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A307720 Lexicographically earliest sequence of positive integers in which, for all positive k, there are exactly k contiguous pairs whose product is k. 9
1, 1, 2, 1, 3, 1, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 2, 4, 2, 4, 2, 4, 2, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 5, 1, 5, 1, 5, 1, 7, 1, 7, 1, 7, 1, 7, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 7, 3, 7, 3, 7, 3, 7, 3, 7, 3, 7, 3, 7, 3, 7, 3, 7, 3, 7, 3, 8, 2, 8 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

All natural integers will appear sooner or later in the sequence - but mostly “later”! Indeed, the sequence increases very slowly: after 100000 terms the smallest term not yet present is 32.

Here is, in the same range, a sample of the count {term, occurrences} so far:

{1,192},{2,396},{3,618},{4,796},{5,1160},{6,1296},{7,2294},{8,2080},{9,2489},{10,2826},{11,3487},{12,1596},{13,2295},{14,1960},{15,2370},{16,2640},{17,4097},{18,2214},{19,4598},{20,2770},{21,3759},{22,4477},{23,5612},{24,4884},{25,5825},{26,6006},{27,6359},{28,4676},{29,5481},{30,3060},{31,1411},{32,0},{33,182},{34,0},{35,315},{36,0},{37,1221},{38,0},{39,214},{40,0},{41,1353},{42,0},{43,1183},{44,0},{45,0},{46,0},{47,1058},{48,0},{49,172},{50,0},{51,0},{52,0},{53,580}...

After 100000 terms, the first products that are not yet present are (the primes): 59, 61, 67, 71, 73, 79, 83, 89, 97, 101,... and (the composites) 118, 122, 134,...

Here is again a sample so far (100000 terms computed) of {product, number of occurrences of the product}:

{1,1},{2,2},{3,3},{4,4},{5,5},{6,6},{7,7},{8,8},{9,9},{10,10},{11,11},{12,12},{13,13},{14,14},{15,15},{16,16},{17,17},{18,18},{19,19},{20,20},{21,21},{22,22},{23,23},{24,24},{25,25},{26,26},{27,27},{28,28},{29,29},{30,30},{31,31},{32,32},{33,33},{34,34},{35,35},{36,36},{37,37},{38,38},{39,39},{40,40},{41,41},{42,42},{43,43},{44,44},{45,45},{46,46},{47,47},{48,48},{49,49},{50,50},{51,51},{52,52},{53,53},{54,54},{55,55},{56,56},{57,57},{58,58},{59,0},{60,60},{61,0},{62,62},{63,63},{64,64},{65,65},{66,66},{67,0},{68,68},{69,69},{70,70},{71,0},{72,72},{73,0},{74,74},{75,75},{76,76},{77,77},{78,78},{79,0},{80,80},{81,81},{82,82},{83,0},{84,84},{85,85},{86,86},{87,87},{88,88},{89,0},{90,90},{91,91},{92,92},{93,93},{94,94},{95,95},{96,96},{97,0},{98,98},{99,99},{100,100},{101,0},{102,102},{103,0},{104,104},{105,105},{106,106},{107,0},{108,108},{109,0},{110,110},{111,111},{112,112},{113,0},{114,114},{115,115},{116,116},{117,117},{118,0},{119,119},{120,120},{121,121},{122,0},{123,123},{124,124},{125,125},{126,126},{127,0},{128,128},{129,129},{130,130},{131,0},{132,132},{133,133},{134,0},{135,135},{136,136},{137,0},{138,138},{139,0},{140,140},{141,141},{142,0},...

LINKS

Jean-Marc Falcoz, Table of n, a(n) for n = 1..28446

Rémy Sigrist, Scatterplot of the first 10000000 terms

Rémy Sigrist, PARI program for A307720

EXAMPLE

The sequence starts with 1,1,2,1,3,1,3,2,2,2,2,2,3,...

a(1) = 1 means that the product [a(n)*a(n+1)] = 1 is true only once [this is here the product a(1) * a(2) = 1 * 1 = 1];

a(3) = 2 means that the product [a(n)*a(n+1)] = 2 is true only twice [those are here the products a(2) * a(3) = 1 * 2 = 2, and a(3) * a(4) = 2 * 1 = 2];

a(5) = 3 means that the product [a(n)*a(n+1)] = 3 is true only three times [those are the products a(4) * a(5) = 1 * 3 = 3 ; a(5) * a(6) = 3 * 1 = 3 and a(6) * a(7) = 1 * 3 = 3];

...

a(27) = 4 means that the product [a(n)*a(n+1)] = 4 is true only four times [those are the products a(8) * a(9) = 2 * 2 = 4 ; a(9) * a(10) = 2 * 2 = 4 ; a(10) * a(11) = 2 * 2 = 4 ; a(11) * a(12) = 2 * 2 = 4] ; etc.

PROG

(PARI) See Links section.

CROSSREFS

Cf. A307707 (same idea, but with the sum of contiguous terms instead of the product), A307730 (the products), A307630 (when n appears), A307631 (indices of records), A307632 (indices of primes) A307633 and A307634 (RUNS transforms of bisections).

See also A307747.

Sequence in context: A249617 A304091 A278801 * A191350 A304795 A036459

Adjacent sequences:  A307717 A307718 A307719 * A307721 A307722 A307723

KEYWORD

nonn,look,nice

AUTHOR

Eric Angelini and Jean-Marc Falcoz, Apr 24 2019

EXTENSIONS

Definition revised slightly by Allan C. Wechsler, Apr 24 2019

STATUS

approved

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Last modified August 23 01:10 EDT 2019. Contains 326211 sequences. (Running on oeis4.)