cp's OEIS Frontend

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.

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A116038 n+p(n)+p(p(n)) is a brilliant number (A078972), where p(n) denotes the n-th prime.

Original entry on oeis.org

1, 2, 11, 29, 33, 43, 53, 77, 81, 103, 335, 441, 497, 547, 623, 631, 691, 693, 709, 729, 749, 795, 909, 913, 941, 949, 969, 983, 1025, 1035, 1123, 1137, 1173, 1177, 1181, 1189, 1229, 1243, 1297, 1323, 1349, 1375, 1389, 1463, 1489, 1561, 1621, 1639
Offset: 1

Views

Author

Giovanni Resta, Feb 13 2006

Keywords

Examples

			1123+p(1123)+p(p(1123)) = 103793 = 271*383.

Crossrefs

Cf. A078972, A116039.

A116062 Brilliant numbers (A078972) made of nontrivial runs of identical digits.

Original entry on oeis.org

22999, 55577, 1166111, 2211133, 2224499, 3377333, 4466333, 5551177, 5555533, 5566999, 5588899, 6622277, 6644333, 6644777, 6660077, 6667799, 7711177, 7711199, 7744111, 7766333, 7771199, 7777999, 7799777, 8866699, 8888777
Offset: 1

Views

Author

Giovanni Resta, Feb 13 2006

Keywords

Comments

A run of length 1 is trivial.

Examples

			8866699 = 2287*3877.

Crossrefs

Cf. A033023, A078972.

A338473 Numbers that can be written as the sum of two brilliant numbers (A078972).

Original entry on oeis.org

8, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 31, 34, 35, 36, 39, 40, 41, 42, 44, 45, 46, 49, 50, 53, 55, 56, 58, 59, 60, 63, 64, 70, 74, 84, 98, 125, 127, 130, 131, 135, 136, 142, 146, 147, 149, 152, 153, 156, 157, 158, 164, 168, 170
Offset: 1

Views

Author

Marius A. Burtea, Dec 06 2020

Keywords

Comments

The sequence is infinite.
There are an infinite number of term pairs (a(k), a(k + 1)) that are consecutive numbers. Indeed, if p is a prime number, then 9 + p^2 and 10 + p^2 are terms. Also, numbers 14 + p^2 and 15 + p^2 are terms.
There are also larger sequences of consecutive numbers that are terms. For example, the 21 consecutive numbers 780, 781, ..., 800 or 4184, 4185, ..., 4204 are terms.

Examples

			8 = 4 + 4 = A078972(1) + A078972(1), so 8 is a term.
10 = 4 + 6 = A078972(1) + A078972(2), so 10 is a term.
15 = 6 + 9 = A078972(2) + A078972(3), so 15 is a term.

Crossrefs

Cf. A078972, A338474.

Programs

  • Magma
    f:=Factorization; br:=func; [k:k in [4..200]|exists(i){m:m in [4..k-4]|br(m) and br(k-m)}];
  • Mathematica
    m = 200; brils = Select[Range[m], (f = FactorInteger[#])[[;; , 2]] == {2} || f[[;; , 2]] == {1, 1} && Equal @@ IntegerLength@f[[;; , 1]] &]; Select[Range[m], Length[IntegerPartitions[#, {2}, brils]] > 0 &] (* Amiram Eldar, Dec 06 2020 *)

A338474 a(n) is the smallest number that can be partitioned into n ways as the sum of two brilliant numbers (A078972).

Original entry on oeis.org

1, 8, 18, 338, 462, 542, 638, 660, 918, 858, 924, 1260, 1140, 1122, 1428, 1326, 1740, 1710, 2520, 2070, 1938, 3150, 3330, 27342, 27810, 29190, 30600, 35754, 32700, 31710, 35310, 32760, 35952, 35790, 35910, 39450, 40950, 41160, 39060, 45990, 40680, 42510, 44520
Offset: 0

Views

Author

Marius A. Burtea, Nov 02 2020

Keywords

Comments

Except for 1, all terms are even numbers.

Examples

			8 = 4 + 4 = A078972(1) + A078972(1);
18 = 4 + 14 = A078972(1) + A078972(5) and 18 = 9 + 9 = A078972(3) + A078972(3).
18 = 15 + 323 = A078972(6) + A078972(22), 338 = 49 + 289 = A078972(10) + A078972(19) and 338 = 169 + 169 = A078972(13) + A078972(13).

Crossrefs

Cf. A078972, A100592, A338473.

