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.

Showing 1-10 of 11 results. Next

A017377 a(n) = 10*n + 9.

Original entry on oeis.org

9, 19, 29, 39, 49, 59, 69, 79, 89, 99, 109, 119, 129, 139, 149, 159, 169, 179, 189, 199, 209, 219, 229, 239, 249, 259, 269, 279, 289, 299, 309, 319, 329, 339, 349, 359, 369, 379, 389, 399, 409, 419, 429, 439, 449, 459, 469, 479, 489, 499, 509, 519, 529, 539
Offset: 0

Views

Author

Keywords

Comments

Numbers k such that k^k ends with 9. - Bruno Berselli, Dec 11 2018

References

  • Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pp. 126-127.

Crossrefs

Programs

Formula

a(n) = 10*n + 9; a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, May 29 2011
G.f.: (9+x)/(x-1)^2. - R. J. Mathar, Oct 16 2015
From Elmo R. Oliveira, Apr 05 2025: (Start)
E.g.f.: exp(x)*(9 + 10*x).
a(n) = A016897(2*n+1). (End)

A008579 Coordination sequence for planar net 3.6.3.6. Spherical growth function for a certain reflection group in plane.

Original entry on oeis.org

1, 4, 8, 14, 18, 22, 28, 30, 38, 38, 48, 46, 58, 54, 68, 62, 78, 70, 88, 78, 98, 86, 108, 94, 118, 102, 128, 110, 138, 118, 148, 126, 158, 134, 168, 142, 178, 150, 188, 158, 198, 166, 208, 174, 218, 182, 228, 190, 238, 198, 248, 206, 258, 214, 268, 222, 278
Offset: 0

Views

Author

Keywords

Comments

Interesting because coefficients never become monotonic.
Also the coordination sequence for a planar net made of densely packed circles. - Yuriy Sibirmovsky, Sep 11 2016
Described by J.-G. Eon (2014) as the coordination sequence of the Kagome net. - N. J. A. Sloane, Jan 03 2018

References

  • P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 161 (but beware errors).

Crossrefs

List of coordination sequences for uniform planar nets: A008458 (the planar net 3.3.3.3.3.3), A008486 (6^3), A008574 (4.4.4.4 and 3.4.6.4), A008576 (4.8.8), A008579 (3.6.3.6), A008706 (3.3.3.4.4), A072154 (4.6.12), A219529 (3.3.4.3.4), A250120 (3.3.3.3.6), A250122 (3.12.12).

Programs

  • Haskell
    a008579 0 = 1
    a008579 1 = 4
    a008579 n = (10 - 2*m) * n' + 8*m - 2 where (n',m) = divMod n 2
    a008579_list = 1 : 4 : concatMap (\x -> map (* 2) [5*x-1,4*x+3]) [1..]
    -- Reinhard Zumkeller, Nov 12 2012
  • Maple
    f := n->if n mod 2 = 0 then 10*(n/2)-2 else 8*(n-1)/2+6 fi;
  • Mathematica
    a[n_?EvenQ] := 10*n/2-2; a[n_?OddQ] := 8*(n-1)/2+6; a[0] = 1; a[1] = 4; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Nov 18 2011, after Maple *)
    CoefficientList[Series[(1+2x)(1+2x+2x^2+2x^3-x^4)/(1-x^2)^2,{x,0,50}],x] (* or *) LinearRecurrence[{0,2,0,-1},{1,4,8,14,18,22},50] (* Harvey P. Dale, Sep 05 2018 *)

Formula

G.f.: (1 + 2*x)*(1 + 2*x + 2*x^2 + 2*x^3 - x^4)/(1 - x^2)^2.
From R. J. Mathar, Nov 26 2014: (Start)
a(2n) = A017365(n), n > 0.
a(2n+1) = A017137(n), n > 0. (End)
From Stefano Spezia, Aug 07 2022: (Start)
a(n) = (9 + (-1)^n)*n/2 - 2*(-1)^n for n > 1.
E.g.f.: 3 - 2*x + (4*x - 2)*cosh(x) + (5*x + 2)*sinh(x). (End)

A269100 a(n) = 13*n + 11.

