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 1-7 of 7 results.

A102306 Numbers k such that A083500(k) differs from A102305(k).

Original entry on oeis.org

7, 9, 13, 14, 18, 19, 21, 26, 27, 28, 31, 35, 36, 37, 39, 42, 43, 45, 49, 52, 54, 56, 57, 61, 62, 63, 65, 67, 70, 72, 73, 74, 76, 77, 78, 79, 81, 84, 86, 90, 91, 93, 95, 97, 98, 99, 103, 104, 105, 108, 109, 111, 112, 114, 117, 119, 122, 124, 126, 127, 129, 130, 133
Offset: 1

Views

Author

Ralf Stephan, Jan 03 2005

Keywords

Comments

What is the cardinality of a(n) with respect to its complement?

A083500 Smallest k such that n*(n+k) + 1 is a cube.

Original entry on oeis.org

6, 11, 18, 27, 38, 51, 2, 83, 29, 123, 146, 171, 43, 38, 258, 291, 326, 1, 51, 443, 174, 531, 578, 627, 678, 2, 10, 530, 902, 963, 473, 1091, 1158, 1227, 3, 25, 438, 1523, 66, 1683, 1766, 330, 1042, 2027, 46, 2211, 2306, 2403, 70, 2603, 2706, 417, 2918, 73
Offset: 1

Views

Author

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 03 2003

Keywords

Comments

For all n, n*(n+k) + 1 is a square for k = 2.
Differs from A102305 at positions listed by A102306.

Examples

			a(7) = 2 as 7*9 + 1 = 64 = 4^3, though 66 also qualifies but 2<66.

Crossrefs

Cf. A083501.

Programs

  • Mathematica
    Do[k = 0; While[i = n(n + k) + 1; !IntegerQ[i^(1/3)], k++ ]; Print[k], {n, 1, 55}]

Extensions

Edited and extended by Robert G. Wilson v, May 11 2003

A129388 Primes that are equal to the mean of 5 consecutive squares.

Original entry on oeis.org

11, 83, 227, 443, 1091, 1523, 2027, 3251, 6563, 9803, 11027, 12323, 13691, 15131, 21611, 29243, 47963, 50627, 56171, 59051, 62003, 65027, 74531, 88211, 91811, 95483, 103043, 119027, 123203, 131771, 136163, 140627, 149771, 173891, 178931
Offset: 1

Views

Author

Zak Seidov, Apr 12 2007

Keywords

Comments

The sum of 5 consecutive squares starting with k^2 is equal to 5*(6 + 4*k + k^2) and the mean is (6 + 4*k + k^2) = (k+2)^2 + 2. Hence a(n)= A056899(n+2).

Examples

			11 = (1^2 + ... + 5^2)/5;
83 = (7^2 + ... + 11^2)/5;
227 = (13^2 + ... + 17^2)/5.

Links

Crossrefs

Cf. A056899, A067201, A102305, A129389.

Programs

  • Magma
    [a: n in [1..600] | IsPrime(a) where a is  n^2 + 2*n + 3 ]; // Vincenzo Librandi, Mar 22 2013
  • Mathematica
    Select[Table[n^2 + 2 n + 3, {n, 1, 600}], PrimeQ] (* Vincenzo Librandi, Mar 22 2013 *)
  • SageMath
    A102305=[n^2+2*n+3 for n in range(1,1001)]
[n^2+2*n+3 for n in (1..600) if is_prime(A102305[n-1])] # G. C. Greubel, Feb 03 2024

A197985 a(n) = round((n+1/n)^2).

Original entry on oeis.org

4, 6, 11, 18, 27, 38, 51, 66, 83, 102, 123, 146, 171, 198, 227, 258, 291, 326, 363, 402, 443, 486, 531, 578, 627, 678, 731, 786, 843, 902, 963, 1026, 1091, 1158, 1227, 1298, 1371, 1446, 1523, 1602, 1683, 1766, 1851, 1938, 2027
Offset: 1

Views

Author

Vincenzo Librandi, Oct 20 2011

Keywords

Comments

Shifted variant of A102305. - R. J. Mathar, Oct 20 2011

Links

Crossrefs

Cf. A059100, A102305, A183199, A160842, A010000.