Programs

  • Magma
    f:=Factorisation; brnumber:=func; v:=[m:m in [2..50000]|brnumber(m)]; a:=[]; for n in [0..32] do k:=1; while  #RestrictedPartitions(k,2,Set(v)) ne n do k:=k+1; end while ; Append(~a,k); end for; a;
  • Mathematica
    m = 46000; brils = Select[Range[m], (f = FactorInteger[#])[[;; , 2]] == {2} || f[[;; , 2]] == {1, 1} && Equal @@ IntegerLength@f[[;; , 1]] &]; a[n_] := Length[IntegerPartitions[n, {2}, brils]]; mx = 43; s = Table[-1, {mx}]; c = 0; n = 1; While[c < mx, i = a[n] + 1; If[i <= mx && s[[i]] < 0, c++; s[[i]] = n]; n++]; s (* Amiram Eldar, Nov 03 2020 *)

A085650 Least k such that 2^n+k is a brilliant number (A078972).

Original entry on oeis.org

2, 0, 1, 5, 3, 57, 15, 33, 5, 3, 11, 75, 441, 75, 81, 31, 21, 15, 429, 2013, 51, 87, 5, 21, 57, 33, 711, 87, 65, 57, 255, 231, 1397, 519, 39, 157, 57, 315, 521, 37, 341, 1545, 389, 877, 1127, 1417, 2841, 247, 675, 147, 71, 507, 10799, 2779, 213, 375, 1149, 739, 8685
Offset: 1

Views

Author

Jason Earls, Amarnath Murthy, Jul 11 2003

Keywords

Extensions

More terms from David Wasserman, Feb 08 2005

A097435 Numbers n such that n and its reversal are distinct brilliant numbers (A078972).

Original entry on oeis.org

143, 169, 187, 319, 341, 781, 913, 961, 1273, 1343, 1691, 1843, 1961, 3431, 3481, 3721, 10807, 11413, 12769, 15049, 15347, 15707, 15857, 16171, 16837, 16867, 17161, 18203, 18437, 19939, 30227, 30281, 31411, 31439, 31979, 32639, 33017, 33109
Offset: 1

Views

Author

Jason Earls, Aug 22 2004

Keywords

Comments

A subset of A097393.
Sometimes called "Tnaillirb" numbers. - Jonathan Vos Post, Nov 22 2004

Examples

			30227 is in the sequence because 30227=167*181 and 72203=103*701.

Crossrefs

Cf. A001358.

A114080 Numbers k such that sigma(k) times the k-th prime is a brilliant number (A078972).

Original entry on oeis.org

2, 4, 9, 16, 25, 64, 729, 65536
Offset: 1

Views

Author

Giovanni Resta, Feb 13 2006

Keywords

Comments

No additional terms up to 1 million. - Harvey P. Dale, Nov 13 2013

Examples

			sigma(25) * p(25) = 3007 = 31*97.

Programs

  • Mathematica
    brilQ[n_]:=Module[{pf=Transpose[FactorInteger[n]][[1]]},PrimeOmega[n] == 2&&IntegerLength[First[pf]]==IntegerLength[Last[pf]]]; Select[Range[ 100000], brilQ[DivisorSigma[1,#]Prime[#]]&] (* Harvey P. Dale, Nov 13 2013 *)

A114325 Number of partitions of n into brilliant numbers (A078972).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 0, 2, 1, 3, 2, 3, 1, 5, 3, 5, 4, 6, 4, 9, 7, 8, 8, 12, 10, 15, 12, 15, 16, 21, 19, 24, 22, 27, 30, 34, 31, 40, 40, 46, 49, 54, 52, 65, 68, 74
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Examples

			a(18)=5 since 18 has the following 5 "brilliant" partitions: {14,4}, {10,4,4}, {9,9}, {6,6,6}, {6,4,4,4}.

Crossrefs

Cf. A101048, A078972.

A115681 Brilliant numbers (A078972) whose digit reversal is the product of 2 palindromes greater than 1.

Original entry on oeis.org

4, 6, 9, 21, 121, 253, 407, 451, 559, 583, 667, 671, 803, 869, 2173, 2537, 5063, 5183, 5893, 10201, 13231, 15251, 16171, 18281, 19291, 22523, 22733, 24743, 25283, 26563, 27383, 28583, 28783, 31613, 35653, 37673, 38683, 40567, 45349, 46217
Offset: 1

Views

Author

Giovanni Resta, Jan 31 2006

Keywords

Examples

			22523=101*223 is brilliant and 32522=2*16261.

Crossrefs

Cf. A078972, A115683.

A115917 Numbers k such that sigma(k) - phi(k) is a brilliant number (A078972).

Original entry on oeis.org

6, 10, 49, 242, 289, 512, 578, 800, 900, 1250, 2048, 5041, 15625, 17424, 22201, 22472, 26450, 28900, 48400, 60025, 62001, 65536, 69169, 91592, 131072, 144722, 146689, 151321, 160801, 201601, 212521, 236196, 242064, 252050, 253009
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Examples

			sigma(62001) - phi(62001) = 49813 = 109*457.

Crossrefs

Cf. A078972, A115916.
Previous Showing 31-40 of 88 results. Next

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