A078972 -id:A078972 - cp's OEIS frontend

A113940 Triangular numbers that are also brilliant (A078972).

6, 10, 15, 21, 253, 703, 1081, 1711, 1891, 2701, 3403, 25651, 34453, 38503, 49141, 60031, 64261, 73153, 79003, 88831, 104653, 108811, 114481, 126253, 146611, 158203, 171991, 188191, 218791, 226801, 258121, 269011, 286903, 351541, 371953, 385003, 392941

Offset: 1

Giovanni Resta, Jan 31 2006

Smallest value where each factor has n digits for n = 1, 2, 3, 4, 5, are: 6 = 2 * 3; 253 = 11 * 23; 25651 = 113 * 227; 2035153 = 1009 * 2017; 202457503 = 10061 * 20123. [From Jonathan Vos Post, Apr 04 2009]

			253 = T(22) and 253 = 11*23 is brilliant.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A068443, A078972, A000217.

brilQ[n_]:=Module[{fin=FactorInteger[n]},Total[Transpose[fin][[2]]]==2&& Length[Union[IntegerLength[Transpose[fin][[1]]]]]==1]
Intersection[Accumulate[Range[850]],Select[Range[362000],brilQ]]  (* Harvey P. Dale, Feb 06 2011 *)

A000217 INTERSECTION A078972. Subset of A068443. [From Jonathan Vos Post, Apr 04 2009]

A115652 Brilliant numbers (A078972) which are the sum of distinct double factorials (A006882).

Original entry on oeis.org

4, 6, 9, 10, 14, 15, 21, 25, 49, 121, 169, 403, 407, 437, 451, 493, 517, 551, 949, 961, 1003, 1007, 1067, 1073, 1079, 1121, 1333, 1343, 1349, 1357, 1387, 1403, 1457, 1501, 3869, 3901, 3953, 4331, 4891, 5183, 5293, 10403, 10807, 11413, 11449

Offset: 1

Giovanni Resta, Jan 28 2006

Double factorials 0!! and 1!! are not considered distinct. Note that double factorial (n!!) is different from (n!)!.

			949 = 13*73 = 9!! + 3!! + 1!!.

Cf. A006882, A078972, A115651, A115653, A115654.

A115669 Numbers the reversal of whose square is a brilliant number (A078972).

Original entry on oeis.org

2, 3, 11, 13, 20, 30, 31, 59, 61, 101, 110, 112, 113, 128, 130, 131, 178, 200, 300, 301, 307, 310, 311, 388, 590, 599, 610, 839, 854, 875, 949, 989, 1003, 1010, 1021, 1031, 1100, 1102, 1103, 1105, 1112, 1120, 1124, 1130, 1175, 1201, 1216, 1280, 1300

Offset: 1

Giovanni Resta, Jan 31 2006

			3481=59^2 is a square and 1843=19*97 is brilliant.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A078972, A115667, A115668.

sbnQ[n_]:=Module[{fi=FactorInteger[IntegerReverse[n^2]]},(Length[fi]==1 && fi[[1,2]]==2)||(Length[fi]==2&&fi[[All,2]]=={1,1})&&Length[Union[ IntegerLength[ fi[[All,1]]]]]==1]; Select[Range[1300],sbnQ] (* Harvey P. Dale, Mar 23 2018 *)

A115916 Numbers k such that sigma(k) + phi(k) is a brilliant number (A078972).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 128, 144, 400, 576, 1296, 8100, 11664, 13225, 15129, 34969, 40000, 44944, 55112, 60025, 63368, 67712, 69938, 91809, 103058, 135424, 157922, 203401, 861184, 1034289, 1258884, 1270418, 1435204, 1440000, 1575025

Offset: 1

Giovanni Resta, Feb 06 2006

			sigma(55112) + phi(55112) = 131819 = 193*683.

Cf. A078972, A115914, A115915.

A239585 Prime factor <= other prime factor of n-th brilliant number, cf. A078972.

Original entry on oeis.org

2, 2, 3, 2, 2, 3, 3, 5, 5, 7, 11, 11, 13, 11, 11, 13, 13, 11, 17, 13, 11, 17, 11, 19, 13, 17, 13, 11, 19, 11, 11, 13, 17, 11, 17, 23, 13, 19, 13, 11, 19, 13, 17, 11, 23, 11, 13, 17, 19, 23, 17, 11, 13, 19, 11, 13, 17, 11, 19, 29, 23, 11, 13, 19, 29, 17, 11

Offset: 1

Reinhard Zumkeller, Mar 22 2014

a(n) = A020639(A078972(n)) = A078972(n) / A239586(n).

