A001749 -id:A001749 - cp's OEIS frontend

A383961 Square array read by upward antidiagonals: T(n,k) is the n-th number whose largest odd divisor is its k-th divisor, n >= 1, k >= 1.

1, 2, 3, 4, 5, 6, 8, 7, 9, 15, 16, 11, 10, 20, 18, 32, 13, 12, 21, 50, 36, 64, 17, 14, 27, 81, 45, 30, 128, 19, 22, 28, 88, 63, 42, 105, 256, 23, 24, 33, 98, 75, 54, 135, 60, 512, 29, 25, 35, 104, 99, 66, 165, 84, 120, 1024, 31, 26, 39, 136, 117, 70, 189, 108, 140, 90

Offset: 1

Omar E. Pol, May 16 2025

This is a permutation of the positive integers.
From Peter Munn, May 18 2025: (Start)
Numbers with the same factorization pattern of their sequence of divisors (see A290110) and the same parity appear here in the same column.
For example, each column k > 2 includes the subsequence 2^(k-2) * p for all prime p > 2^(k-2).
(End)

			The corner 15 X 15 of the square array is as follows:
      1,  3,  6,  15,  18,  36,  30, 105,  60, 120,  90, 315,  816, 1360, 180, ...
      2,  5,  9,  20,  50,  45,  42, 135,  84, 140, 126, 324,  880, 1520, 210, ...
      4,  7, 10,  21,  81,  63,  54, 165, 108, 168, 150, 432,  912, 1632, 252, ...
      8, 11, 12,  27,  88,  75,  66, 189, 132, 220, 198, 440, 1040, 1760, 270, ...
     16, 13, 14,  28,  98,  99,  70, 195, 156, 240, 216, 495, 1056, 1824, 300, ...
     32, 17, 22,  33, 104, 117,  72, 200, 162, 260, 234, 520, 1104, 1840, 330, ...
     64, 19, 24,  35, 136, 147,  78, 231, 204, 308, 264, 525, 1120, 1904, 378, ...
    128, 23, 25,  39, 152, 153, 100, 255, 225, 340, 280, 528, 1144, 2000, 390, ...
    256, 29, 26,  40, 176, 171, 102, 273, 228, 364, 294, 560, 1232, 2080, 396, ...
    512, 31, 34,  44, 184, 175, 110, 285, 276, 380, 306, 585, 1248, 2128, 462, ...
   1024, 37, 38,  51, 208, 207, 114, 297, 348, 405, 312, 616, 1392, 2208, 468, ...
   2048, 41, 46,  52, 232, 243, 130, 345, 372, 460, 336, 624, 1456, 2288, 510, ...
   4096, 43, 48,  55, 242, 245, 138, 351, 400, 476, 342, 675, 1458, 2320, 546, ...
   8192, 47, 49,  56, 248, 261, 144, 357, 441, 480, 350, 680, 1488, 2464, 570, ...
  16384, 53, 58,  57, 296, 272, 154, 375, 444, 500, 408, 693, 1496, 2480, 588, ...
  ...

Column 1 gives A000079.
Column 2 gives A065091.
Column 3 consists of (A001248 U A091629 U A100484)\{4}.
Column 4 consists of numbers >= 15 in (A001749 U A030078 U A046388 U A070875).
Row 1 gives A383402.
Cf. A000005, A000265, A001227, A027750, A038547, A174090, A182469, A290110, A383401.

f[n_] := If[OddQ[n], DivisorSigma[0, n], FirstPosition[Divisors[n], n/2^IntegerExponent[n, 2]][[1]]]; seq[m_] := Module[{t = Table[0, {m}, {m}], v = Table[0, {m}], c = 0, k = 1, i, j}, While[c < m*(m + 1)/2, i = f[k]; If[i <= m, j = v[[i]] + 1; If[j <= m - i + 1, t[[i]][[j]] = k; v[[i]]++; c++]]; k++]; Table[t[[j]][[i - j + 1]], {i, 1, m}, {j, 1, i}] // Flatten]; seq[11] (* Amiram Eldar, May 16 2025 *)

A175880 a(1)=1, a(2)=2. If n >= 3: if n/2 is in the sequence, a(n)=0, otherwise a(n)=n.

