A236536 -id:A236536 - cp's OEIS frontend

A236542 Array T(n,k) read along descending antidiagonals: row n contains the primes with n steps in the prime index chain.

2, 7, 3, 13, 17, 5, 19, 41, 59, 11, 23, 67, 179, 277, 31, 29, 83, 331, 1063, 1787, 127, 37, 109, 431, 2221, 8527, 15299, 709, 43, 157, 599, 3001, 19577, 87803, 167449, 5381, 47, 191, 919, 4397, 27457, 219613, 1128889, 2269733, 52711

Offset: 1

R. J. Mathar, Jan 28 2014

Row n contains the primes A000040(j) for which A049076(j) = n.

			The array starts:
    2,    7,   13,   19,   23,   29,   37,   43,   47,   53,...
    3,   17,   41,   67,   83,  109,  157,  191,  211,  241,...
    5,   59,  179,  331,  431,  599,  919, 1153, 1297, 1523,...
   11,  277, 1063, 2221, 3001, 4397, 7193, 9319,10631,12763,...
   31, 1787, 8527,19577,27457,42043,72727,96797,112129,137077,...

N. Fernandez, An order of primeness, F(p).

Cf. A007821 (row 1), A049078 (row 2), A049079 (row 3), A007097 (column 1), A058010 (diagonal), A057456 - A057457 (columns), A135044, A236536.

A236542 := proc(n,k)
    option remember ;
    if n = 1 then
        A007821(k) ;
    else
        ithprime(procname(n-1,k)) ;
    end if:
end proc:
for d from 2 to 10 do
    for k from d-1 to 1 by -1 do
            printf("%d,",A236542(d-k,k)) ;
    end do:
end do:

A007821 = Prime[Select[Range[15], !PrimeQ[#]&]];
T[n_, k_] := T[n, k] = If[n == 1, If[k <= Length[A007821], A007821[[k]], Print["A007821 must be extended"]; Abort[]], Prime[T[n-1, k]]];
Table[T[n-k+1, k], {n, 1, 9}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Apr 16 2020 *)

T(1,k) = A007821(k).

T(n,k) = prime( T(n-1,k) ), n>1 .

A238538 A fourth-order linear divisibility sequence: a(n) = (2^n + 1)(2^(3n) - 1)/ ( (2 + 1)*(2^3 - 1) ).

Original entry on oeis.org

1, 15, 219, 3315, 51491, 811395, 12882499, 205321155, 3278747331, 52408827075, 838132189379, 13406842675395, 214483303960771, 3431523432591555, 54902699475185859, 878429788032676035, 14054769379960303811, 224875452250864496835, 3598000373385828511939

Offset: 1

Peter Bala, Feb 28 2014

This is a fourth-order linear divisibility sequence, that is, the sequence satisfies a linear recurrence of order 4 and if n | m then a(n) | a(m). This is a consequence of the following more general result: The polynomials P(n,x,y) := (x^n + y^n)*(x^(3*n) - y^(3*n)) form a fourth-order linear divisibility sequence in the polynomial ring Z[x,y]. See the Bala link.
Hence, for a fixed integers M and N, the normalized sequence (M^n + N^n)*(M^(3*n) - N^(3*n))/ ( (M + N)*(M^3 - N^3) ) for n = 1,2,3,... is a linear divisibility sequence of order 4. It has the rational o.g.f. x*(1 - 2*M*N*(M^2 - M*N + N^2)*x + (M*N)^4*x^2)/( (1 - M^4*x)*(1 - M^3*N*x)*(1 - M*N^3*x)*(1 - N^4*x) ). This is the case M = 2, N = 1. For other cases see A238539(M = 2, N = -1), A238540(M = 3, N = 1) and A238541(M = 3, N = 2). See also A238536, A238537 and A215466.
Note, these sequences do not belong to the family of linear divisibility sequences of the fourth order studied by Williams and Guy, which have o.g.f.s of the form x*(1 - q*x^2)/Q(x), Q(x) a quartic polynomial and q an integer parameter.

Michael De Vlieger, Table of n, a(n) for n = 1..831
Peter Bala, A family of linear divisibility sequences of order four
E. L. Roettger and H. C. Williams, Appearance of Primes in Fourth-Order Odd Divisibility Sequences, J. Int. Seq., Vol. 24 (2021), Article 21.7.5.
Wikipedia, Divisibility sequence
Wikipedia, Lucas Sequence
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
Index entries for linear recurrences with constant coefficients, signature (27,-202,432,-256).

