OFFSET
1,2
COMMENTS
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.
From Jon E. Schoenfield, May 22 2021: (Start)
a(16) <= 13399626241.
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)
The sequence is infinite as for any n >= 0, A038507(n) starts a product crescendo of length >= n. - Rémy Sigrist, May 22 2021
EXAMPLE
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:
1 * 181 = 181
2 * 91 = 182
3 * 61 = 183
4 * 46 = 184
5 * 37 = 185
6 * 31 = 186
11 * 17 = 187
47 * 4 = 188
63 * 3 = 189
95 * 2 = 190
191 * 1 = 191
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).
Additional example: the crescendo from 2885373 (length 27) goes:
1 * 2885373 = 2885373
2 * 1442687 = 2885374
5 * 577075 = 2885375
6 * 480896 = 2885376
11 * 262307 = 2885377
19 * 151862 = 2885378
21 * 137399 = 2885379
89 * 32420 = 2885380
859 * 3359 = 2885381
1458 * 1979 = 2885382
4817 * 599 = 2885383
12437 * 232 = 2885384
19365 * 149 = 2885385
33551 * 86 = 2885386
93077 * 31 = 2885387
131154 * 22 = 2885388
221953 * 13 = 2885389
288539 * 10 = 2885390
320599 * 9 = 2885391
360674 * 8 = 2885392
412199 * 7 = 2885393
480899 * 6 = 2885394
577079 * 5 = 2885395
721349 * 4 = 2885396
961799 * 3 = 2885397
1442699 * 2 = 2885398
2885399 * 1 = 2885399
PROG
(PARI)
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))))}
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
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Jon Wild, May 20 2021
EXTENSIONS
a(13)-a(14) from Rémy Sigrist, May 21 2021
a(15) from Jon E. Schoenfield, May 21 2021
STATUS
approved