A255413 -id:A255413 - cp's OEIS frontend

A007310 Numbers congruent to 1 or 5 mod 6.

1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 121, 125, 127, 131, 133, 137, 139, 143, 145, 149, 151, 155, 157, 161, 163, 167, 169, 173, 175

Offset: 1

C. Christofferson (Magpie56(AT)aol.com)

Numbers n such that phi(4n) = phi(3n). - Benoit Cloitre, Aug 06 2003
Or, numbers relatively prime to 2 and 3, or coprime to 6, or having only prime factors >= 5; also known as 5-rough numbers. (Edited by M. F. Hasler, Nov 01 2014: merged with comments from Zak Seidov, Apr 26 2007 and Michael B. Porter, Oct 09 2009)
Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 38 ).
Numbers k such that k mod 2 = 1 and (k+1) mod 3 <> 1. - Klaus Brockhaus, Jun 15 2004
Also numbers n such that the sum of the squares of the first n integers is divisible by n, or A000330(n) = n*(n+1)*(2*n+1)/6 is divisible by n. - Alexander Adamchuk, Jan 04 2007
Numbers n such that the sum of squares of n consecutive integers is divisible by n, because A000330(m+n) - A000330(m) = n*(n+1)*(2*n+1)/6 + n*(m^2+n*m+m) is divisible by n independent of m. - Kaupo Palo, Dec 10 2016
A126759(a(n)) = n + 1. - Reinhard Zumkeller, Jun 16 2008
Terms of this sequence (starting from the second term) are equal to the result of the expression sqrt(4!*(k+1) + 1) - but only when this expression yields integral values (that is when the parameter k takes values, which are terms of A144065). - Alexander R. Povolotsky, Sep 09 2008
For n > 1: a(n) is prime if and only if A075743(n-2) = 1; a(2*n-1) = A016969(n-1), a(2*n) = A016921(n-1). - Reinhard Zumkeller, Oct 02 2008
A156543 is a subsequence. - Reinhard Zumkeller, Feb 10 2009
Numbers n such that ChebyshevT(x, x/2) is not an integer (is integer/2). - Artur Jasinski, Feb 13 2010
If 12*k + 1 is a perfect square (k = 0, 2, 4, 10, 14, 24, 30, 44, ... = A152749) then the square root of 12*k + 1 = a(n). - Gary Detlefs, Feb 22 2010
A089128(a(n)) = 1. Complement of A047229(n+1) for n >= 1. See A164576 for corresponding values A175485(a(n)). - Jaroslav Krizek, May 28 2010
Cf. property described by Gary Detlefs in A113801 and in Comment: more generally, these numbers are of the form (2*h*n+(h-4)*(-1)^n-h)/4 (with h, n natural numbers), therefore ((2*h*n+(h-4)*(-1)^n-h)/4)^2-1 == 0 (mod h); in this case, a(n)^2 - 1 == 0 (mod 6). Also a(n)^2 - 1 == 0 (mod 12). - Bruno Berselli, Nov 05 2010 - Nov 17 2010
Numbers n such that ( Sum_{k = 1..n} k^14 ) mod n = 0. (Conjectured) - Gary Detlefs, Dec 27 2011
From Peter Bala, May 02 2018: (Start)
The above conjecture is true. Apply Ireland and Rosen, Proposition 15.2.2. with m = 14 to obtain the congruence 6*( Sum_{k = 1..n} k^14 )/n = 7 (mod n), true for all n >= 1.  Suppose n is coprime to 6, then 6 is a unit in Z/nZ, and it follows from the congruence that ( Sum_{k = 1..n} k^14 )/n is an integer. On the other hand, if either 2 divides n or 3 divides n then the congruence shows that ( Sum_{k = 1..n} k^14 )/n cannot be integral. (End)
A126759(a(n)) = n and A126759(m) < n for m < a(n). - Reinhard Zumkeller, May 23 2013
(a(n-1)^2 - 1)/24 = A001318(n), the generalized pentagonal numbers. - Richard R. Forberg, May 30 2013
Numbers k for which A001580(k) is divisible by 3. - Bruno Berselli, Jun 18 2014
Numbers n such that sigma(n) + sigma(2n) = sigma(3n). - Jahangeer Kholdi and Farideh Firoozbakht, Aug 15 2014
a(n) are values of k such that Sum_{m = 1..k-1} m*(k-m)/k is an integer. Sums for those k are given by A062717. Also see Detlefs formula below based on A062717. - Richard R. Forberg, Feb 16 2015
a(n) are exactly those positive integers m such that the sequence b(n) = n*(n + m)*(n + 2*m)/6 is integral, and also such that the sequence c(n) = n*(n + m)*(n + 2*m)*(n + 3*m)/24 is integral. Cf. A007775. - Peter Bala, Nov 13 2015
Along with 2, these are the numbers k such that the k-th Fibonacci number is coprime to every Lucas number. - Clark Kimberling, Jun 21 2016
This sequence is the Engel expansion of 1F2(1; 5/6, 7/6; 1/36) + 1F2(1; 7/6, 11/6; 1/36)/5. - Benedict W. J. Irwin, Dec 16 2016
The sequence a(n), n >= 4 is generated by the successor of the pair of polygonal numbers {P_s(4) + 1, P_(2*s - 1)(3) + 1}, s >= 3. - Ralf Steiner, May 25 2018
The asymptotic density of this sequence is 1/3. - Amiram Eldar, Oct 18 2020
Also, the only vertices in the odd Collatz tree A088975 that are branch values to other odd nodes t == 1 (mod 2) of A005408. - Heinz Ebert, Apr 14 2021
From Flávio V. Fernandes, Aug 01 2021: (Start)
For any two terms j and k, the product j*k is also a term (the same property as p^n and smooth numbers).
From a(2) to a(phi(A033845(n))), or a((A033845(n))/3), the terms are the totatives of the A033845(n) itself. (End)
Also orders n for which cyclic and semicyclic diagonal Latin squares exist (see A123565 and A342990). - Eduard I. Vatutin, Jul 11 2023
If k is in the sequence, then k*2^m + 3 is also in the sequence, for all m > 0. - Jules Beauchamp, Aug 29 2024