Programs

  • Magma
    [Round((n+1/n)^2): n in [1..60]];
  • Mathematica
    Table[Floor[(n+1/n)^2+1/2],{n,50}] (* Harvey P. Dale, Aug 12 2012 *)
Join[{4}, 2+Range[2,50]^2] (* G. C. Greubel, Feb 04 2024 *)
  • SageMath
    [4]+[n^2+2 for n in range(2,51)] # G. C. Greubel, Feb 04 2024

Formula

a(n) = n^2 + 2, n > 1.
a(n) = a(n-1) + 2*n - 1, n > 2.
From G. C. Greubel, Feb 04 2024: (Start)
G.f.: x*(4 - 6*x + 5*x^2 - x^3)/(1 - x)^3.
E.g.f.: -2 + x + (2 + x + x^2)*exp(x). (End)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4. - Chai Wah Wu, May 09 2024

A292915 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)/(2 - exp(x)).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 6, 13, 1, 4, 11, 26, 75, 1, 5, 18, 51, 150, 541, 1, 6, 27, 94, 299, 1082, 4683, 1, 7, 38, 161, 582, 2163, 9366, 47293, 1, 8, 51, 258, 1083, 4294, 18731, 94586, 545835, 1, 9, 66, 391, 1910, 8345, 37398, 189171, 1091670, 7087261, 1, 10, 83, 566, 3195, 15666, 74067, 378214, 2183339, 14174522, 102247563
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 26 2017

Keywords

Comments

A(n,k) is the k-th binomial transform of A000670 evaluated at n.

Examples

			E.g.f. of column k: A_k(x) = 1 + (k + 1)*x/1! + (k^2 + 2*k + 3)*x^2/2! + (k^3 + 3*k^2 + 9*k + 13)*x^3/3! +  (k^4 + 4*k^3 + 18*k^2 + 52*k + 75) x^4/4! + ...
Square array begins:
    1,     1,     1,     1,     1,      1,  ...
    1,     2,     3,     4,     5,      6,  ...
    3,     6,    11,    18,    27,     38,  ...
   13,    26,    51,    94,   161,    258,  ...
   75,   150,   299,   582,  1083,   1910,  ...
  541,  1082,  2163,  4294,  8345,  15666,  ...

Links

Crossrefs

Columns k=0..4 give A000670, A000629, A007047, A259533, A368317.
Rows n=0..2 give A000012, A000027, A102305.
Main diagonal gives A292916.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 50);
T:= func< n,k | Coefficient(R!(Laplace( Exp(k*x)/(2-Exp(x)) )), n) >;
A292915:= func< n,k | T(k,n-k) >;
[A292915(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 12 2024
  • Maple
    A:= proc(n, k) option remember; k^n +add(
       binomial(n, j)*A(j, k), j=0..n-1)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 27 2017
  • Mathematica
    Table[Function[k, n! SeriesCoefficient[Exp[k x]/(2 - Exp[x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, HurwitzLerchPhi[1/2, -n, k]/2][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
  • PARI
    a000670(n) = sum(k=0, n, k!*stirling(n, k, 2));
A(n, k) = 2^k*a000670(n)-sum(j=0, k-1, 2^j*(k-1-j)^n); \\ Seiichi Manyama, Dec 25 2023
  • SageMath
    def T(n,k): return factorial(n)*( exp(k*x)/(2-exp(x)) ).series(x, n+1).list()[n]
def A292915(n,k): return T(k,n-k)
flatten([[A292915(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jun 12 2024

Formula

E.g.f. of column k: exp(k*x)/(2 - exp(x)).
A(n,k) = 2^k*A000670(n) - Sum_{j=0..k-1} 2^j*(k-1-j)^n. - Seiichi Manyama, Dec 25 2023

A350510 Square array read by descending antidiagonals: A(n,k) is the least number m such that the base-n expansion of m contains the base-n expansions of 1..k as substrings.