A055642(a(n)) = A055642(A239586(n)).

			      n  | A239585(n) | A239586(n) | A078972(n)   Lengths of factors
  -------+------------+------------+-----------   ------------------
      1  |        2   |        2   |        4          1
      5  |        2   |        7   |       14
     10  |        7   |        7   |       49
         |.........................|              ..................
     11  |       11   |       11   |      121          2
     78  |       11   |       97   |     1067
    100  |       37   |       37   |     1369
    241  |       97   |       97   |     9409
         |.........................|              ..................
    242  |      101   |      101   |    10201          3
   1000  |      193   |      263   |    50759
   2530  |      101   |      997   |   100697
  10000  |      743   |      937   |   696191
  10537  |      997   |      997   |   994009
         |.........................|              ..................
  10538  |     1009   |     1009   |  1018081          4

Reinhard Zumkeller, Table of n, a(n) for n = 1..20000
Dario Alpern, Brilliant Numbers

Subsequence of A084126.

Cf. A020639, A055642, A078972, A239586.

Haskell
```
a239585 = a020639 . a078972
```

Table[With[{f = FactorInteger[k]}, If[Total[f[[All, 2]]] == 2 && Length[Union[IntegerLength[f[[All, 1]]]]] == 1, f[[1, 1]], Nothing]], {k, 1000}] (* Paolo Xausa, Oct 02 2024 *)
dlist2[d_] := Union[Times @@@ Tuples[Prime[Range[PrimePi[10^(d-1)] + 1, PrimePi[10^d]]], 2]]; (* Generates terms with d-digits prime factors -- faster but memory intensive *)
Map[FactorInteger[#][[1, 1]]&, Flatten[Array[dlist2, 2]]] (* Paolo Xausa, Oct 09 2024 *)

A239586 Prime factor >= other prime factor of n-th brilliant number, cf. A078972.

Original entry on oeis.org

2, 3, 3, 5, 7, 5, 7, 5, 7, 7, 11, 13, 13, 17, 19, 17, 19, 23, 17, 23, 29, 19, 31, 19, 29, 23, 31, 37, 23, 41, 43, 37, 29, 47, 31, 23, 41, 29, 43, 53, 31, 47, 37, 59, 29, 61, 53, 41, 37, 31, 43, 67, 59, 41, 71, 61, 47, 73, 43, 29, 37, 79, 67, 47, 31, 53, 83

Offset: 1

Reinhard Zumkeller, Mar 22 2014

a(n) = A006530(A078972(n)) = A078972(n) / A239585(n).

A055642(a(n)) = A055642(A239585(n)).

			See A239585.

Reinhard Zumkeller, Table of n, a(n) for n = 1..20000
Dario Alpern, Brilliant Numbers

Subsequence of A084127.

Cf. A006530, A055642, A078972, A239585.

Haskell
```
a239586 n = a078972 n `div` a239585 n
```

Table[With[{f = FactorInteger[k]}, If[Total[f[[All, 2]]] == 2 && Length[Union[IntegerLength[f[[All, 1]]]]] == 1, f[[-1, 1]], Nothing]], {k, 1000}] (* Paolo Xausa, Oct 02 2024 *)
dlist2[d_] := Union[Times @@@ Tuples[Prime[Range[PrimePi[10^(d-1)] + 1, PrimePi[10^d]]], 2]]; (* Generates terms with d-digits prime factors -- faster but memory intensive *)
Map[FactorInteger[#][[-1,1]]&,Flatten[Array[dlist2,2]]] (* Paolo Xausa, Oct 08 2024 *)

A083289 Least k such that 10^n+k is a brilliant number (cf. A078972).

Original entry on oeis.org

3, 0, 21, 3, 201, 13, 18081, 43, 140049, 81, 600009, 147, 6000009, 73, 380000361, 3, 1400000049, 831, 14000000049, 49, 380000000361, 987, 600000000009, 691, 78000000001521, 183, 740000000001369, 4153, 6200000000000961, 279

Offset: 0

Jason Earls, Jun 03 2003

If n is an even positive exponent, then a(n) is the first prime greater than 10^(n/2) squared less 10^n.

