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 A344488 Numbers that start a product crescendo of record length. 0
 1, 2, 3, 7, 47, 181, 1307, 2503, 40973, 46833, 109177, 2885373, 11744311, 192968969, 899988745 (list; graph; refs; listen; history; text; internal format)
 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 LINKS Table of n, a(n) for n=1..15. 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 Cf. A038507. Sequence in context: A030090 A260298 A283807 * A257108 A274385 A075461 Adjacent sequences: A344485 A344486 A344487 * A344489 A344490 A344491 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

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Last modified August 15 12:24 EDT 2024. Contains 375173 sequences. (Running on oeis4.)