Original entry on oeis.org

11, 24, 37, 50, 63, 76, 89, 102, 115, 128, 141, 154, 167, 180, 193, 206, 219, 232, 245, 258, 271, 284, 297, 310, 323, 336, 349, 362, 375, 388, 401, 414, 427, 440, 453, 466, 479, 492, 505, 518, 531, 544, 557, 570, 583, 596, 609, 622, 635, 648, 661, 674, 687, 700, 713, 726, 739
Offset: 0

Views

Author

Bruno Berselli, Feb 19 2016

Keywords

Comments

Any square mod 13 is one of 0, 1, 3, 4, 9, 10 or 12 (A010376) but not 11, and for this reason there are no squares in the sequence. Likewise, any cube mod 13 is one of 0, 1, 5, 8 or 12, therefore no a(k) is a cube.
Sequences of the type 13*n + k, for k = 0..12, without squares and cubes:
k = 2: A153080,
k = 6: A186113,
k = 7: A269044,
k = 11: this case.
The sum of the sixth powers of any two terms of the sequence is also a term of the sequence. Example: a(3)^6 + a(8)^6 = a(179129674278) = 2328685765625.
The primes of the sequence are listed in A140373.

Crossrefs

Subsequence of A094784, A106389.
Cf. A140373.
Similar sequences of the type k*n+k-2: A023443 (k=1), A005843 (k=2), A016777 (k=3), A016825 (k=4), A016885 (k=5), A016957 (k=6), A017041 (k=7), A017137 (k=8), A017245 (k=9), A017365 (k=10), A017497 (k=11), A017641 (k=12).
Sequences of the form 13*n+q: A008595 (q=0), A190991 (q=1), A153080 (q=2), A127547 (q=4), A154609 (q=5), A186113 (q=6), A269044 (q=7), this sequence (q=11).

Programs

  • Magma
    [13*n+11: n in [0..60]];
  • Mathematica
    13 Range[0,60] + 11
    Range[11, 800, 13]
    Table[13 n + 11, {n, 0, 60}] (* Bruno Berselli, Feb 22 2016 *)
    LinearRecurrence[{2,-1},{11,24},60] (* Harvey P. Dale, Jun 14 2023 *)
  • Maxima
    makelist(13*n+11, n, 0, 60);
    
  • PARI
    vector(60, n, n--; 13*n+11)
    
  • Python
    [13*n+11 for n in range(61)]
    
  • Sage
    [13*n+11 for n in range(61)]
    

Formula

G.f.: (11 + 2*x)/(1 - x)^2.
a(n) = -A153080(-n-1).
Sum_{i = h..h+13*k} a(i) = a(h*(13*k + 1) + k*(169*k + 35)/2).
Sum_{i >= 0} 1/a(i)^2 = .012486605016510955990... = polygamma(1, 11/13)/13^2.
E.g.f.: (11 + 13*x)*exp(x). - G. C. Greubel, May 31 2024

A166729 Positive integers with English names ending in "t".

Original entry on oeis.org

8, 28, 38, 48, 58, 68, 78, 88, 98, 108, 128, 138, 148, 158, 168, 178, 188, 198, 208, 228, 238, 248, 258, 268, 278, 288, 298, 308, 328, 338, 348, 358, 368, 378, 388, 398, 408, 428, 438, 448, 458, 468, 478, 488, 498, 508, 528, 538, 548, 558, 568, 578, 588, 598
Offset: 1

Views

Author

Rick L. Shepherd, Oct 20 2009

Keywords

Examples

			Fifty-eight (58) is a term; eighteen (18) is not a term (but is a term of A060228).
		

Crossrefs

Formula

A017365 MINUS {n | n = 18 mod 100}.

A306277 Numbers congruent to 1 or 8 mod 10.

Original entry on oeis.org

1, 8, 11, 18, 21, 28, 31, 38, 41, 48, 51, 58, 61, 68, 71, 78, 81, 88, 91, 98, 101, 108, 111, 118, 121, 128, 131, 138, 141, 148, 151, 158, 161, 168, 171, 178, 181, 188, 191, 198, 201, 208, 211, 218, 221, 228, 231, 238, 241, 248, 251, 258, 261, 268, 271, 278, 281, 288, 291, 298, 301, 308, 311, 318, 321
Offset: 1

Views

Author

Davis Smith, Feb 02 2019

Keywords

Comments

A007310(a(n)+1) is always a multiple of 5.
a(1) = 1, a(n+1) = a(n)+7 when n is odd, a(n+1) = a(n)+3 when n is even.
a(n) mod 6 follows the following pattern: 1,2,5,0,3,4,1,2,5,0,3,4, and so on.
A020639(A007310(a(n)+1)) = 5.

Crossrefs

Cf. A017281, A017365 (bisections).
One less than A273669.