Original entry on oeis.org

1, 2, 1, 6, 5, 1, 12, 11, 6, 1, 44, 38, 27, 7, 1, 44, 95, 75, 38, 8, 1, 92, 285, 331, 194, 51, 9, 1, 184, 933, 1115, 694, 310, 66, 10, 1, 1208, 2805, 4455, 3819, 1865, 466, 83, 11, 1, 1256, 7179, 17799, 16444, 8345, 3267, 668, 102, 12, 1
Offset: 2

Views

Author

Davis Smith, Jan 02 2022

Keywords

Examples

			Square array begins:
n/k|| 1 |  2 |   3 |    4 |     5 |      6 |       7 |        8 |
================================================================|
2  || 1 |  2 |   6 |   12 |    44 |     44 |      92 |      184 |
3  || 1 |  5 |  11 |   38 |    95 |    285 |     933 |     2805 |
4  || 1 |  6 |  27 |   75 |   331 |   1115 |    4455 |    17799 |
5  || 1 |  7 |  38 |  194 |   694 |   3819 |   16444 |    82169 |
6  || 1 |  8 |  51 |  310 |  1865 |   8345 |   55001 |   289577 |
7  || 1 |  9 |  66 |  466 |  3267 |  22875 |  123717 |   947260 |
8  || 1 | 10 |  83 |  668 |  5349 |  42798 |  342391 |  2177399 |
9  || 1 | 11 | 102 |  922 |  8303 |  74733 |  672604 |  6053444 |
10 || 1 | 12 | 123 | 1234 | 12345 | 123456 | 1234567 | 12345678 |
11 || 1 | 13 | 146 | 1610 | 17715 | 194871 | 2143588 | 23579476 |

Links

Crossrefs

Cf. A023811, A035239, A049363, A056744, A057544, A061845, A102305.
The first n - 1 terms of rows: 2: A047778, 3: A048435, 4: A048436, 5: A048437, 6: A048438, 7: A048439, 8: A048440, 9: A048441, 10: A007908, 11: A048442, 12: A048443, 13: A048444, 14: A048445, 15: A048446, 16: A048447.

Programs

  • Mathematica
    T[n_,k_]:=(m=0;While[!ContainsAll[Subsequences@IntegerDigits[++m,n],IntegerDigits[Range@k,n]]];m);Flatten@Table[T[1+i,j+1-i],{j,9},{i,j}] (* Giorgos Kalogeropoulos, Jan 09 2022 *)
  • PARI
    A350510_rows(n,k,N=0)= my(L=List(concat(apply(z->fromdigits([1..z],n),[1..n-1]),if(n>2,fromdigits(concat([1,0],[2..n-1]),n),[]))),T1(x)=digits(x,n),T2(x)=fromdigits(x,n),A(x)=my(S=T1(x));setbinop((y,z)->T2(S[y..z]),[1..#S]),N=if(N,N,L[#L]),A1=A(N));while(#Lsetsearch(A1,z),[1..#L+1])),A1=A(N++));listput(L,N));Vec(L)

Formula

For k < n, A(n,k) = A(n,k - 1)*n + k = Sum_{i=1..k} i*(n^(k - i)).
A(n,n) = A049363(n).
A(n,2) = A057544(n).
For n > 3, A(n,3) = A102305(n).
A(n,n - 1) = A023811(n).

A183199 Least integer k such that Floor(k*f(n+1))>k*f(n), where f(n)=(n^2)/(1+n^2).

Original entry on oeis.org

3, 6, 11, 18, 27, 38, 51, 66, 83, 102, 123, 146, 171, 198, 227, 258, 291, 326, 363, 402, 443, 486, 531, 578, 627, 678, 731, 786, 843, 902, 963, 1026, 1091, 1158, 1227, 1298, 1371, 1446, 1523, 1602, 1683, 1766, 1851, 1938, 2027, 2118, 2211, 2306, 2403, 2502
Offset: 1

Views

Author

Clark Kimberling, Dec 29 2010

Keywords

Comments

Appears to be essentially the same as A102305, A059100 and A010000. - R. J. Mathar, Jun 07 2011

Crossrefs

Cf. A183162.

Programs

  • Mathematica
    Table[k=1; While[Floor[k*((n+1)^2)/(1+(n+1)^2)]<=k*(n^2)/(1+(n^2)), k++]; k, {n,100}]
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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