A114537 -id:A114537 - cp's OEIS frontend

A245815 Permutation of natural numbers induced when A245821 is restricted to nonprime numbers: a(n) = A062298(A245821(A018252(n))).

1, 2, 5, 3, 4, 7, 9, 59, 11, 6, 20, 125, 18, 25, 15, 10, 16, 26, 32, 31, 103, 8, 12, 35, 41, 50, 13, 39, 85, 64, 43, 164, 29, 38, 17, 66, 19, 24, 21, 45, 132, 37, 105, 139, 82, 33, 65, 27, 507, 52, 14, 180, 161, 96, 46, 22, 190, 141, 87, 1603, 80, 36, 143, 107, 54, 670, 34, 47, 23, 68, 177, 1337, 40

Offset: 1

Antti Karttunen, Aug 02 2014

nonn

This permutation is induced when A245821 is restricted to nonprimes, A018252, the first column of A114537, but equally, when it is restricted to column 2 (A007821), column 3 (A049078), etc. of that square array, or alternatively, to the successive rows of A236542.

The sequence of fixed points f(n) begins as 1, 2, 15, 142, 548, 1694, 54681. A018252(f(n)) gives the nonprime terms of A245823.

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245816.
Related permutations: A245813, A245819, A245821.
Cf. A007821, A018252, A049078, A062298, A114537, A236542, A245823.

allocatemem(234567890);
v014580 = vector(2^18);
v091226 = vector(2^22);
v091242 = vector(2^22);
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
i=0; j=0; n=2; while((n < 2^22), if(isA014580(n), i++; v014580[i] = n; v091226[n] = v091226[n-1]+1, j++; v091242[j] = n; v091226[n] = v091226[n-1]); n++);
A002808(n)={my(k); for(k=0, primepi(n), isprime(n++)&&k--); n}; \\ This function from M. F. Hasler
A062298(n) = n-primepi(n);
A018252(n) = if(1==n, 1, A002808(n-1));
A014580(n) = v014580[n];
A091226(n) = v091226[n];
A091242(n) = v091242[n];
A091205(n) = if(n<=1, n, if(isA014580(n), prime(A091205(A091226(n))), {my(irfs, t); irfs=subst(lift(factor(Mod(1, 2)*Pol(binary(n)))), x, 2); irfs[,1]=apply(t->A091205(t), irfs[,1]); factorback(irfs)}));
A245703(n) = if(1==n, 1, if(isprime(n), A014580(A245703(primepi(n))), A091242(A245703(n-primepi(n)-1))));
A245821(n) = A091205(A245703(n));
A245815(n) = A062298(A245821(A018252(n)));
for(n=1, 10001, write("b245815.txt", n, " ", A245815(n)));

(define (A245815 n) (A062298 (A245821 (A018252 n))))

a(n) = A062298(A245821(A018252(n))).
As a composition of related permutations:
a(n) = A245813(A245819(n)).
Also following holds for all n >= 1:
a(n) = A062298(A049084(A245821(A007821(n)))).
a(n) = A062298(A049084(A049084(A245821(A049078(n))))).

A245816 Permutation of natural numbers induced when A245822 is restricted to nonprime numbers: a(n) = A062298(A245822(A018252(n))).

Original entry on oeis.org

1, 2, 4, 5, 3, 10, 6, 22, 7, 16, 9, 23, 27, 51, 15, 17, 35, 13, 37, 11, 39, 56, 69, 38, 14, 18, 48, 78, 33, 120, 20, 19, 46, 67, 24, 62, 42, 34, 28, 73, 25, 103, 31, 206, 40, 55, 68, 92, 300, 26, 76, 50, 99, 65, 157, 281, 165, 184, 8, 121, 134, 277, 423, 30, 47, 36, 223, 70, 514, 75, 101, 116, 236, 139, 74

Offset: 1

Antti Karttunen, Aug 02 2014

nonn

This permutation is induced when A245822 is restricted to nonprimes, A018252, the first column of A114537, but equally, when it is restricted to column 2 (A007821), column 3 (A049078), etc. of that square array, or alternatively, to the successive rows of A236542.

The sequence of fixed points f(n) begins as 1, 2, 15, 142, 548, 1694, 54681. A018252(f(n)) gives the nonprime terms of A245823.

