User:M. F. Hasler/Work in progress/Factorization of A173426 = 123...321

This page is about formulae and factorizations concerning sequence A173426 = concatenation(1, 2, 3, ..., n, n-1, ..., 1).

Formulae

Using summation in decimal length clades, one can obtain analytical expressions for the terms a(n)=A173426(n):

a(n) = A002275(n)^2, for 1 <= n < 10. Here A002275(n)=(10^n-1)/9 (repunits 1....1).

a(n) = (120999998998*10^(4*n-28) - 2*10^(2*n-9) + 879*10^10 + 121)/99^2, for 10 <= n < 10^2;

a(n) = (120999998998*10^(6*n-227) - (1099022*10^(6*n-406) + 242*10^(3*n-108) - 1087789*10^191)/111^2 + 8790000000121)/99^2, for 10^2 <= n < 10^3; etc. - Serge Batalov, Jul 29 2015

Factorizations

The motivation for work on this sequence lies in the fact that until 2015, a(10) was the only known prime in the sequence. On July 20, 2015, S.S.Gupta announced that he found a(2446) was again a probable prime. Serge Batalov checked that there are no other (PR)prime members in this sequence for n < 50000 and, heuristically, there might be not more than these two primes in the sequence.

Related sequences: A075023 = smallest prime factor, A075024 = greatest prime factor,

A260587 = ω (omega = distinct prime factors), A260588 = Ω (bigomega = prime factors with repetition),

A260589 = table of prime factors, A260597 = primes in the order they occur for the first time as a factor.

From the formulas we see that a(1..9) are squares, correspondingly Ω(a(n)) &geq; 2 ω(a(n)), with equality except for n=9 where a(n) has a factor 3^4. For larger n this is no more true.

The main data is in A260589 = table of prime factors, written here as valid mathematical expressions (product of powers):

   n  | ω |  Ω | factorization
   1  | 0 |  0 | (1)
   2  | 1 |  2 | 11^2
   3  | 2 |  4 | 3^2 * 37^2
   4  | 2 |  4 | 11^2 * 101^2
   5  | 2 |  4 | 41^2 * 271^2
   6  | 5 | 10 | 3^2 * 7^2 * 11^2 * 13^2 * 37^2
   7  | 2 |  4 | 239^2 * 4649^2
   8  | 4 |  8 | 11^2 * 73^2 * 101^2 * 137^2
   9  | 3 |  8 | 3^4 * 37^2 * 333667^2
  10  | 1 |  1 | 12345678910987654321
  11  | 2 |  2 | 7 * 17636684157301569664903
  12  | 3 |  5 | 3^2 * 7^2 * 2799473675762179389994681
  13  | 3 |  3 | 1109 * 4729 * 2354041513534224607850261
  14  | 6 |  6 | 7 * 571 * 3167 * 10723 * 439781 * 2068140300159522133
  15  | 6 |  7 | 3 * 3 * 7 * 75401 * 687437 * 759077450603 * 498056174529497 
  16  | 4 |  4 | 71 * 18427 * 84072523705132513 * 112240064764214229701 
  17  | 4 |  5 | 7^2 * 31 * 19405011688367 * 4188353169004802474320231191377 
  18  | 4 |  8 | 3^5 * 7 * 8087 * 89747677939149063905891825989378400337330283
  19  | 6 |  6 | 251 * 281 * 5519 * 96601 * 3234653 * 101499808219558125890487026338898113
  20  | 2 |  2 | 7 * 17636684157301733059308816884574168816593059017301569664903 
  21  | 5 |  6 | 3^2 * 7 * 828703 * 94364768151913037621 * 250591098443370396365457961250972909
  22  | 3 |  3 | 70607 * 2070589723 * 84444849128440941966622214924855780389173179337962861
  23  | 4 |  4 | 7 * 15913 * 26474626905689 * 41863450981885380285553603078024640631870627809294279
  24  | 8 |  9 | 3^2 * 7 * 659 * 56383 * 860297 * 1593071 * 3099587
      |   |    | * 124151926361487052832634228773669372337884921119
  25  | 2 |  2 | 989931671244066864878631629 * 12471243489559387823527232424981012432152516319410549
  26  | 6 |  6 | 7 * 3209 * 17627 * 1322221 * 554840431325362973971
      |   |    | * 425007569210708911287454339430638873033275602731
  27  | 8 | 11 | 3^4 * 7 * 223 * 28807 * 108727 * 5439394515032275997
      |   |    | * 361855463775135800641 * 1583820370053082897093702337733277
  28  | 2 |  2 | 149 * 82856905436988007661182361202698806929028608914581377465249745095179303029618262489850029 
  29  | 4 |  4 | 7 * 317310923 * 296879723071339
      |   |    | * 187219628406423274641796678374294157135482589263663245469064512096386399
  30  | 9 | 10 | 3 * 3 * 7 * 167 * 761 * 133337 * 431911 * 273884231501 * 4950715302671
      |   |    | * 1974664753038296742475217254720689919163714869541093307653
  31  | 4 |  4 | 827 * 1141296551 * 10940622359204560200188943089306257
      |   |    | * 1195553280682910880337446470831673384752842314104497852789
  32  | 9 |  9 | 7 * 31 * 5537737 * 42583813 * 62231909 * 19871693507 * 1441602757913
      |   |    | * 15884064847039967 * 85196766451630865226923357575951268652918901
  33  | 6 |  8 | 3^2 * 7^2 * 281 * 743580875118413 * 177233764237488717892587862569137279765057
      |   |    | * 75595404542176366142496672372284175472002192879461
  34  | 6 |  6 | 197 * 509 * 17780359481 * 34117699655579 * 22315348168833851
      |   |    | * 9095090459568178362056606866951887331526450025789748268486739806360273
  35  | 6 |  6 | 7 * 10243 * 73778819 * 217751506979 * 815234955828637451
      |   |    | * 131466197983570230758712242832342445534290787689788875589041140940590332185871
  36  | 7 | 12 | 3^6 * 7 * 2399 * 1048199731 * 148677268927651979881001
      |   |    | * 501084089458936713560190133492008915991 * 1291397464203883365973624616866510728560214133
  37  | 3 |  3 | 39907897297 * 4487024637709913481144687156591679497977441809976700526003
      |   |    | * 68944190609423342665826899266824308649818482165892994600731
  38  | 5 |  6 | 7^2 · 313 · 21671333 · 2785755911621699171251156092325505793170741 · 1333355244...61<78>
  39  | 7 |  8 | 3^2 · 7 · 733 · 2777 · 190063 · 108446941 · 4670675415...89<114>

Notes on programming

In order to (re)produce this table in PARI without using the excessive time required for some of the factorizations, the prime factors larger than default(primelimit) may be added to an internal list using addprimes(). It is not needed to add the largest of the prime factors, for a given number. Thus, e.g.,

addprimes([273884231501, 4950715302671])

is enough to factor a(30) immediately.

External links

• FactorDB: (121*10^(4*n-19) - 1002*10^(4*n-28) - 2*10^(2*n-9) + 879*10^10 + 121)/99^2

History

Created. — MFH 15:55, 30 July 2015 (UTC)