A192506 -id:A192506 - cp's OEIS frontend

A255127 Ludic array: square array A(row,col), where row n lists the numbers removed at stage n in the sieve which produces Ludic numbers. Array is read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

2, 4, 3, 6, 9, 5, 8, 15, 19, 7, 10, 21, 35, 31, 11, 12, 27, 49, 59, 55, 13, 14, 33, 65, 85, 103, 73, 17, 16, 39, 79, 113, 151, 133, 101, 23, 18, 45, 95, 137, 203, 197, 187, 145, 25, 20, 51, 109, 163, 251, 263, 281, 271, 167, 29, 22, 57, 125, 191, 299, 325, 367, 403, 311, 205, 37, 24, 63, 139, 217, 343, 385, 461, 523, 457, 371, 253, 41

Offset: 2

Antti Karttunen, Feb 22 2015

The starting offset of the sequence giving the terms of square array is 2. However, we can tacitly assume that a(1) = 1 when the sequence is used as a permutation of natural numbers. However, term 1 itself is out of the array.

The choice of offset = 2 for the terms starting in rows >= 1 is motivated by the desire to have a permutation of the integers n -> a(n) with a(n) = A(A002260(n-1), A004736(n-1)) for n > 1 and a(1) := 1. However, since this sequence is declared as a "table", offset = 2 would mean that the first *row* (not element) has index 2. I think the sequence should have offset = 1 and the permutation of the integers would be n -> a(n-1) with a(0) := 1 (if a(1) = A(1,1) = 2). Or, the sequence could have offset 0, with an additional row 0 of length 1 with the only element a(0) = A(0,1) = 1, the permutation still being n -> a(n-1) if a(n=0, 1, 2, ...) = (1, 2, 4, ...). This would be in line with considering 1 as the first ludic number, and A(n, 1) = A003309(n+1) for n >= 0. - M. F. Hasler, Nov 12 2024

			The top left corner of the array:
   2,   4,   6,   8,  10,  12,   14,   16,   18,   20,   22,   24,   26
   3,   9,  15,  21,  27,  33,   39,   45,   51,   57,   63,   69,   75
   5,  19,  35,  49,  65,  79,   95,  109,  125,  139,  155,  169,  185
   7,  31,  59,  85, 113, 137,  163,  191,  217,  241,  269,  295,  323
  11,  55, 103, 151, 203, 251,  299,  343,  391,  443,  491,  539,  587
  13,  73, 133, 197, 263, 325,  385,  449,  511,  571,  641,  701,  761
  17, 101, 187, 281, 367, 461,  547,  629,  721,  809,  901,  989, 1079
  23, 145, 271, 403, 523, 655,  781,  911, 1037, 1157, 1289, 1417, 1543
  25, 167, 311, 457, 599, 745,  883, 1033, 1181, 1321, 1469, 1615, 1753
  29, 205, 371, 551, 719, 895, 1073, 1243, 1421, 1591, 1771, 1945, 2117
...

Antti Karttunen, Table of n, a(n) for n = 2..10441; the first 144 antidiagonals of array, flattened

Transpose: A255129.
Inverse: A255128. (When considered as a permutation of natural numbers with a(1) = 1).
Cf. A260738 (index of the row where n occurs), A260739 (of the column).
Main diagonal: A255410.
Column 1: A003309 (without the initial 1). Column 2: A254100.
Row 1: A005843, Row 2: A016945, Row 3: A255413, Row 4: A255414, Row 5: A255415, Row 6: A255416, Row 7: A255417, Row 8: A255418, Row 9: A255419.
A192607 gives all the numbers right of the leftmost column, and A192506 gives the composites among them.
Cf. A272565, A271419, A271420 and permutations A269379, A269380, A269384.
Cf. also related or derived arrays A260717, A257257, A257258 (first differences of rows), A276610 (of columns), A276580.
Analogous arrays for other sieves: A083221, A255551, A255543.
Cf. A376237 (ludic factorials), A377469 (ludic analog of A005867).

