A344488 Numbers that start a product crescendo of record length.

1, 2, 3, 7, 47, 181, 1307, 2503, 40973, 46833, 109177, 2885373, 11744311, 192968969, 899988745

Offset: 1

Author

Jon Wild, May 20 2021

Keywords

• nonn
• hard
• more

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

Examples

			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
		

Crossrefs

Cf. A038507.

Programs

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

Extensions

a(13)-a(14) from Rémy Sigrist, May 21 2021

a(15) from Jon E. Schoenfield, May 21 2021