Programs

  • Maple
    seq(seq(10*i+j, j=[1, 8]), i=0..350);
  • Mathematica
    Select[Range[350], MemberQ[{1, 8}, Mod[#, 10]] &]
  • PARI
    for(n=1, 350, if((n%10==1) || (n%10==8), print1(n, ", ")))
    
  • PARI
    Vec(x*(1 + 7*x + 2*x^2) / ((1 - x)^2*(1 + x)) + O(x^40)) \\ Colin Barker, Feb 09 2019

Formula

a(n) = 5*n - 2*A000034(n+1).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>3.
a(n) = A273669(n) - 1. - Antti Karttunen, Feb 07 2019
G.f.: x*(1 + 7*x + 2*x^2) / ((1 - x)^2*(1 + x)). - Colin Barker, Feb 09 2019
E.g.f.: 2 + (5*x - 3)*exp(x) + exp(-x). - David Lovler, Sep 07 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = (5+sqrt(5))^(3/2)*phi*Pi/(100*sqrt(2)) + log(phi)/(2*sqrt(5)) + log(2)/5, where phi is the golden ratio (A001622). - Amiram Eldar, Apr 15 2023

A262389 Numbers whose last digit is composite.

Original entry on oeis.org

4, 6, 8, 9, 14, 16, 18, 19, 24, 26, 28, 29, 34, 36, 38, 39, 44, 46, 48, 49, 54, 56, 58, 59, 64, 66, 68, 69, 74, 76, 78, 79, 84, 86, 88, 89, 94, 96, 98, 99, 104, 106, 108, 109, 114, 116, 118, 119, 124, 126, 128, 129, 134, 136, 138, 139, 144, 146, 148, 149
Offset: 1

Views

Author

Wesley Ivan Hurt, Sep 21 2015

Keywords

Comments

Numbers ending in 4, 6, 8 or 9.
Union of A017317, A017341, A017365 and A017377.
Subsequence of A118951 (numbers containing at least one composite digit).
Complement of (A197652 Union A260181).

Crossrefs

Cf. A118951, A197652, A260181 (last digit is prime).

Programs

  • Magma
    [(5*n+1-(-1)^n+(3+(-1)^n)*(-1)^((2*n-3-(-1)^n) div 4) div 2) div 2: n in [1..70]]; // Vincenzo Librandi, Sep 21 2015
  • Maple
    A262389:=n->(5*n+1-(-1)^n+(3+(-1)^n)*(-1)^((2*n-3-(-1)^n)/4)/2)/2: seq(A262389(n), n=1..100);
  • Mathematica
    Table[(5n+1-(-1)^n+(3+(-1)^n)*(-1)^((2n-3-(-1)^n)/4)/2)/2, {n, 100}]
    LinearRecurrence[{1, 0, 0, 1, -1}, {4, 6, 8, 9, 14}, 80] (* Vincenzo Librandi, Sep 21 2015 *)
    CoefficientList[Series[(4 + 2*x + 2*x^2 + x^3 + x^4)/((x - 1)^2*(1 + x + x^2 + x^3)), {x, 0, 80}], x] (* Wesley Ivan Hurt, Sep 21 2015 *)
    Select[Range[200],CompositeQ[Mod[#,10]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 21 2019 *)

Formula

G.f.: x*(4+2*x+2*x^2+x^3+x^4)/((x-1)^2*(1+x+x^2+x^3)).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (5*n+1-(-1)^n+(3+(-1)^n)*(-1)^((2*n-3-(-1)^n)/4)/2)/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(10-2*sqrt(5))*Pi - sqrt(5)*arccoth(3/sqrt(5)) - 4*log(2))/20. - Amiram Eldar, Jul 30 2024

Extensions

Name edited by Jon E. Schoenfield, Feb 15 2018

A306289 The smallest prime factor of numbers greater than 1 and coprime to 6.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 23, 5, 29, 31, 5, 37, 41, 43, 47, 7, 53, 5, 59, 61, 5, 67, 71, 73, 7, 79, 83, 5, 89, 7, 5, 97, 101, 103, 107, 109, 113, 5, 7, 11, 5, 127, 131, 7, 137, 139, 11, 5, 149, 151, 5, 157, 7, 163, 167, 13, 173, 5, 179, 181, 5, 11, 191, 193
Offset: 1