rows = 12; cols = 12; t = Range[2, 3000]; r = {1}; n = 1; While[n <= rows, k = First[t]; AppendTo[r, k]; t0 = t; t = Drop[t, {1, -1, k}]; ro[n++] = Complement[t0, t][[1 ;; cols]]]; A = Array[ro, rows]; Table[ A[[n - k + 1, k]], {n, 1, rows}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Mar 14 2016, after Ray Chandler *)

a255127 = lambda n: A255127(A002260(k-1), A004736(k-1))
def A255127(n, k):
    A = A255127; R = A.rows
    while len(R) <= n or len(R[n]) < min(k, A.P[n]): A255127_extend(2*n)
    return R[n][(k-1) % A.P[n]] + (k-1)//A.P[n] * A.S[n]
A=A255127; A.rows=[[1],[2],[3]]; A.P=[1]*3; A.S=[0,2,6]; A.limit=30
def A255127_extend(rMax=9, A=A255127):
    A.limit *= 2; L = [x+5-x%2 for x in range(0, A.limit, 3)]
    for r in range(3, rMax):
        if len(A.P) == r:
            A.P += [ A.P[-1] * (A.rows[-1][0] - 1) ]  # A377469
            A.rows += [[]]; A.S += [ A.S[-1] * L[0] ] # ludic factorials
        if len(R := A.rows[r]) < A.P[r]: # append more terms to this row
            R += L[ L[0]*len(R) : A.S[r] : L[0] ]
        L = [x for i, x in enumerate(L) if i%L[0]] # M. F. Hasler, Nov 17 2024

(define (A255127 n) (if (<= n 1) n (A255127bi (A002260 (- n 1)) (A004736 (- n 1)))))
(define (A255127bi row col) ((rowfun_n_for_A255127 row) col))
;; definec-macro memoizes its results:
(definec (rowfun_n_for_A255127 n) (if (= 1 n) (lambda (n) (+ n n)) (let* ((rowfun_for_remaining (rowfun_n_for_remaining_numbers (- n 1))) (eka (rowfun_for_remaining 0))) (COMPOSE rowfun_for_remaining (lambda (n) (* eka (- n 1)))))))
(definec (rowfun_n_for_remaining_numbers n) (if (= 1 n) (lambda (n) (+ n n 3)) (let* ((rowfun_for_prevrow (rowfun_n_for_remaining_numbers (- n 1))) (off (rowfun_for_prevrow 0))) (COMPOSE rowfun_for_prevrow (lambda (n) (+ 1 n (floor->exact (/ n (- off 1)))))))))

From M. F. Hasler, Nov 12 2024: (Start)
A(r, c) = A(r, c-P(r)) + S(r) = A(r, ((c-1) mod P(r)) + 1) + floor((c-1)/P(r))*S(r) with periods P = (1, 1, 2, 8, 48, 480, 5760, ...) = A377469, and shifts S = (2, 6, 30, 210, 2310, 30030, 510510) = A376237(2, 3, ...). For example:
A(1, c) = A(1, c-1) + 2 = 2 + (c-1)*2 = 2*c,
A(2, c) = A(2, c-1) + 6 = 3 + (c-1)*6 = 6*c - 3,
A(3, c) = A(3, c-2) + 30 = {5 if c is odd else 19} + floor((c-1)/2)*30 = 15*c - 11 + (c mod 2),
A(4, c) = A(4, c-8) + 210 = A(4, ((c-1) mod 8)+1) + floor((c-1)/8)*210, etc. (End)

A192607 Nonludic numbers: complement of A003309.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 19, 20, 21, 22, 24, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 84, 85, 86, 87

Offset: 1

Reinhard Zumkeller, Jul 05 2011

nonn

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

Cf. A192505 (not ludic but prime), A192506 (neither ludic nor prime).

a192607 n = a192607_list !! (n-1)
a192607_list = filter ((== 0) . a192490) [1..]

a3309[nmax_] := a3309[nmax] = Module[{t = Range[2, nmax], k, r = {1}}, While[Length[t] > 0, k = First[t]; AppendTo[r, k]; t = Drop[t, {1, -1, k}]]; r];
nmax = 1000;
Complement[Range[nmax], a3309[nmax]] (* Jean-François Alcover, Dec 10 2021, after Ray Chandler in A003309 *)

A192490(a(n)) = 0.