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245815.
Related permutations: A245814, A245820, A245822.
Cf. A007821, A018252, A049078, A062298, A114537, A236542, A245823.

allocatemem(123456789);
default(primelimit, 2^22)
A014580 = vector(2^18);
A091226 = vector(2^22);
A002808(n)={my(k); for(k=0, primepi(n), isprime(n++)&&k--); n}; \\ This function from M. F. Hasler, Oct 31 2008
A062298(n) = n-primepi(n);
A018252(n) = if(1==n,1,A002808(n-1));
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
i=0; j=0; n=2; while((n < 2^22), if(isA014580(n), i++; A014580[i] = n; A091226[n] = A091226[n-1]+1, A091226[n] = A091226[n-1]); n++)
A091204(n) = if(n<=1, n, if(isprime(n), A014580[A091204(primepi(n))], {my(pfs,t,bits,i); pfs=factor(n); pfs[, 1]=apply(t->Pol(binary(A091204(t))), pfs[, 1]); sum(i=1, #bits=Vec(factorback(pfs))%2, bits[i]<<(#bits-i))}));
A091245(n) = ((n-A091226[n])-1);
A245704(n) = if(1==n, 1, if(isA014580(n), prime(A245704(A091226[n])), A002808(A245704(A091245(n)))));
A245822(n) = A245704(A091204(n));
A245816(n) = A062298(A245822(A018252(n)));
for(n=1, 10001, write("b245816.txt", n, " ", A245816(n)));

(define (A245816 n) (A062298 (A245822 (A018252 n))))

a(n) = A062298(A245822(A018252(n))).
As a composition of related permutations:
a(n) = A245820(A245814(n)).
Also following holds for all n >= 1:
a(n) = A062298(A049084(A245822(A007821(n)))).
a(n) = A062298(A049084(A049084(A245822(A049078(n))))).
etc.

A191442 Dispersion of ([n*sqrt(3)+1/2]), where [ ]=floor, by antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 7, 6, 5, 12, 10, 8, 9, 21, 17, 14, 11, 16, 36, 29, 24, 19, 13, 28, 62, 50, 42, 33, 23, 15, 48, 107, 87, 73, 57, 40, 26, 18, 83, 185, 151, 126, 99, 69, 45, 31, 20, 144, 320, 262, 218, 171, 120, 78, 54, 35, 22, 249, 554, 454, 378, 296, 208, 135, 94

Offset: 1

Clark Kimberling, Jun 04 2011

Background discussion:  Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1.  The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...).  Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers.  The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence.  Examples:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.

			Northwest corner:
  1....2....3....5....9
  4....7....12...21...36
  6....10...17...29...50
  8....14...24...42...73
  11...19...33...57...99

Cf. A114537, A035513, A035506.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;  x = Sqr[3];
f[n_] := Floor[n*x+1/2] (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191442 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191442 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191451 Dispersion of (3*n-2), for n>=2, by antidiagonals.

Original entry on oeis.org

1, 4, 2, 13, 7, 3, 40, 22, 10, 5, 121, 67, 31, 16, 6, 364, 202, 94, 49, 19, 8, 1093, 607, 283, 148, 58, 25, 9, 3280, 1822, 850, 445, 175, 76, 28, 11, 9841, 5467, 2551, 1336, 526, 229, 85, 34, 12, 29524, 16402, 7654, 4009, 1579, 688, 256, 103, 37, 14, 88573

Offset: 1

Clark Kimberling, Jun 05 2011

Row 1:  A003462
Row 2:  A060816
Background discussion:  Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1.  The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...).  Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers.  The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence.  Examples:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.

			Northwest corner:
  1...4....13...40...121
  2...7....22...67...202
  3...10...31...94...283
  5...16...49...148..445
  6...19...58...175..526

Cf. A114537, A035513, A035506, A191449.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;
f[n_] :=3n+1 (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191451 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191451 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A322027 Maximum order of primeness among the prime factors of n; a(1) = 0.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 1, 1, 2, 3, 4, 2, 1, 1, 3, 1, 2, 2, 1, 3, 2, 4, 1, 2, 3, 1, 2, 1, 1, 3, 5, 1, 4, 2, 3, 2, 1, 1, 2, 3, 2, 2, 1, 4, 3, 1, 1, 2, 1, 3, 2, 1, 1, 2, 4, 1, 2, 1, 3, 3, 1, 5, 2, 1, 3, 4, 2, 2, 2, 3, 1, 2, 1, 1, 3, 1, 4, 2, 1, 3, 2, 2, 2, 2, 3, 1, 2

Offset: 1

Gus Wiseman, Nov 24 2018

nonn

The order of primeness (A078442) of a prime number p is the number of times one must apply A000720 to obtain a nonprime number.

			a(105) = 3 because the prime factor of 105 = 3*5*7 with maximum order of primeness is 5, with order 3.