Original entry on oeis.org

1, 2, 3, 0, 5, 0, 7, 8, 9, 0, 11, 12, 13, 0, 15, 0, 17, 0, 19, 20, 21, 0, 23, 0, 25, 0, 27, 28, 29, 0, 31, 32, 33, 0, 35, 36, 37, 0, 39, 0, 41, 0, 43, 44, 45, 0, 47, 48, 49, 0, 51, 52, 53, 0, 55, 0, 57, 0, 59, 60, 61, 0, 63, 0, 65, 0, 67, 68, 69, 0, 71, 0, 73, 0, 75, 76, 77, 0, 79, 80

Offset: 1

Adriano Caroli, Dec 05 2010

If n > 0 and n is in the sequence, then a(2*n) = 0. Example: 5 is in the sequence, so a(2*5) = a(10) = 0.

Is this a(n) = n*A039982(n-1), n > 1? [R. J. Mathar, Dec 07 2010]

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A053661.

A000040, A001749, A002001, A002042, A002063, A002089, A003947, A004171 and A081294 are subsequences.

import Data.List (delete)
a175880 n = a175880_list !! (n-1)
a175880_list = 1 : f [2..] [2..] where
   f (x:xs) (y:ys) | x == y    = x : (f xs $ delete (2*x) ys)
                   | otherwise = 0 : (f xs (y:ys))
for_bFile = take 10000 a175880_list
-- Reinhard Zumkeller, Feb 09 2011

A110654 := proc(n) 2*n+1-(-1)^n ; %/4 ;end proc:
A175880 := proc(n) if n <=2 then n; else if type(n,'even') then n-2*procname(A110654(n)) ; else n; end if; end if; end proc:
seq(A175880(n),n=1..40) ; # R. J. Mathar, Dec 07 2010

a(n) = n - (1 + (-1)^n) * a((2*n + 1 - (-1)^n)/4), n >= 3.

a(n) = n - A010673(n+1)*a(A110654(n)).

A370010 a(n) is the greatest prime less than 4*prime(n).

Original entry on oeis.org

7, 11, 19, 23, 43, 47, 67, 73, 89, 113, 113, 139, 163, 167, 181, 211, 233, 241, 263, 283, 283, 313, 331, 353, 383, 401, 409, 421, 433, 449, 503, 523, 547, 547, 593, 601, 619, 647, 661, 691, 709, 719, 761, 769, 787, 787, 839, 887, 907, 911, 929, 953, 953, 997

Offset: 1

Clark Kimberling, Feb 09 2024

nonn

			7 < 4*2 < 11 < 4*3 < 13 < 17 < 19  < 5*3 < 23, so (a(1), a(2), a(3)) = (7,11,19).

Cf. A000040, A001749, A059786, A059788, A370008, A370011.

Table[Prime[PrimePi[4*Prime[n]]], {n,1,200}]

a(n) = precprime(4*prime(n)); \\ Michel Marcus, Feb 10 2024

A370011 a(n) is the least prime greater than 4*prime(n).

Original entry on oeis.org

11, 13, 23, 29, 47, 53, 71, 79, 97, 127, 127, 149, 167, 173, 191, 223, 239, 251, 269, 293, 293, 317, 337, 359, 389, 409, 419, 431, 439, 457, 509, 541, 557, 557, 599, 607, 631, 653, 673, 701, 719, 727, 769, 773, 797, 797, 853, 907, 911, 919, 937, 967, 967

Offset: 1

Clark Kimberling, Feb 09 2024

nonn

			7 < 4*2 < 11 < 4*3 < 13 < 17 < 19 < 4*5 < 23, so (a(1), a(2), a(3)) = (11,13,23).

Cf. A000040, A001749, A059786, A059788, A370008, A370010.

Table[NextPrime[Prime[PrimePi[4*Prime[n]]]], {n, 1, 200}]

a(n) = nextprime(4*prime(n)); \\ Michel Marcus, Feb 10 2024

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.

cp's OEIS Frontend

A383961 Square array read by upward antidiagonals: T(n,k) is the n-th number whose largest odd divisor is its k-th divisor, n >= 1, k >= 1.

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A175880 a(1)=1, a(2)=2. If n >= 3: if n/2 is in the sequence, a(n)=0, otherwise a(n)=n.

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A370010 a(n) is the greatest prime less than 4*prime(n).

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A370011 a(n) is the least prime greater than 4*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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