A175526 A000120-abundant numbers.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 112, 114, 115, 116, 117, 118, 120

Offset: 1

Vladimir Shevelev, Dec 03 2010

nonn

Definition see in A175522. All even numbers > 2 are in the sequence.

A192895(a(n)) > 0. Reinhard Zumkeller, Jul 12 2011

Reinhard Zumkeller (first 1000 terms) & Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A175522 (perfect version), A175524 (deficient version), A257691 (complement, non-abundant version).
Cf. A000120, A192895.
Cf. also A005100, A005101.
a(n) differs from A091212(n) and from A205783(n+1) for the first time at n=37, where a(37) = 55, while 55 is missing from both A091212 and A205783.
Differs from A192506 for the first time at n=54, where a(54) = 77, while 77 is missing from A192506.

import Data.List (findIndices)
a175526 n = a175526_list !! (n-1)
a175526_list = map (+ 1) $ findIndices (> 0) a192895_list
-- Reinhard Zumkeller, Jul 12 2011

isA175526 := proc(n) s := 0 ; for d in (numtheory[divisors](n) minus {n}) do s := s+A000120(d) ; end do: evalb(s> A000120(n)) ; end proc:
for n from 1 to 120 do if isA175526(n) then printf("%d,",n); end if; end do: # R. J. Mathar, Jul 11 2011

okQ[n_] := DivisorSum[n, Total[IntegerDigits[#, 2]]*(-1)^Boole[#==n]&]>0; Select[Range[120], okQ] (* Jean-François Alcover, Dec 06 2015 *)

A192895(n) = sumdiv(n, d, hammingweight(d)*(-1)^(d==n)); \\ Charles R Greathouse IV, Feb 07 2013
isA175526(n) = (A192895(n) > 0);
n = 0; i = 0; while(i < 10000, n++; if(isA175526(n), i++; write("b175526.txt", i, " ", n)));
\\ Antti Karttunen, May 11 2015

is(n)=sumdiv(n,d,hammingweight(d))>2*hammingweight(n) \\ Charles R Greathouse IV, Jan 28 2016

is_A175526 = lambda x: sum(A000120(d) for d in divisors(x)) > 2*A000120(x)
A175526 = filter(is_A175526, IntegerRange(1, 10**4))
# D. S. McNeil, Dec 04 2010

A091212 Composite numbers whose binary representation encodes a polynomial reducible over GF(2).

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 88, 90, 92, 93, 94, 95, 96, 98, 99

Offset: 1

Antti Karttunen, Jan 03 2004

nonn

"Encoded in binary representation" means that a polynomial a(n)*X^n+...+a(0)*X^0 over GF(2) is represented by the binary number a(n)*2^n+...+a(0)*2^0 in Z (where each coefficient a(k) = 0 or 1).
From Reinhard Zumkeller, Jul 05-12 2011, values for maximum n corrected by Antti Karttunen, May 18 2015: (Start)
a(n) = A192506(n) for n <= 36.
a(n) = A175526(n) for n <= 36.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..55429
A. Karttunen, Scheme-program for computing beginning of this sequence.
Index entries for sequences related to binary encoded polynomials over GF(2)

Intersection of A002808 and A091242.
Cf. A257688 (complement, either 1, irreducible in GF(2)[X] or prime), A091206 (prime and irreducible), A091209 (prime and reducible), A091214 (nonprime and irreducible).
Cf. A091213, A236861, A235036 (a subsequence, apart from 1).
Differs from both A175526 and A192506 for the first time at n=37, where a(37) = 56, while A175526(37) = A192506(37) = 55, a term missing from here (as 55 encodes a polynomial which is irreducible in GF(2)[X]).
Differs from its subsequence A205783(n+1) for the first time at n=47, where a(47) = 69, while 69 is missing from A205783.

isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
isA091212(n) = ((n > 1) && !isprime(n) && !isA014580(n));
n = 0; i = 0; while(n < 2^16, n++; if(isA091212(n), i++; write("b091212.txt", i, " ", n)));

a(n) = A091242(A091213(n)).