Cf. A215466, A236536, A236537, A238539, A238540, A238541.

seq(1/21*(2^n + 1)*(2^(3*n) - 1), n = 1..20);

LinearRecurrence[{27,-202,432,-256},{1,15,219,3315},20] (* Harvey P. Dale, Jul 04 2019 *)

a(n) = (1/21)*(2^n + 1)*(2^(3*n) - 1) = A000051(n)*A024088(n)/21.
a(n) = (1/21)*(4^n - 1)*(8^n - 1)/(2^n - 1).
O.g.f.: x*(1 - 12*x + 16*x^2)/((1 - x)*(1 - 2*x)*(1 - 8*x)*(1 - 16*x)).
Recurrence equation: a(n) = 27*a(n-1) - 202*a(n-2) + 432*a(n-4) - 256*a(n-4).

A135044 a(1)=1, then a(c) = p and a(p) = c, where c = T_c(r,k) and p = T_p(r,k), and where T_p contains the primes arranged in rows by the prime index chain and T_c contains the composites arranged in rows by the order of compositeness. See Formula.

Original entry on oeis.org

1, 4, 9, 2, 16, 7, 6, 13, 3, 19, 26, 17, 8, 23, 41, 5, 12, 67, 10, 29, 59, 37, 14, 83, 179, 11, 43, 331, 20, 47, 39, 109, 277, 157, 53, 431, 22, 1063, 31, 191, 15, 2221, 27, 61, 211, 71, 30, 599, 1787, 919, 241, 3001, 35, 73, 8527, 127, 1153, 79, 21, 19577, 44, 89, 283

Offset: 1

Katarzyna Matylla, Feb 11 2008

nonn

Exchanges primes with composites, primeth primes with composith composites, etc.
Exchange the k-th prime of order j with the k-th composite of order j and vice versa.
Self-inverse permutation of positive integers.
If n is the composite number A236536(r,k), then a(n) is the corresponding prime A236542(r,k) at the same position (r,k). Vice versa, if n is the prime A236542(r,k), then a(n) is the corresponding composite A236536(r,k) at the same position. - Andrew Weimholt, Jan 28 2014
The original name for this entry did not produce this sequence, but instead A236854, which differs from this permutation for the first time at n=8, where A236854(8)=23, while here a(8)=13. - Antti Karttunen, Feb 01 2014

			From _Andrew Weimholt_, Jan 29 2014: (Start)
More generally, takes the primes organized in an array according to the sieving process described in the Fernandez paper:
        Row[1](n) = 2, 7, 13, 19, 23, ...
        Row[2](n) = 3, 17, 41, 67, 83, ...
        Row[3](n) = 5, 59, 179, ...
        Row[4](n) = 11, 277, ...
        Lets call this  T_p (n, k)
Also take the composites organized in a similar manner, except we use "composite" numbered positions in our sieve:
        Row[1](n) = 4, 6, 8, 10, 14, 20, 22, ...
        Row[2](n) = 9, 12, 15, 18, 24, ...
        Row[3](n) = 16, 21, 25, ...
        Lets call this T_c (n, k)
If we now take the natural numbers and swap each number (except for 1) with the number which holds the same spot in the other array, then we get the sequence: 1, 4, 9, 2, 16, 7, 6, 13, with for example a(8) = 13 (13 holds the same position in the 'prime' table as 8 does in the 'composite' table). (End)

R. J. Mathar, Table of n, a(n) for n = 1..197
N. Fernandez, An order of primeness, F(p).
N. Fernandez, An order of primeness [cached copy, included with permission of the author]
Index to permutations of positive integers

Cf. A000040, A007097, A049076, A049078 - A049081, A058322, A058324 - A058328, A093046, A002808, A006508, A059981, A078442, A236854.

A135044 := proc(n)
    if n = 1 then
        1;
    elif isprime(n) then
        idx := -1 ;
        for r from 1 do
            for c from 1 do
                if A236542(r,c) = n then
                    idx := [r,c] ;
                end if;
                if A236542(r,c) >= n then
                    break;
                end if;
            end do:
            if type(idx,list)  then
                break;
            end if;
        end do:
        A236536(r,c) ;
    else
        idx := -1 ;
        for r from 1 do
            for c from 1 do
                if A236536(r,c) = n then
                    idx := [r,c] ;
                end if;
                if A236536(r,c) >= n then
                    break;
                end if;
            end do:
            if type(idx,list)  then
                break;
            end if;
        end do:
        A236542(r,c) ;
    end if;
end proc: # R. J. Mathar, Jan 28 2014