Views

Author

Davis Smith, Feb 03 2019

Keywords

Comments

a(n) is the least prime factor of the n-th number that is greater than 1 and congruent to 1 or 5 (mod 6).
a(n) = 5 when n is congruent to {1, 8} (mod 10) (n is a term in A017281, A017365, or A306277). a(n) = 7 when n is congruent to {2, 11} (mod 14) but not {1, 8} (mod 10). a(n) = 11 when n is congruent to {3, 18} (mod 22) but not a case where it equals 5 or 7. a(n) = 13 when n is congruent to {4, 21} (mod 26) (n is a term in A306285) but not a case where it equals 5, 7, or 11. a(n) = 17 when n is congruent to {5, 28} (mod 34) but not a case where it equals 5, 7, 11, or 13. a(n) = 19 when n is congruent to {6, 31} (mod 38) (n is a term in A306331) but not a case where it equals 5, 7, 11, 13, or 17.
Conjecture: This pattern continues indefinitely. a(n) = A007310(m + 1) when n is congruent to {m, A306277(m + 1)} (mod A091999(m + 1)) but not congruent to {k, A306277(k + 1)} (mod A091999(k + 1)), m > k >= 1. The indices of the first appearance of a number in this sequence supports this conjecture in that they are never, for m > 0, congruent to A306277(m + 1) mod A091999(m + 1).

Examples

			a(n) is the least term, other than 0, in n-th row of the array A(m,n), where A(m,n) is A007310(m + 1) when A007310(n + 1) mod A007310(m + 1) is congruent to 0, otherwise 0.
Table begins
  \m  1 2  3  4  5  6  7  8  9  10  11  12  13  14  15  16 ...
  n\
   1| 5 0  0  0  0  0  0  0  0   0   0   0   0   0   0   0 ...
   2| 0 7  0  0  0  0  0  0  0   0   0   0   0   0   0   0 ...
   3| 0 0 11  0  0  0  0  0  0   0   0   0   0   0   0   0 ...
   4| 0 0  0 13  0  0  0  0  0   0   0   0   0   0   0   0 ...
   5| 0 0  0  0 17  0  0  0  0   0   0   0   0   0   0   0 ...
   6| 0 0  0  0  0 19  0  0  0   0   0   0   0   0   0   0 ...
   7| 0 0  0  0  0  0 23  0  0   0   0   0   0   0   0   0 ...
   8| 5 0  0  0  0  0  0 25  0   0   0   0   0   0   0   0 ...
   9| 0 0  0  0  0  0  0  0 29   0   0   0   0   0   0   0 ...
  10| 0 0  0  0  0  0  0  0  0  31   0   0   0   0   0   0 ...
  11| 5 7  0  0  0  0  0  0  0   0  35   0   0   0   0   0 ...
  12| 0 0  0  0  0  0  0  0  0   0   0  37   0   0   0   0 ...
  13| 0 0  0  0  0  0  0  0  0   0   0   0  41   0   0   0 ...
  14| 0 0  0  0  0  0  0  0  0   0   0   0   0  43   0   0 ...
  15| 0 0  0  0  0  0  0  0  0   0   0   0   0   0  47   0 ...
  16| 0 7  0  0  0  0  0  0  0   0   0   0   0   0   0  49 ...
For the n-th row of this square array, the leftmost terms, other than 0, are the factors of A(n,n). A(n,n) = A007310(n + 1). If for every m, m < n, A(m,n) = 0, then a(n) = A007310(n + 1) and A007310(n + 1) is prime.
		

References

  • G. Pólya and G. Szegő, Problems and Theorems in Analysis II (Springer 1924, reprinted 1976), Part Eight, Chap. 2, Section 2, Problems 96 and 105.

Crossrefs

Programs

  • Maple
    seq(min(op(numtheory[factorset] (6*ceil(n/2)+(-1)^n))), n=1..64) ;
  • Mathematica
    FactorInteger[Rest@ Flatten@ Array[6 # + {1, 5} &, 33, 0]][[All, 1, 1]] (* Michael De Vlieger, Feb 15 2019 *)
    FactorInteger[#][[1,1]]&/@Select[Range[2,200],CoprimeQ[#,6]&] (* Harvey P. Dale, Jul 10 2020 *)
  • PARI
    for(n=2, 211, if((n%6==1)||(n%6==5), print1(factor(n)[1,1], ", ")))
    
  • PARI
    vector(64,n,factor(6*ceil(n/2)+(-1)^n)[1,1])
    
  • PARI
    a(n) = n++; factor(n\2*6-(-1)^n)[1,1]; \\ Michel Marcus, Feb 06 2019

Formula

a(n) = A020639(A007310(n + 1)).
a(n) = A020639(3n + A000034(n + 1)).
a(n) = A020639(6*ceiling(n/2) + (-1)^n).
a(floor(prime(n + 2)/3)) = prime(n + 2).