A192503 Ludic prime numbers.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 23, 29, 37, 41, 43, 47, 53, 61, 67, 71, 83, 89, 97, 107, 127, 131, 149, 157, 173, 179, 181, 193, 211, 223, 227, 233, 239, 257, 277, 283, 307, 313, 331, 337, 353, 359, 383, 389, 397, 419, 421, 431, 433, 463, 467, 503, 509, 541, 577

Offset: 1

Reinhard Zumkeller, Jul 05 2011

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Intersection of A000040 and A003309.
Cf. A010051, A192490.
Cf. A192506 (neither ludic nor prime).

a192503 n = a192503_list !! (n-1)
a192503_list = filter ((== 1) . a010051) a003309_list

a3309[nmax_] := a3309[nmax] = Module[{t = Range[2, nmax], k, r = {1}}, While[Length[t] > 0, k = First[t]; AppendTo[r, k]; t = Drop[t, {1, -1, k}]]; r];
ludicQ[n_, nmax_] /; 1 <= n <= nmax := MemberQ[a3309[nmax], n];
terms = 1000;
f[nmax_] := f[nmax] = Select[Range[nmax], ludicQ[#, nmax] && PrimeQ[#]&] // PadRight[#, terms]&;
f[nmax = terms];
f[nmax = 2 nmax];
While[f[nmax] != f[nmax/2], nmax = 2 nmax];
seq = f[nmax] (* Jean-François Alcover, Dec 10 2021, after Ray Chandler in A003309 *)

A192503(maxn,bflag=0)={my(Vw=vector(maxn, x, x+1), Vl=Vec([1]), vwn=#Vw,i,vj,L=List());
while(vwn>0, i=Vw[1]; Vl=concat(Vl,[i]);
      Vw=vector((vwn*(i-1))\i,x,Vw[(x*i+i-2)\(i-1)]); vwn=#Vw);
kill(Vw); vwn=#Vl;
for(j=1,vwn, vj=Vl[j]; if(isprime(vj),listput(L,vj))); kill(Vw); vwn=#L;
if(bflag, for(i=1,vwn, print(i," ",L[i]))); if(!bflag, return(Vec(L)));
} \\ Anatoly E. Voevudko, Feb 28 2016

A010051(a(n))*A192490(a(n)) = 1.

A257689 Numbers that are either ludic or prime.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 115, 119, 121, 127, 131, 137, 139, 143, 149, 151, 157, 161, 163, 167, 173, 175, 179, 181, 191, 193, 197, 199, 209, 211, 221, 223, 227, 229, 233, 235, 239, 241, 247, 251, 257, 263, 265

Offset: 1

Antti Karttunen, May 07 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10181

Union of primes (A000040) and ludic numbers (A003309).
Cf. A192506 (complement, neither ludic nor prime), A192503 (ludic and prime), A192504 (ludic and nonprime), A192505 (nonludic and prime).
Differs from A206074(n-1), A186891(n) and A257688(n) for the first time at n=19, where a(19) = 59, while A206074(18) = A186891(19) = A257688(19) = 55, a term missing from here.
Differs from A257691 for the first time at n=24, where a(24) = 77, while A257691(24) = 79.

a3309[nmax_] := a3309[nmax] = Module[{t = Range[2, nmax], k, r = {1}}, While[Length[t] > 0, k = First[t]; AppendTo[r, k]; t = Drop[t, {1, -1, k}]]; r];
ludicQ[n_, nmax_] /; 1 <= n <= nmax := MemberQ[a3309[nmax], n];
terms = 1000;
f[nmax_] := f[nmax] = Select[Range[nmax], ludicQ[#, nmax] || PrimeQ[#]&] // PadRight[#, terms]&;
f[nmax = terms];
f[nmax = 2 nmax];
While[f[nmax] != f[nmax/2], nmax = 2 nmax];
seq = f[nmax] (* Jean-François Alcover, Dec 10 2021, after Ray Chandler in A003309 *)

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A255127 Ludic array: square array A(row,col), where row n lists the numbers removed at stage n in the sieve which produces Ludic numbers. Array is read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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A192607 Nonludic numbers: complement of A003309.

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A175526 A000120-abundant numbers.

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A091212 Composite numbers whose binary representation encodes a polynomial reducible over GF(2).

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A192503 Ludic prime numbers.

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A257689 Numbers that are either ludic or prime.

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