Chai Wah Wu, Table of n, a(n) for n = 0..42
Dario Alejandro Alpern, Brilliant numbers

Cf. A078972, A084475, A084476.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; LengthBase10[n_] := Floor[ Log[10, n] + 1]; f[n_] := Block[{k = 0}, If[ EvenQ[n] && n > 1, NextPrim[ 10^(n/2)]^2 - 10^n, While[fi = FactorInteger[10^n + k]; Plus @@ Flatten[ Table[ # [[2]], {1}] & /@ fi] != 2 || Length[ Union[ LengthBase10 /@ Flatten[ Table[ # [[1]], {1}] & /@ fi]]] != 1, k++ ]; k]]; Table[ f[n], {n, 0, 30}]

from sympy import nextprime, factorint
def A083289(n):
    a, b = divmod(n,2)
    c, d = 10**n, 10**a
    if b == 0: return nextprime(d)**2-c
    k = 0
    while True:
        fs = factorint(c+k,multiple=True)
        if len(fs) == 2 and min(fs) >= d:
            return k
        k += 1 # Chai Wah Wu, Sep 28 2021

Edited and extended by Robert G. Wilson v, Jun 27 2003

A113941 Pentagonal numbers (A000326) that are also brilliant numbers (A078972).

Original entry on oeis.org

35, 247, 1247, 2501, 4187, 15251, 17767, 33227, 49051, 63551, 68587, 71177, 76501, 81317, 96647, 112477, 118301, 128627, 147737, 159251, 182527, 241001, 250717, 265651, 302177, 318551, 438751, 485357, 563347, 655051, 1563151, 1600117

Offset: 1

Giovanni Resta, Jan 31 2006

This is to pentagonal numbers A000326 as A113940 is to triangular numbers A000217. These may be seen as the 5th and 3rd row of an infinite array of k-gonal numbers which are also brilliant numbers, where the 4th row is A001248 squares of primes. - Jonathan Vos Post, Apr 05 2009

			a(1) = 35 = 5th pentagonal number = 5*(3*5-1)/2 = 5 * 7, with the two prime factors each being one digit in length. a(2) = 247 = 13th pentagonal number = 13*(3*13-1)/2 = 13 * 19, with the two prime factors each being two digits in length. a(6) = 15251 = 101 * 151, with the two prime factors each being three digits in length. - _Jonathan Vos Post_, Apr 05 2009
17767 is the 109th pentagonal number and 17767=109*163 is brilliant.

Cf. A000326, A078972, A159190.

Cf. A001222, A001248, A001358, A113940, A114439.

A000326 INTERSECTION A078972.

Edited by N. J. A. Sloane, Apr 07 2009 at the suggestion of R. J. Mathar

Two more terms from R. J. Mathar, Apr 06 2009

A115644 Brilliant numbers (A078972) that are sums of distinct factorials.

Original entry on oeis.org

6, 9, 25, 121, 841, 871, 5041, 5767, 363721, 368761, 409111, 3633841, 3992431, 3992551, 4032121, 4037791, 39962281, 39962311, 39963031, 40279711, 40279801, 43585921, 43591687, 43909207, 479047801, 479365321, 479370271, 482631271

Offset: 1

Giovanni Resta, Jan 27 2006

nonn

			39962281 = 11! + 8! + 7! + 5!+ 1! = 4861*8221.

Cf. A078972, A025494, A115645, A115646, A089359.

brillQ[n_] := Block[{d = FactorInteger[n]}, Plus@@Last/@d==2 && (Last/@d=={2} || Length@IntegerDigits@((First/@d)[[1]])==Length@IntegerDigits@((First/@d)[[2]]))]; fac=Range[20]!;lst={}; Do[ n = Plus@@(fac*IntegerDigits[k, 2, 20]); If[brillQ[n], AppendTo[lst, n]], {k, 2^20-1}]; lst

A115667 Squares whose digit reversal is a brilliant number (A078972).

Original entry on oeis.org

4, 9, 121, 169, 400, 900, 961, 3481, 3721, 10201, 12100, 12544, 12769, 16384, 16900, 17161, 31684, 40000, 90000, 90601, 94249, 96100, 96721, 150544, 348100, 358801, 372100, 703921, 729316, 765625, 900601, 978121, 1006009, 1020100

Offset: 1

Giovanni Resta, Jan 31 2006

			3481=59^2 is a square and 1843=19*97 is brilliant.

Cf. A078972, A115668, A115669.

cp's OEIS Frontend

A113940 Triangular numbers that are also brilliant (A078972).

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A115652 Brilliant numbers (A078972) which are the sum of distinct double factorials (A006882).

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A115669 Numbers the reversal of whose square is a brilliant number (A078972).

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A115916 Numbers k such that sigma(k) + phi(k) is a brilliant number (A078972).

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A239585 Prime factor <= other prime factor of n-th brilliant number, cf. A078972.

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Haskell

Mathematica

A239586 Prime factor >= other prime factor of n-th brilliant number, cf. A078972.

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

Haskell

Mathematica

A083289 Least k such that 10^n+k is a brilliant number (cf. A078972).

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A113941 Pentagonal numbers (A000326) that are also brilliant numbers (A078972).

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Extensions

A115644 Brilliant numbers (A078972) that are sums of distinct factorials.

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