Alois P. Heinz, Table of n, a(n) for n = 1..65536
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included at A006450 with permission of the author]

Cf. A000720, A006450, A007097, A007821, A049076, A076610, A078442, A109082, A114537, A184155, A276625, A279065, A322028, A322030.

with(numtheory):
p:= proc(n) option remember;
      `if`(isprime(n), 1+p(pi(n)), 0)
    end:
a:= n-> max(0, map(p, factorset(n))):
seq(a(n), n=1..120);  # Alois P. Heinz, Nov 24 2018

Table[If[n==1,0,Max@@(Length[NestWhileList[PrimePi,PrimePi[#],PrimeQ]]&/@FactorInteger[n][[All,1]])],{n,100}]

A138947 Square array T[i+1,j] = prime(T[i,j]), T[1,j] = j-th nonprime = A018252(j); read by upward antidiagonals.

Original entry on oeis.org

1, 4, 2, 6, 7, 3, 8, 13, 17, 5, 9, 19, 41, 59, 11, 10, 23, 67, 179, 277, 31, 12, 29, 83, 331, 1063, 1787, 127, 14, 37, 109, 431, 2221, 8527, 15299, 709, 15, 43, 157, 599, 3001, 19577, 87803

Offset: 1

M. F. Hasler, Apr 28 2008

For i>1, T[i,j] = A018252(j)-th number among those not occurring in rows < i.
A permutation of the integers > 0.
Transpose of A114537. See that sequence and the link for more information and references.

			The first row (1,4,6,8,9,10...) of the array gives the nonprime numbers A018252.
The 2nd row (2,7,13,19,23,29,37,...) of the array gives the primes with nonprime index, A000040(A018252(j)) = A007821(j).
The i-th row is { A000040(k) | A049076(k)=i-1 } = A078442^{-1}(i-1).
Column j is the sequence b(n+1)=prime(b(n)) starting with b(j)=A018252(j): A007097, A057450, A057451, A057452, A057453, A057456, A057457, ...

Alexandrov, Lubomir. "On the nonasymptotic prime number distribution." arXiv preprint math/9811096 (1998). (See Appendix.)

N. Fernandez, An order of primeness, F(p).
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A018252, A007821, A006450, A049076, A007097.

If the antidiagonals are read in the opposite direction we get A114537.

t[1, 1] = 1; t[1, 2] = 4; t[1, k_] := (p = t[1, k-1]; If[ PrimeQ[p+1], p+2, p+1]); t[n_ /; n > 1, k_] := Prime[t[n-1, k]]; Flatten[ Table[ t[n, k-n+1], {k, 1, 9}, {n, 1, k}]] (* Jean-François Alcover, Oct 03 2011 *)

p=c=0; T=matrix( 10,10, i,j, if( i==1, while( isprime(c++),); p=c, p=prime(p))); A138947=concat( vector( vecmin( matsize( T )),i, vector( i,j, T[ j,i+1-j ])))

T[i,j] = j-th number for which A078442 equals i-1.

A288469 a(n) = n if n is a nonprime, otherwise take the prime index of n and repeat until you get a nonprime which is then a(n).

Original entry on oeis.org

1, 1, 1, 4, 1, 6, 4, 8, 9, 10, 1, 12, 6, 14, 15, 16, 4, 18, 8, 20, 21, 22, 9, 24, 25, 26, 27, 28, 10, 30, 1, 32, 33, 34, 35, 36, 12, 38, 39, 40, 6, 42, 14, 44, 45, 46, 15, 48, 49, 50, 51, 52, 16, 54, 55, 56, 57, 58, 4, 60, 18, 62, 63, 64, 65, 66, 8, 68, 69, 70, 20, 72, 21, 74, 75, 76, 77, 78, 22, 80, 81, 82, 9, 84, 85, 86

Offset: 1

Peter Weiss, Jun 09 2017

nonn

a(n) = 1 for n in A007097. - Robert Israel, Jun 09 2017

			For n = 17:  17 is a prime, so you take the prime index of 17 which is 7. 7 is a prime, so you take the prime index of 7 which is 4. 4 is a nonprime, so a(17) = 4.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000720, A007097, A010051, A114537.

f:= proc(n) option remember; if isprime(n) then procname(numtheory:-pi(n)) else n fi end proc:
map(f, [$1..100]); # Robert Israel, Jun 09 2017

Table[If[!PrimeQ@ n, n, NestWhile[PrimePi, n, PrimeQ]], {n, 86}] (* Michael De Vlieger, Jun 09 2017 *)

a(n)=while(isprime(n), n=primepi(n)); n \\ Charles R Greathouse IV, Jun 09 2017

From Robert Israel, Jun 09 2017: (Start)
a(n) = n + A010051(n)*(a(A000720(n))-n).
a(A114537(n,k)) = A114537(n,1). (End)

A322028 Number of distinct orders of primeness among the prime factors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2

Offset: 1

Gus Wiseman, Nov 24 2018

nonn

The order of primeness (A078442) of a prime number p is the number of times one must apply A000720 to obtain a nonprime number.

			a(105) = 3 because the prime factors of 105 = 3*5*7 have 3 different orders of primeness, namely 2, 3, and 1 respectively.