			G.f. = x + 5*x^2 + 7*x^3 + 11*x^4 + 13*x^5 + 17*x^6 + 19*x^7 + 23*x^8 + ...

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, 1980.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Andreas Enge, William Hart, and Fredrik Johansson, Short addition sequences for theta functions, arXiv:1608.06810 [math.NT], 2016-2018.
B. W. J. Irwin, Constants Whose Engel Expansions are the k-rough Numbers.
L. B. W. Jolley, Summation of Series, Dover, 1961
Cedric A. B. Smith, Prime factors and recurring duodecimals, Math. Gaz. 59 (408) (1975) 106-109.
William A. Stein's The Modular Forms Database, PARI-readable dimension tables for Gamma_0(N).
Eric Weisstein's World of Mathematics, Rough Number.
Eric Weisstein's World of Mathematics, Pi Formulas. [_Jaume Oliver Lafont_, Oct 23 2009]
Index entries for sequences related to smooth numbers [_Michael B. Porter_, Oct 09 2009]
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

A005408 \ A016945. Union of A016921 and A016969; union of A038509 and A140475. Essentially the same as A038179. Complement of A047229. Subsequence of A186422.
Cf. A000330, A001580, A002194, A019670, A032528 (partial sums), A038509 (subsequence of composites), A047209, A047336, A047522, A056020, A084967, A090771, A091998, A144065, A175885-A175887.
For k-rough numbers with other values of k, see A000027, A005408, A007775, A008364-A008366, A166061, A166063.
Cf. A126760 (a left inverse).
Row 3 of A260717 (without the initial 1).
Cf. A105397 (first differences).

