A138636 -id:A138636 - cp's OEIS frontend

A352025 a(n) = floor(sqrt(6*prime(n))).

3, 4, 5, 6, 8, 8, 10, 10, 11, 13, 13, 14, 15, 16, 16, 17, 18, 19, 20, 20, 20, 21, 22, 23, 24, 24, 24, 25, 25, 26, 27, 28, 28, 28, 29, 30, 30, 31, 31, 32, 32, 32, 33, 34, 34, 34, 35, 36, 36, 37, 37, 37, 38, 38, 39, 39, 40, 40, 40, 41, 41, 41, 42, 43, 43, 43, 44

Offset: 1

Zak Seidov, Feb 28 2022

nonn

Cf. A000006, A000196, A138636.

Mathematica
```
Table[Floor[Sqrt[6*Prime[k]]], {k, 50}]
```

a(n) = sqrtint(6*prime(n)); \\ Michel Marcus, Mar 19 2022

a(n) = A000196(A138636(n)). - Michel Marcus, Mar 19 2022

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

Werner Schulte, Jan 05 2024

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.

			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.

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

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)) }

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

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.

A367449 Numbers k for which there are exactly k pairs (i, j), 1 <= i < j < k, such that i + j is a divisor of k.

Original entry on oeis.org

30, 42, 54, 66, 78, 102, 114, 138, 174, 186, 208, 222, 246, 258, 282, 318, 354, 366, 402, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 678, 762, 786, 822, 834, 894, 906, 942, 978, 1002, 1038, 1074, 1086, 1146, 1158, 1182, 1194, 1266, 1312, 1338, 1362, 1374

Offset: 1

Marius A. Burtea, Dec 10 2023

nonn

Numbers k >= 1 for which A367588(k) = Sum_{d|k} floor((d-1)/2) = k;
Numbers k >= 1 for which A000203(k) - A000005(k) - A183063(k) = 2*k.
The sequence is infinite because all numbers of the form m = 6*p, p >= 5 prime (A138636), are terms.
Indeed: sigma(6*p) - tau(6*p) - A183063(6*p) = 3*4*(p + 1) - 8 - 4 = 12*p = 2*m.
If m = 2^k*p, p = 2^(k + 1) - 4*k - 3 prime number, then m is a term. Indeed: sigma(m) - tau(m) - A183063(m) = (2^(k + 1) - 1)*(p + 1) - 2*(k + 1) - 2*k = 2*m.

			30 is a term since it has exactly 30 pairs (i,j): (1, 2), (2, 3), (1, 4), (2, 4), (1, 5), (4, 6), (3, 7), (2, 8), (7, 8), (1,9), (6, 9), (5, 10), (4, 11), (3, 12), (2, 13), (1, 14), (14, 16), (13, 17),(12, 18), (11, 19), (10, 20), (9, 21), (8, 22), (7, 23), (6, 24), (5, 25), (4,26), (3, 27), (2, 28), (1, 29).

Fixed points of A367588.

Cf. A000005, A000203, A138636, A183063.

[k:k in [1..1000]|(DivisorSigma(1,k)-#Divisors(k)-#[d:d in Divisors(k)| IsEven(d)]) eq 2*k ];

filter:= proc(n) uses numtheory;
  sigma(n) - tau(n) - `if`(n::even, tau(n/2),0) = 2*n
end proc:
select(filter, [$1..10000]); # Robert Israel, Dec 12 2023

f1[p_, e_] := e+1; f1[2, e_] := 2*e+1; f2[p_, e_] := (p^(e+1)-1)/(p-1); s[1] = 0; s[n_] := Module[{fct = FactorInteger[n]}, Times @@ f2 @@@ fct - Times @@ f1 @@@ fct]; Select[Range[1400], s[#] == 2*# &] (* Amiram Eldar, Dec 16 2023 *)

isok(k) = sumdiv(k, d, (d-1)\2) == k; \\ Michel Marcus, Dec 19 2023

cp's OEIS Frontend

A352025 a(n) = floor(sqrt(6*prime(n))).

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

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A367449 Numbers k for which there are exactly k pairs (i, j), 1 <= i < j < k, such that i + j is a divisor of k.

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