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 1-3 of 3 results.

A272400 Square array read by antidiagonals upwards in which T(n,k) is the product of the n-th noncomposite number and the sum of the divisors of k, n>=1, k>=1.

Original entry on oeis.org

1, 2, 3, 3, 6, 4, 5, 9, 8, 7, 7, 15, 12, 14, 6, 11, 21, 20, 21, 12, 12, 13, 33, 28, 35, 18, 24, 8, 17, 39, 44, 49, 30, 36, 16, 15, 19, 51, 52, 77, 42, 60, 24, 30, 13, 23, 57, 68, 91, 66, 84, 40, 45, 26, 18, 29, 69, 76, 119, 78, 132, 56, 75, 39, 36, 12, 31, 87, 92, 133, 102, 156, 88, 105, 65, 54, 24, 28
Offset: 1

Views

Author

Omar E. Pol, Apr 28 2016

Keywords

Examples

			The corner of the square array begins:
1,   3,   4,   7,   6,  12,   8,  15,  13,  18...
2,   6,   8,  14,  12,  24,  16,  30,  26,  36...
3,   9,  12,  21,  18,  36,  24,  45,  39,  54...
5,  15,  20,  35,  30,  60,  40,  75,  65,  90...
7,  21,  28,  49,  42,  84,  56, 105,  91, 126...
11, 33,  44,  77,  66, 132,  88, 165, 143, 198...
13, 39,  52,  91,  78, 156, 104, 195, 169, 234...
17, 51,  68, 119, 102, 204, 136, 255, 221, 306...
19, 57,  76, 133, 114, 228, 152, 285, 247, 342...
23, 69,  92, 161, 138, 276, 184, 345, 299, 414...
...

Crossrefs

Rows 1-3: A000203, A074400, A272027.
Columns 1-2: A008578, A112773.
The diagonal 2, 9, 20... is A272211, the main diagonal of A272214.
Cf. A272173.

Programs

  • Mathematica
    Table[If[# == 1, 1, Prime[# - 1]] DivisorSigma[1, k] &@(n - k + 1), {n, 12}, {k, n}] // Flatten (* Michael De Vlieger, Apr 28 2016 *)

Formula

T(n,k) = A008578(n)*A000203(k), n>=1, k>=1.
T(n,k) = A272214(n-1,k), n>=2.

A195896 Numbers of the form 2*p-1 or 3*p-1 where p is 1 or a prime.

Original entry on oeis.org

1, 2, 3, 5, 8, 9, 13, 14, 20, 21, 25, 32, 33, 37, 38, 45, 50, 56, 57, 61, 68, 73, 81, 85, 86, 92, 93, 105, 110, 117, 121, 122, 128, 133, 140, 141, 145, 157, 158, 165, 176, 177, 182, 193, 200, 201, 205, 212, 213, 217, 218, 225, 236, 248, 253, 261, 266, 273, 277, 290, 297, 301, 302, 308
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Sep 24 2011

Keywords

Examples

			a(1)=1 because p=1 and 2*1 - 1 = 1;
a(2)=2 because p=1 and 3*1 - 1 = 2;
a(3)=3 because p=2 and 2*2 - 1 = 3;
a(4)=5 because p=2 and 3*3 - 1 = 5 or p=3 and p=2 and 3*2 - 1 = 5;
a(5)=8 because p=3 and 3*3 - 1 = 8.

Crossrefs

Cf. A196127, A008578.

Programs

  • Maple
    isA195896 := proc(n)
        for p in {(n+1)/2,(n+1)/3} do
        if type(p,'integer') then
                if isprime(p) or p = 1 then
                        return true;
                end if;
        end if;
        end do;
        false ;
end proc:
for n from 1 to 400 do
        if isA195896(n) then
                printf("%d,",n) ;
        end if;
end do: # R. J. Mathar, Oct 15 2011
  • Mathematica
    Union[Flatten[Join[{1,2},{2#-1,3#-1}&/@Prime[Range[50]]]]] (* Harvey P. Dale, Mar 27 2015 *)

Formula

Union of A076274 and A112773.

A363473 Triangle read by rows: T(n, k) = k * prime(n - k + A061395(k)) for 1 < k <= n, and T(n, 1) = A008578(n).

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 5, 10, 15, 8, 7, 14, 21, 12, 25, 11, 22, 33, 20, 35, 18, 13, 26, 39, 28, 55, 30, 49, 17, 34, 51, 44, 65, 42, 77, 16, 19, 38, 57, 52, 85, 66, 91, 24, 27, 23, 46, 69, 68, 95, 78, 119, 40, 45, 50, 29, 58, 87, 76, 115, 102, 133, 56, 63, 70, 121, 31, 62, 93, 92, 145, 114, 161, 88, 99, 110, 143, 36
Offset: 1

Views

Author

Werner Schulte, Jan 05 2024

Keywords

Comments

Conjecture: this is a permutation of the natural numbers.
Generalized conjecture: Let T(n, k) = b(k) * prime(n - k + A061395(b(k))) for 1 < k <= n, and T(n, 1) = A008578(n), where b(n), n > 0, is a permutation of the natural numbers with b(1) = 1, then T(n, k), read by rows, is a permutation of the natural numbers.

Examples

			Triangle begins:
n\k :   1    2    3    4    5    6    7    8    9   10   11   12   13
=====================================================================
 1  :   1
 2  :   2    4
 3  :   3    6    9
 4  :   5   10   15    8
 5  :   7   14   21   12   25
 6  :  11   22   33   20   35   18
 7  :  13   26   39   28   55   30   49
 8  :  17   34   51   44   65   42   77   16
 9  :  19   38   57   52   85   66   91   24   27
10  :  23   46   69   68   95   78  119   40   45   50
11  :  29   58   87   76  115  102  133   56   63   70  121
12  :  31   62   93   92  145  114  161   88   99  110  143   36
13  :  37   74  111  116  155  138  203  104  117  130  187   60  169
etc.

Crossrefs

Cf. A000040, A001747, A001749, A001750, A008578, A112773, A138636, A253560, A272470, A061395.

Programs

  • PARI
    T(n, k) = { if(k==1, if(n==1, 1, prime(n-1)), i=floor((k+1)/2);
            while(k % prime(i) != 0, i=i-1); k*prime(n-k+i)) }
  • SageMath
    def prime(n): return sloane.A000040(n)
def A061395(n): return prime_pi(factor(n)[-1][0]) if n > 1 else 0
def T(n, k):
     if k == 1: return prime(n - 1) if n > 1 else 1
     return k * prime(n - k + A061395(k))
for n in range(1, 11): print([T(n,k) for k in range(1, n+1)])
# Peter Luschny, Jan 07 2024

Formula

T(n, n) = A253560(n) for n > 0.
T(n, 1) = A008578(n) for n > 0.
T(n, 2) = A001747(n) for n > 1.
T(n, 3) = A112773(n) for n > 2.
T(n, 4) = A001749(n-3) for n > 3.
T(n, 5) = A001750(n-2) for n > 4.
T(n, 6) = A138636(n-4) for n > 5.
T(n, 7) = A272470(n-3) for n > 6.
Showing 1-3 of 3 results.

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