A154414 Primes of the form 20*k^2 + 32*k + 13.

Original entry on oeis.org

13, 157, 461, 673, 1217, 1549, 2333, 4993, 6337, 7069, 7841, 11329, 12301, 13313, 17761, 18973, 21517, 25633, 30109, 36637, 41953, 45697, 47629, 51613, 62273, 69149, 78877, 81409, 97441, 105997, 108929, 114913, 137117, 140449, 143821, 161281, 164893, 191297, 195229
Offset: 1

Views

Author

Vincenzo Librandi, Jan 09 2009

Keywords

Crossrefs

Cf. A017365.

Programs

  • Magma
    [a: n in [0..100] | IsPrime(a) where a is 20*n^2+32*n+13]; // Vincenzo Librandi, Jul 23 2012
    
  • Mathematica
    Select[Table[20n^2+32n+13,{n,0,6001}],PrimeQ] (* Vincenzo Librandi, Jul 23 2012 *)
  • PARI
    {for(n=0, 100, if(isprime(k=20*n^2+32*n+13), print1(k, ", ")))}; \\ Vincenzo Librandi, Jul 23 2012

A385623 Array read by ascending antidiagonals: A(n,k) is the number obtained by concatenation of n with k in that order, with k >= 0.

Original entry on oeis.org

0, 10, 1, 20, 11, 2, 30, 21, 12, 3, 40, 31, 22, 13, 4, 50, 41, 32, 23, 14, 5, 60, 51, 42, 33, 24, 15, 6, 70, 61, 52, 43, 34, 25, 16, 7, 80, 71, 62, 53, 44, 35, 26, 17, 8, 90, 81, 72, 63, 54, 45, 36, 27, 18, 9, 100, 91, 82, 73, 64, 55, 46, 37, 28, 19, 10, 110, 101, 92, 83, 74, 65, 56, 47, 38, 29, 110, 11
Offset: 0

Views

Author

Stefano Spezia, Jul 05 2025

Keywords

Examples

			Array begins as:
   0,  1,  2,  3,  4,  5,  6,  7, ...
  10, 11, 12, 13, 14, 15, 16, 17, ...
  20, 21, 22, 23, 24, 25, 26, 27, ...
  30, 31, 32, 33, 34, 35, 36, 37, ...
  40, 41, 42, 43, 44, 45, 46, 47, ...
  50, 51, 52, 53, 54, 55, 56, 57, ...
  60, 61, 62, 63, 64, 65, 66, 67, ...
  ...
		

Crossrefs

Cf. A001477 (1st row), A020338 (main diagonal), A055642, A385624 (antidiagonal sums).

Programs

  • Mathematica
    A[n_,k_]:=FromDigits[Join[IntegerDigits[n],IntegerDigits[k]]]; Table[A[n,k],{n,0,6},{k,0,7}] (* or *)
    A[n_,k_]:=If[k==0,10n,n*10^(Floor[Log10[k]]+1)+k]; Table[A[n-k,k],{n,0,11},{k,0,n}]//Flatten
  • PARI
    T(n, k) = fromdigits(concat(digits(n), digits(k))); \\ Michel Marcus, Jul 06 2025

Formula

A(n,0) = 10*n and A(n,k) = n*10^(floor(log_10(k)) + 1) + k for k > 0.

A017372 (10*n+8)^8.

Original entry on oeis.org

16777216, 11019960576, 377801998336, 4347792138496, 28179280429056, 128063081718016, 457163239653376, 1370114370683136, 3596345248055296, 8507630225817856, 18509302102818816, 37588592026706176, 72057594037927936, 131532383853732096, 230193853492166656
Offset: 0

Views

Author

Keywords

Crossrefs

Cf. A001016 (n^8), A017365 (10n+8).

Programs

Formula

From Wesley Ivan Hurt, Oct 31 2014: (Start)
G.f.: 256*(65536 + 42456897*x + 1090727863*x^2 + 5245638469*x^3 + 6743985795*x^4 + 2426203459*x^5 + 199242373*x^6 + 1679607*x^7 + x^8)/(1 - x)^9.
a(n) = 9*a(n-1)-36*a(n-2)+84*a(n-3)-126*a(n-4)+126*a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-9).
a(n) = (10*n+8)^8 = A001016(A017365(n)). (End)
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