Happy Anniversary Rule 30

(Image: By Richard Ling - Own work; Location: Cod Hole, Great Barrier Reef, Australia, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=293495)

This year my wife and I are celebrating our 40th anniversary. This year Stephen Wolfram also celebrates the 40th anniversary of his favorite scientific discovery as he wrote in My All-Time Favorite Science Discovery

Also, see a prior post -https://www.blogger.com/blog/post/edit/8711601538345502519/636715033944644578.

This discovery led to Wolfram's publication of A New Kind of Science (now available online).

A375508 Contribution to the OEIS

 

A375508 Begin A160649 with n instead of 2; a(n) is the position in the new sequence at which it generates the same numbers as A160649 or a(n)=0 if it doesn't. 0

1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 5, 4, 1, 3, 1, 1, 3, 2, 1, 2, 1, 1, 6, 2, 5, 1, 5, 4, 1, 1, 3, 3, 2, 2, 1, 1, 5, 1, 4, 3, 2, 1, 2, 2, 1, 1, 3, 2, 2, 5, 1, 1, 4, 2, 3, 1, 2, 1, 3, 2, 7, 1, 7, 6, 6, 5, 5, 1, 4, 3, 1, 1, 4, 1, 2, 3, 1, 1, 2, 9, 9, 8, 1, 8, 1, 7 (list; graph; refs; listen; history; edit; text; internal format)

OFFSET 1,4 COMMENTS The indices of the matching entries of A160649 and this sequence do not necessarily have to be the same (see Examples). LINKS James C. McMahon, Table of n, a(n) for n = 1..10000 Michael De Vlieger, Histogram of the frequency of a(n) = k, n = 1..2^20, k = 1..14. Maximum in the dataset is k = 41. Michael De Vlieger, Histogram of the frequency of a(n) = k, n = 1..2^24, k = 1..14. Maximum in the dataset is k = 57. Wikipedia, Kruskal count EXAMPLE Using () to indicate the point at which the new sequence generates the same numbers as A160649: A160649: 2, 3, 4, 6, 8, 11, 12... a(1)=1 Start=3: (3), 4, 6, 8, 11, 12... a(2)=1 Start=4: (4), 6, 8, 11, 12, 15... a(3)=1 Start=5: 5, (6), 8, 11, 12, 15... a(4)=2 MATHEMATICA Lim=88; pseq1=NestList[#+PrimeOmega[#]&, 2, Lim] (* pseq1 is base sequence A160649 *); pseq={}; Do[ i=1; s=n; While[!MemberQ[pseq1, s], s=s+PrimeOmega[s]; i++]; AppendTo[pseq, i], {n, 2, Lim}]; pseq (* pseq is A375508 *) CROSSREFS Cf. A160649. Sequence in context: A326194 A331251 A309858 * A022921 A080763 A245920 Adjacent sequences: A375505 A375506 A375507 * A375509 A375511 A375512 KEYWORD nonn,new AUTHOR James C. McMahon, Aug 18 2024 STATUS approved

Additional work on this sequence is given in the post: https://jamesmacmath.blogspot.com/2024/08/kruskal-count-with-prime-omega.html

A375606 Contribution to the OEIS

A375606 a(1) = 1; a(n+1) = a(n) + floor(harmonic mean of previous terms). 0

1, 2, 3, 4, 5, 7, 9, 11, 14, 17, 20, 23, 27, 31, 35, 40, 45, 50, 55, 61, 67, 73, 80, 87, 94, 102, 110, 118, 126, 135, 144, 153, 163, 173, 183, 193, 204, 215, 226, 238, 250, 262, 275, 288, 301, 314, 328, 342, 356, 371, 386, 401, 417, 433, 449, 465, 482, 499, 516 (list; graph; refs; listen; history; edit; text; internal format)

OFFSET 1,2 LINKS James C. McMahon, Table of n, a(n) for n = 1..10000 EXAMPLE a(6) = 5 + floor(harmonic mean(1,2,3,4,5)) = 5 + floor(300/377) = 7. MATHEMATICA Nest[Append[#, Last[#]+Floor@HarmonicMean[#]]&, {1}, 58] CROSSREFS Cf. A065094, A114829. Sequence in context: A274197 A286267 A337334 * A008750 A076677 A029001 Adjacent sequences: A375603 A375604 A375605 * A375607 A375608 A375609 KEYWORD nonn,new AUTHOR James C. McMahon, Aug 20 2024 STATUS approved