Filtered([1..150],n->n mod 6=1 or n mod 6=5); # Muniru A Asiru, Dec 19 2018

a007310 n = a007310_list !! (n-1)
a007310_list = 1 : 5 : map (+ 6) a007310_list
-- Reinhard Zumkeller, Jan 07 2012

[n: n in [1..250] | n mod 6 in [1, 5]]; // Vincenzo Librandi, Feb 12 2016

seq(seq(6*i+j,j=[1,5]),i=0..100); # Robert Israel, Sep 08 2014

Select[Range[200], MemberQ[{1, 5}, Mod[#, 6]] &] (* Harvey P. Dale, Aug 27 2013 *)
a[n_] := (6 n + (-1)^n - 3)/2; a[rem156, 60] (* Robert G. Wilson v, May 26 2014 from a suggestion by N. J. A. Sloane *)
Flatten[Table[6n + {1, 5}, {n, 0, 24}]] (* Alonso del Arte, Feb 06 2016 *)
Table[2*Floor[3*n/2] - 1, {n, 1000}] (* Mikk Heidemaa, Feb 11 2016 *)

isA007310(n) = gcd(n,6)==1 \\ Michael B. Porter, Oct 09 2009

A007310(n)=n\2*6-(-1)^n \\ M. F. Hasler, Oct 31 2014

\\ given an element from the sequence, find the next term in the sequence.
nxt(n) = n + 9/2 - (n%6)/2 \\ David A. Corneth, Nov 01 2016

