This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A342767 #15 Apr 08 2021 10:48:40 %S A342767 1,1,1,1,2,1,1,2,2,1,1,4,3,4,1,1,2,4,4,2,1,1,4,3,8,3,4,1,1,2,6,4,4,6, %T A342767 2,1,1,8,3,8,5,8,3,8,1,1,4,8,4,6,6,4,8,4,1,1,4,9,16,5,12,5,16,9,4,1,1, %U A342767 2,6,8,8,6,6,8,8,6,2,1,1,8,3,8,9,16,7,16,9,8,3,8,1 %N A342767 Array T(n, k), n, k > 0, read by antidiagonals; a variant of lunar multiplication (A087062) based on prime factorizations of numbers (see Comments section for precise definition). %C A342767 To compute T(n, k): %C A342767 - write the prime factors of n and of k in ascending order with multiplicities on two lines, right aligned, %C A342767 - to "multiply" two prime numbers: take the smallest, %C A342767 - to "add" two prime numbers: take the largest, %C A342767 - for example, for T(12, 14): %C A342767 12 -> 2 2 3 %C A342767 14 -> x 2 7 %C A342767 ------- %C A342767 2 2 3 %C A342767 + 2 2 2 %C A342767 --------- %C A342767 2 2 2 3 -> 24 = T(12, 14) %C A342767 This sequence is closely related to lunar multiplication (A087062): %C A342767 - let n and k be two p-smooth numbers, %C A342767 - let f be the function that associates to a p-smooth number, say m, the unique number whose (p+1)-base digits are prime, nondecreasing and whose product is m, %C A342767 - let g be the inverse of f, %C A342767 - then for any p-smooth numbers n and k, T(n, k) = g(f(n) "*" f(k)) where "*" denotes lunar product in base p+1, %C A342767 - as T(n, p) = n for any prime number >= A006530(n), we don't have prime numbers here, %C A342767 - however, if we consider only p-smooth numbers (for some prime number p), then p is the "unit" and the semiprimes p*q (with q <= p) are "prime". %H A342767 Rémy Sigrist, <a href="/A342767/b342767.txt">Table of n, a(n) for n = 1..10011</a> %H A342767 Rémy Sigrist, <a href="/A342767/a342767.gp.txt">PARI program for A342767</a> %H A342767 <a href="/index/Di#dismal">Index entries for sequences related to dismal (or lunar) arithmetic</a> %F A342767 T(n, k) = T(k, n). %F A342767 T(n, n) = A342768(n). %F A342767 T(n, 1) = 1. %F A342767 T(n, 2) = A061142(n). %F A342767 T(n, 3) = A079065(n). %F A342767 T(n, p) = n for any prime number p >= A006530(n). %e A342767 Array T(n, k) begins: %e A342767 n\k| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 %e A342767 ---+------------------------------------------------------ %e A342767 1| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 %e A342767 2| 1 2 2 4 2 4 2 8 4 4 2 8 2 4 -> A061142 %e A342767 3| 1 2 3 4 3 6 3 8 9 6 3 12 3 6 -> A079065 %e A342767 4| 1 4 4 8 4 8 4 16 8 8 4 16 4 8 %e A342767 5| 1 2 3 4 5 6 5 8 9 10 5 12 5 10 %e A342767 6| 1 4 6 8 6 12 6 16 18 12 6 24 6 12 %e A342767 7| 1 2 3 4 5 6 7 8 9 10 7 12 7 14 %e A342767 8| 1 8 8 16 8 16 8 32 16 16 8 32 8 16 %e A342767 9| 1 4 9 8 9 18 9 16 27 18 9 36 9 18 %e A342767 10| 1 4 6 8 10 12 10 16 18 20 10 24 10 20 %e A342767 11| 1 2 3 4 5 6 7 8 9 10 11 12 11 14 %e A342767 12| 1 8 12 16 12 24 12 32 36 24 12 48 12 24 %e A342767 13| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 %e A342767 14| 1 4 6 8 10 12 14 16 18 20 14 24 14 28 %o A342767 (PARI) See Links section. %Y A342767 Cf. A006530, A061142, A079065, A087062, A342765, A342768. %K A342767 nonn,tabl %O A342767 1,5 %A A342767 _Rémy Sigrist_, Apr 02 2021