beginning at any smaller number. Here is the crescendo:
%e 1 * 181 = 181
%e 2 * 91 = 182
%e 3 * 61 = 183
%e 4 * 46 = 184
%e 5 * 37 = 185
%e 6 * 31 = 186
%e 11 * 17 = 187
%e 47 * 4 = 188
%e 63 * 3 = 189
%e 95 * 2 = 190
%e 191 * 1 = 191
%e This set of 11 products forms a longer crescendo than the previous record (which started at 47), and is the longest until the set of 13 products it is possible to write starting from 1307 (the next entry in the sequence).
%e Additional example: the crescendo from 2885373 (length 27) goes:
%e 1 * 2885373 = 2885373
%e 2 * 1442687 = 2885374
%e 5 * 577075 = 2885375
%e 6 * 480896 = 2885376
%e 11 * 262307 = 2885377
%e 19 * 151862 = 2885378
%e 21 * 137399 = 2885379
%e 89 * 32420 = 2885380
%e 859 * 3359 = 2885381
%e 1458 * 1979 = 2885382
%e 4817 * 599 = 2885383
%e 12437 * 232 = 2885384
%e 19365 * 149 = 2885385
%e 33551 * 86 = 2885386
%e 93077 * 31 = 2885387
%e 131154 * 22 = 2885388
%e 221953 * 13 = 2885389
%e 288539 * 10 = 2885390
%e 320599 * 9 = 2885391
%e 360674 * 8 = 2885392
%e 412199 * 7 = 2885393
%e 480899 * 6 = 2885394
%e 577079 * 5 = 2885395
%e 721349 * 4 = 2885396
%e 961799 * 3 = 2885397
%e 1442699 * 2 = 2885398
%e 2885399 * 1 = 2885399
%o (PARI)
%o b(n)={if(n==1, 1, my(m=1); for(k=1, oo, fordiv(n+k, d, if(d>m, m=d; break)); if(m==n+k, return(k+1))))}
%o lista(lim)={my(m=0); for(n=1, lim, my(t=b(n)); if(t > m, print1(n, ", "); m=t))} \\ _Andrew Howroyd_, May 21 2021
%Y Cf. A038507.
%K nonn,hard,more
%O 1,2
%A _Jon Wild_, May 20 2021
%E a(13)-a(14) from _Rémy Sigrist_, May 21 2021
%E a(15) from _Jon E. Schoenfield_, May 21 2021