def A007310(n): return (n+(n>>1)<<1)-1 # Chai Wah Wu, Oct 10 2023

[i for i in range(150) if gcd(6,i) == 1] # Zerinvary Lajos, Apr 21 2009

a(n) = (6*n + (-1)^n - 3)/2. - Antonio Esposito, Jan 18 2002
a(n) = a(n-1) + a(n-2) - a(n-3), n >= 4. - Roger L. Bagula
a(n) = 3*n - 1 - (n mod 2). - Zak Seidov, Jan 18 2006
a(1) = 1 then alternatively add 4 and 2. a(1) = 1, a(n) = a(n-1) + 3 + (-1)^n. - Zak Seidov, Mar 25 2006
1 + 1/5^2 + 1/7^2 + 1/11^2 + ... = Pi^2/9 [Jolley]. - Gary W. Adamson, Dec 20 2006
For n >= 3 a(n) = a(n-2) + 6. - Zak Seidov, Apr 18 2007
From R. J. Mathar, May 23 2008: (Start)
Expand (x+x^5)/(1-x^6) = x + x^5 + x^7 + x^11 + x^13 + ...
O.g.f.: x*(1+4*x+x^2)/((1+x)*(1-x)^2). (End)
a(n) = 6*floor(n/2) - 1 + 2*(n mod 2). - Reinhard Zumkeller, Oct 02 2008
1 + 1/5 - 1/7 - 1/11 + + - - ... = Pi/3 = A019670 [Jolley eq (315)]. - Jaume Oliver Lafont, Oct 23 2009
a(n) = ( 6*A062717(n)+1 )^(1/2). - Gary Detlefs, Feb 22 2010
a(n) = 6*A000217(n-1) + 1 - 2*Sum_{i=1..n-1} a(i), with n > 1. - Bruno Berselli, Nov 05 2010
a(n) = 6*n - a(n-1) - 6 for n>1, a(1) = 1. - Vincenzo Librandi, Nov 18 2010
Sum_{n >= 1} (-1)^(n+1)/a(n) = A093766 [Jolley eq (84)]. - R. J. Mathar, Mar 24 2011
a(n) = 6*floor(n/2) + (-1)^(n+1). - Gary Detlefs, Dec 29 2011
a(n) = 3*n + ((n+1) mod 2) - 2. - Gary Detlefs, Jan 08 2012
a(n) = 2*n + 1 + 2*floor((n-2)/2) = 2*n - 1 + 2*floor(n/2), leading to the o.g.f. given by R. J. Mathar above. - Wolfdieter Lang, Jan 20 2012
1 - 1/5 + 1/7 - 1/11 + - ... = Pi*sqrt(3)/6 = A093766 (L. Euler). - Philippe Deléham, Mar 09 2013
1 - 1/5^3 + 1/7^3 - 1/11^3 + - ... = Pi^3*sqrt(3)/54 (L. Euler). - Philippe Deléham, Mar 09 2013
gcd(a(n), 6) = 1. - Reinhard Zumkeller, Nov 14 2013
a(n) = sqrt(6*n*(3*n + (-1)^n - 3)-3*(-1)^n + 5)/sqrt(2). - Alexander R. Povolotsky, May 16 2014
a(n) = 3*n + 6/(9*n mod 6 - 6). - Mikk Heidemaa, Feb 05 2016
From Mikk Heidemaa, Feb 11 2016: (Start)
a(n) = 2*floor(3*n/2) - 1.
a(n) = A047238(n+1) - 1. (suggested by Michel Marcus) (End)
E.g.f.: (2 + (6*x - 3)*exp(x) + exp(-x))/2. - Ilya Gutkovskiy, Jun 18 2016
From Bruno Berselli, Apr 27 2017: (Start)
a(k*n) = k*a(n) + (4*k + (-1)^k - 3)/2 for k>0 and odd n, a(k*n) = k*a(n) + k - 1 for even n. Some special cases:
k=2: a(2*n) = 2*a(n) + 3 for odd n, a(2*n) = 2*a(n) + 1 for even n;
k=3: a(3*n) = 3*a(n) + 4 for odd n, a(3*n) = 3*a(n) + 2 for even n;
k=4: a(4*n) = 4*a(n) + 7 for odd n, a(4*n) = 4*a(n) + 3 for even n;
k=5: a(5*n) = 5*a(n) + 8 for odd n, a(5*n) = 5*a(n) + 4 for even n, etc. (End)
From Antti Karttunen, May 20 2017: (Start)
a(A273669(n)) = 5*a(n) = A084967(n).
a((5*n)-3) = A255413(n).
A126760(a(n)) = n. (End)
a(2*m) = 6*m - 1, m >= 1; a(2*m + 1) = 6*m + 1, m >= 0. - Ralf Steiner, May 17 2018
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sqrt(3) (A002194).
Product_{n>=2} (1 + (-1)^n/a(n)) = Pi/3 (A019670). (End)

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), ...

Original entry on oeis.org

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)

A255407 Permutation of natural numbers: a(n) = A255127(A252460(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 23, 20, 21, 22, 25, 24, 19, 26, 27, 28, 29, 30, 37, 32, 33, 34, 35, 36, 41, 38, 39, 40, 43, 42, 47, 44, 45, 46, 53, 48, 31, 50, 51, 52, 61, 54, 49, 56, 57, 58, 67, 60, 71, 62, 63, 64, 65, 66, 77, 68, 69, 70, 83, 72, 89, 74, 75, 76, 59, 78, 91, 80, 81

Offset: 1

Antti Karttunen, Feb 22 2015

nonn

a(n) tells which number in Ludic array A255127 is at the same position where n is in array A083221, the sieve of Eratosthenes. As both arrays have A005843 (even numbers) and A016945 as their two topmost rows, both sequences are among the fixed points of this permutation.

Equally: a(n) tells which number in array A255129 is at the same position where n is in the array A083140, as they are the transposes of above two arrays.

			A083221(8,1) = 19 and A255127(8,1) = 23, thus a(19) = 23.
A083221(9,1) = 23 and A255127(9,1) = 25, thus a(23) = 25.
A083221(3,2) = 25 and A255127(3,2) = 19, thus a(25) = 19.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Cf. A001248, A003309, A007310, A008578, A083221, A084967, A252460, A254100, A255413, A255127.
Inverse: A255408.
Similar permutations: A249818.

a(n) = A255127(A252460(n)).
Other identities. For all n >= 1:
a(2n) = 2n. [Fixes even numbers.]
a(3n) = 3n. [Fixes multiples of three.]
a(A008578(n)) = A003309(n). [Maps noncomposites to Ludic numbers.]
a(A001248(n)) = A254100(n). [Maps squares of primes to "postludic numbers".]
a(A084967(n)) = a(5*A007310(n)) = A007310((5*n)-3) = A255413(n). [Maps A084967 to A255413.]
(And similarly between other columns and rows of A083221 and A255127.)