Composite[n_Integer] := Block[{k = n + PrimePi@n + 1}, While[k != n + PrimePi@k + 1, k++ ]; k]; Compositeness[n_] := Block[{c = 1, k = n}, While[ !(PrimeQ@k || k == 1), k = k - 1 - PrimePi@k; c++ ]; c]; Primeness[n_] := Block[{c = 1, k = n}, While[ PrimeQ@k, k = PrimePi@k; c++ ]; c];
ckj[k_, j_] := Select[ Table[Composite@n, {n, 10000}], Compositeness@# == j &][[k]]; pkj[k_, j_] := Select[ Table[Prime@n, {n, 3000}], Primeness@# == j &][[k]]; f[0]=0; f[1] = 1;
f[n_] := If[ PrimeQ@ n, pn = Primeness@n; ckj[ Position[ Select[ Table[ Prime@ i, {i, 150}], Primeness@ # == pn &], n][[1, 1]], pn], cn = Compositeness@n; pkj[ Position[ Select[ Table[ Composite@ i, {i, 500}], Compositeness@ # == cn &], n][[1, 1]], cn]]; Array[f, 64] (* Robert G. Wilson v *)

a(1)=1, a(A236536(r,k))=A236542(r,k), a(A236542(r,k))=A236536(r,k)

Edited, corrected and extended by Robert G. Wilson v, Feb 18 2008

Name corrected by Andrew Weimholt, Jan 29 2014

A260621 Let b(k, n) = number obtained when the map x->A002808(x) is applied k times to n; a(n) is the smallest k such that b(k, n) + 1 is prime.

Original entry on oeis.org

1, 1, 12, 2, 1, 1, 3, 11, 1, 1, 7, 9, 1, 2, 10, 4, 2, 1, 1, 6, 8, 3, 3, 1, 9, 3, 1, 1, 18, 3, 1, 5, 7, 2, 2, 1, 4, 8, 2, 14, 1, 1, 6, 17, 2, 6, 1, 4, 6, 1, 1, 2, 2, 3, 7, 1, 13, 6, 1, 4, 16, 5, 16, 1, 5, 31, 35, 3, 5, 2, 1, 2, 3, 1, 1, 2, 6, 1, 1, 12, 5, 1, 2

Offset: 1

Matthew Campbell, Sep 25 2015

nonn

a(n) is also the smallest value of k at which b(k, n+1) - b(k, n) > 1.

			When n = 3, writing Composite(x) for A002808(x):
1. Composite(3) = 8. 8 + 1 = 9 = 3^2. 9 is not prime.
2. Composite(8) = 15. 15 + 1 = 16 = 2^4. 16 is not prime.
3. Composite(15) = 25. 25 + 1 = 26 = 2*13. 26 is not prime.
4. Composite(25) = 38. 38 + 1 = 39 = 3*13. 39 is not prime.
5. Composite(38) = 55. 55 + 1 = 56 = 2^3*7. 56 is not prime.
6. Composite(55) = 77. 77 + 1 = 78 = 2*3*13. 78 is not prime.
7. Composite(77) = 105. 105 + 1 = 106 = 2*53. 106 is not prime.
8. Composite(105) = 140. 140 + 1 = 141 = 3*47. 141 is not prime.
9. Composite(140) = 183. 183 + 1 = 184 = 2^3*23. 184 is not prime.
10. Composite(183) = 235. 235 + 1 = 236 = 2^2*59. 236 is not prime.
11. Composite(235) = 298. 298 + 1 = 299 = 13*23. 299 is not prime.
12. Composite(298) = 372. 372 + 1 = 373. 373 is prime.
--------------------------------------------------------------
Since the composite function was applied 12 times, a(3)=12.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Primes and nonprimes: A000040, A002808, A008578, A018252.
a(1) = p, a(n+1) = a(n)-th composite number: A006508, A022450, A022451, A025010, A025011, A059407, A059408.
Composites with order n > 1: A050435, A050436, A050438, A050439, A050440.
Composites with order n = b, n >= 1: A022449.
Composites with prime subscripts: A065858.
Composites without prime subscripts: A175251.
Order of compositeness: A059981, A236536.
Prime(n)-1: A006093.

c = Select[Range[10^5], CompositeQ]; Table[k = 1; While[! PrimeQ[Nest[c[[#]] &, n, k] + 1], k++]; k, {n, 120}] (* Michael De Vlieger, Jul 15 2016 *)

Terms from a(12) onward from Jon E. Schoenfield, Sep 27 2015

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A236542 Array T(n,k) read along descending antidiagonals: row n contains the primes with n steps in the prime index chain.

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A238538 A fourth-order linear divisibility sequence: a(n) = (2^n + 1)*(2^(3*n) - 1)/ ( (2 + 1)*(2^3 - 1) ).

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A135044 a(1)=1, then a(c) = p and a(p) = c, where c = T_c(r,k) and p = T_p(r,k), and where T_p contains the primes arranged in rows by the prime index chain and T_c contains the composites arranged in rows by the order of compositeness. See Formula.

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A260621 Let b(k, n) = number obtained when the map x->A002808(x) is applied k times to n; a(n) is the smallest k such that b(k, n) + 1 is prime.

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A238538 A fourth-order linear divisibility sequence: a(n) = (2^n + 1)(2^(3n) - 1)/ ( (2 + 1)*(2^3 - 1) ).