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 A356639 #51 Aug 01 2024 09:19:00
%S A356639 1,1,3,17,155,2677,73327,3578339,329652351
%N A356639 Number of integer sequences b with b(1) = 1, b(m) > 0 and b(m+1) - b(m) > 0, of length n which transform under the map S into a nonnegative integer sequence. The transform c = S(b) is defined by c(m) = Product_{k=1..m} b(k) / Product_{k=2..m} (b(k) - b(k-1)).
%C A356639 This sequence can be calculated by a recursive algorithm:
%C A356639 Let B1 be an array of finite length, the "1" denotes that it is the first generation. Let B1' be the reversed version of B1. Let C be the element-wise product C = B1 * B1'. Then B2 is a concatenation of taking each element of B1 and add all divisors of the corresponding element in C. If we start with B1 = {1} then we get this sequence of arrays: B2 = {2}, B3 = {3, 4, 6}, ... . a(n) is the length of the array Bn. In short the length of Bn+1 and so a(n+1) is the sum over A000005(Bn * Bn').
%C A356639 The transform used in the definition of this sequence is its own inverse, so if c = S(b) then b = S(c). The eigensequence is 2^n = S(2^n).
%C A356639 There exist some transformation pairs of infinite sequences in the database:
%C A356639 A000027 <--> A000142; A000079 <--> A000079; A000045 <--> A001654;
%C A356639 A026549 <--> A038754; A100071 <--> A001405; A058295 <--> A------;
%C A356639 A111286 <--> A098011; A093968 <--> A205825; A166447 <--> A------;
%C A356639 A098558 <--> A004277; A010551 <--> A087811; A100538 <--> A329227;
%C A356639 A079352 <--> A------; A082458 <--> A------; A008233 <--> A264635;
%C A356639 A138278 <--> A------; A006501 <--> A264557; A336496 <--> A------;
%C A356639 A019464 <--> A------; A062112 <--> A------; A171647 <--> A359039;
%C A356639 A279312 <--> A------; A031923 <--> A------.
%C A356639 These transformation pairs are conjectured:
%C A356639 A137326 <--> A------; A066332 <--> A300902; A208147 <--> A308546;
%C A356639 A057895 <--> A------; A349080 <--> A------; A019442 <--> A------;
%C A356639 A349079 <--> A------.
%C A356639 ("A------" means not yet in the database.)
%C A356639 Some sequences in the lists above may need offset adjustment to force a beginning with 1,2,... in the transformation.
%C A356639 If we allowed signed rational numbers, further interesting transformation pairs could be observed. For example, 1/n will transform into factorials with alternating sign. 2^(-n) transforms into ones with alternating sign and 1/A000045(n) into A000045 with alternating sign.
%e A356639 a(4) = 17. The 17 transformation pairs of length 4 are:
%e A356639 {1, 2, 3, 4} = S({1, 2, 6, 24}).
%e A356639 {1, 2, 3, 5} = S({1, 2, 6, 15}).
%e A356639 {1, 2, 3, 6} = S({1, 2, 6, 12}).
%e A356639 {1, 2, 3, 9} = S({1, 2, 6, 9}).
%e A356639 {1, 2, 3, 12} = S({1, 2, 6, 8}).
%e A356639 {1, 2, 3, 21} = S({1, 2, 6, 7}).
%e A356639 {1, 2, 4, 5} = S({1, 2, 4, 20}).
%e A356639 {1, 2, 4, 6} = S({1, 2, 4, 12}).
%e A356639 {1, 2, 4, 8} = S({1, 2, 4, 8}).
%e A356639 {1, 2, 4, 12} = S({1, 2, 4, 6}).
%e A356639 {1, 2, 4, 20} = S({1, 2, 4, 5}).
%e A356639 {1, 2, 6, 7} = S({1, 2, 3, 21}).
%e A356639 {1, 2, 6, 8} = S({1, 2, 3, 12}).
%e A356639 {1, 2, 6, 9} = S({1, 2, 3, 9}).
%e A356639 {1, 2, 6, 12} = S({1, 2, 3, 6}).
%e A356639 {1, 2, 6, 15} = S({1, 2, 3, 5}).
%e A356639 {1, 2, 6, 24} = S({1, 2, 3, 4}).
%e A356639 b(1) = 1 by definition, b(2) = 1+1 as 1 has only 1 as divisor.
%e A356639 a(3) = A000005(b(2)*b(2)) = 3.
%e A356639 The divisors of b(2) are 1,2,4. So b(3) can be b(2)+1, b(2)+2 and b(2)+4.
%e A356639 a(4) = A000005((b(2)+1)*(b(2)+4)) + A000005((b(2)+2)*(b(2)+2)) + A000005((b(2)+4)*(b(2)+1)) = 17.
%Y A356639 Cf. A000005.
%Y A356639 Cf. A000027, A000045, A000079, A000142, A001405.
%Y A356639 Cf. A001654, A004277, A006501, A008233, A019442.
%Y A356639 Cf. A010551, A019464, A026549, A031923, A038754.
%Y A356639 Cf. A057895, A058295, A062112, A066332, A100071.
%Y A356639 Cf. A079352, A082458, A087811, A093968, A098011.
%Y A356639 Cf. A098558, A100538, A111286, A166447, A137326.
%Y A356639 Cf. A138278, A171647, A205825, A208147, A264557.
%Y A356639 Cf. A264635, A308546, A329227, A336496, A349079.
%Y A356639 Cf. A349080, A359039.
%K A356639 more,nonn
%O A356639 1,3
%A A356639 _Thomas Scheuerle_, Aug 19 2022