A260738 Row index to A255127: a(1) = 0; for n > 1, a(n) = number of the stage where n is removed in the sieve which produces Ludic numbers.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 4, 1, 2, 1, 5, 1, 6, 1, 2, 1, 7, 1, 3, 1, 2, 1, 8, 1, 9, 1, 2, 1, 10, 1, 4, 1, 2, 1, 3, 1, 11, 1, 2, 1, 12, 1, 13, 1, 2, 1, 14, 1, 3, 1, 2, 1, 15, 1, 5, 1, 2, 1, 4, 1, 16, 1, 2, 1, 3, 1, 17, 1, 2, 1, 18, 1, 6, 1, 2, 1, 19, 1, 3, 1, 2, 1, 20, 1, 4, 1, 2, 1, 21, 1, 22, 1, 2, 1, 3, 1, 23, 1, 2, 1, 7, 1, 5, 1, 2, 1, 24, 1, 3, 1, 2, 1, 4, 1, 25, 1, 2, 1, 26, 1

Offset: 1

Antti Karttunen, Jul 30 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10105

Row index to array A255127.
Cf. A003309, A254100.
Cf. A260739 (corresponding column index).
Cf. A055396, A260438 for row indices to other arrays similar to A255127.
Differs from A055396 for the first time at n=19.

(define (A260738 n) (cond ((= 1 n) 0) ((even? n) 1) (else (let searchrow ((row 2)) (let searchcol ((col 1)) (cond ((>= (A255127bi row col) n) (if (= (A255127bi row col) n) row (searchrow (+ 1 row)))) (else (searchcol (+ 1 col))))))))) ;; Code for A255127bi given in A255127.

Other identities. For all n >= 1:
a(A003309(n)) = n-1. [In Ludic sieve A003309(k+1) (i.e., the k-th Ludic number after 1) is the first among the numbers removed at stage k.]
a(2n) = 1. [All even numbers are removed at the stage one of the sieve.]
a(A016945(n)) = 2, a(A255413(n)) = 3, a(A255414(n)) = 4, ..., a(A255419(n)) = 9.
a(A254100(n)) = n.
For all n >= 2:
A255127(a(n), A260739(n)) = n.

A255419 Row 9 of Ludic array A255127.

Original entry on oeis.org

25, 167, 311, 457, 599, 745, 883, 1033, 1181, 1321, 1469, 1615, 1753, 1903, 2041, 2191, 2339, 2483, 2623, 2773, 2911, 3059, 3211, 3353, 3493, 3637, 3781, 3929, 4067, 4217, 4367, 4507, 4657, 4795, 4937, 5087, 5227, 5377, 5527, 5665, 5813, 5957, 6101, 6241, 6389, 6535, 6683, 6821, 6971, 7111

Offset: 1

Antti Karttunen, Feb 22 2015

nonn

This is the first row in A255127 that starts with a composite number.

Antti Karttunen, Table of n, a(n) for n = 1..10001

Row 9 of A255127. See A255413 - A255418 for rows 3 through 8.

my(L=vector(97847750, x, 3*x+1+x%2), m(n, k)=2^(n\/k*k)\(2^k-1)); for(i=3, 8, L=vecextract(L, 2^#L-m(#L, L[1])-1)); L255419=vecextract(L, m(#L, L[1]));
\\ If only terms up to N < P are needed, the vector L above can be chosen shorter
A255419(n, P=2027520)=n--\P*293543250 + L255419[n%P+1] \\ M. F. Hasler, Nov 17 2024

# if only terms up to a smaller limit S are needed, then S can be decreased
def A255419(n, S=293543250, P=2027520):
    try: n -= 1; return A255419.L[n]
    except IndexError: return A255419.L[n%P] + n//P*S
    except AttributeError: L = [x+5-x%2 for x in range(0, S, 3)]
    while (k:=L[0]) < 25: L = [x for i, x in enumerate(L) if i%k]
    A255419.L = L[::k]; return A255419(n+1) # M. F. Hasler, Nov 17 2024

(define (A255419 n) (A255127bi 9 n)) ;; Code for A255127bi given in A255127.

a(n) = a(n-P) + S = a((n-1)%P + 1) + S*floor((n-1)/P) with period P = 2027520 = A377469(9) and shift S = 293543250 = A376237(10). - M. F. Hasler, Nov 17 2024

A258016 Unlucky numbers removed at the stage three of Lucky sieve.

Original entry on oeis.org

19, 39, 61, 81, 103, 123, 145, 165, 187, 207, 229, 249, 271, 291, 313, 333, 355, 375, 397, 417, 439, 459, 481, 501, 523, 543, 565, 585, 607, 627, 649, 669, 691, 711, 733, 753, 775, 795, 817, 837, 859, 879, 901, 921, 943, 963, 985, 1005, 1027, 1047, 1069, 1089, 1111, 1131, 1153, 1173, 1195, 1215, 1237, 1257, 1279, 1299, 1321, 1341, 1363, 1383, 1405

Offset: 1

Antti Karttunen, Jul 27 2015

Numbers congruent to 19 or 39 modulo 42. - Jianing Song, Apr 27 2022

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1)

Row 3 of A255543. Every seventh term of A047241.

Cf. also A258011.

a(n) = A047241(7*n).
a(n) = A260436(A255413(1+n)).
From Jianing Song, Apr 27 2022: (Start)
a(n) = a(n-2) + 42.
a(n) = a(n-1) + a(n-2) - a(n-3).
G.f.: (19*x+20*x^2+3*x^3)/(1-x-x^2+x^3).
E.g.f.: 3 + (21*x-3)*cosh(x) + (21*x-2)*sinh(x). (End)

A260714 Row 4 of A260717.

Original entry on oeis.org

7, 11, 13, 17, 23, 25, 29, 31, 37, 41, 43, 47, 53, 55, 59, 61, 67, 71, 73, 77, 83, 85, 89, 91, 97, 101, 103, 107, 113, 115, 119, 121, 127, 131, 133, 137, 143, 145, 149, 151, 157, 161, 163, 167, 173, 175, 179, 181, 187, 191, 193, 197, 203, 205, 209, 211, 217, 221, 223, 227, 233, 235, 239, 241, 247, 251, 253, 257

Offset: 1

Antti Karttunen, Jul 30 2015

nonn

Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 1, -1).

Row 4 of A260717.
Setwise difference of A007310(2..) \ A255413.
Cf. A255414 (only every seventh term of this sequence, starting from 7).

(define (A260714 n) (A260717bi 4 n)) ;; Code for A260717bi given in A260717.

Other identities. For all n >= 1:

a(1+(7*(n-1))) = A255414(n).

cp's OEIS Frontend

A007310 Numbers congruent to 1 or 5 mod 6.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Haskell

Magma

Maple

Mathematica

PARI

PARI

PARI

Python

Sage

Formula

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), ...

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Scheme

Formula

A255407 Permutation of natural numbers: a(n) = A255127(A252460(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A260738 Row index to A255127: a(1) = 0; for n > 1, a(n) = number of the stage where n is removed in the sieve which produces Ludic numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

Formula

A255419 Row 9 of Ludic array A255127.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Python

Scheme

Formula

A258016 Unlucky numbers removed at the stage three of Lucky sieve.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A260714 Row 4 of A260717.

Views

Author

Keywords

Links

Crossrefs

Programs