Alois P. Heinz, Table of n, a(n) for n = 1..65536
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included at A006450 with permission of the author]

Cf. A000720, A006450, A007097, A007821, A049076, A076610, A078442, A109082, A114537, A184155, A276625, A279065, A322027, A322030.

with(numtheory):
p:= proc(n) option remember;
      `if`(isprime(n), 1+p(pi(n)), 0)
    end:
a:= n-> nops(map(p, factorset(n))):
seq(a(n), n=1..120);  # Alois P. Heinz, Nov 24 2018

Table[If[n==1,0,Length[Union[Length[NestWhileList[PrimePi,PrimePi[#],PrimeQ]]&/@FactorInteger[n][[All,1]]]]],{n,100}]

A322030 Numbers whose prime factors all have the same order of primeness.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 16, 17, 19, 23, 25, 26, 27, 28, 29, 31, 32, 37, 38, 41, 43, 46, 47, 49, 51, 52, 53, 56, 58, 59, 61, 64, 67, 71, 73, 74, 76, 79, 81, 83, 86, 89, 91, 92, 94, 97, 98, 101, 103, 104, 106, 107, 109, 112, 113, 116, 121, 122, 123

Offset: 1

Gus Wiseman, Nov 24 2018

nonn

The order of primeness (A078442) of a prime number p is the number of times one must apply A000720 to obtain a nonprime number.

			182 is in the sequence because its prime factors 2, 7, 13 all have order of primeness 1.

Alois P. Heinz, Table of n, a(n) for n = 1..20000
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included at A006450 with permission of the author]

Cf. A000720, A006450, A007097, A007821, A049076, A076610, A078442, A109082, A114537, A184155, A276625, A322027, A322028.

with(numtheory):
p:= proc(n) option remember;
      `if`(isprime(n), 1+p(pi(n)), 0)
    end:
a:= proc(n) option remember; local k; for k from 1+`if`(n=1,
      0, a(n-1)) while nops(map(p, factorset(k)))>1 do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 24 2018

ordpri[n_]:=If[!PrimeQ[n],0,Length[NestWhileList[PrimePi,PrimePi[n],PrimeQ]]];
Select[Range[200],SameQ@@ordpri/@FactorInteger[#][[All,1]]&]

A361996 Order array of A361994, read by descending antidiagonals.

Original entry on oeis.org

1, 2, 3, 6, 7, 4, 15, 17, 11, 5, 39, 43, 28, 14, 8, 102, 112, 73, 38, 20, 9, 268, 292, 191, 100, 51, 23, 10, 568, 592, 491, 263, 132, 61, 27, 12, 868, 892, 791, 563, 345, 159, 72, 32, 13, 1168, 1192, 1091, 863, 645, 416, 189, 83, 35, 16, 1468, 1492, 1391

Offset: 1

Clark Kimberling, Apr 05 2023

This array is an interspersion (hence a dispersion, as in A114537 and A163255), so every positive integer occurs exactly once. See A333029 for the definition of order array.

			Corner:
  1    2    6   15   39  102  268 ...
  3    7   17   43  112  292  592 ...
  4   11   28   73  191  491  791 ...
  5   14   38  100  263  563  863 ...
  8   20   51  132  345  645  945 ...
  9   23   61  159  416  716 1016 ...
  ...

Cf. A114537, A163255, A333029, A361993, A361996.

zz = 300; z = 30;
w[n_, k_] := w[n, k] = Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
b[h_, k_] := b[h, k] = w[2 h - 1, 2 k - 1] + w[2 h - 1, 2 k] + w[2 h, 2 k - 1] + w[2 h, 2 k];
s = Flatten[Table[b[h, k], {h, 1, zz}, {k, 1, z}]];
r[h_, k_] := Length[Select[s, # <= b[h, k] &]]
TableForm[Table[r[h, k], {h, 1, 50}, {k, 1, 12}]](*A351996, array*)
v = Table[r[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten  (*A351996, sequence *)

cp's OEIS Frontend

A245815 Permutation of natural numbers induced when A245821 is restricted to nonprime numbers: a(n) = A062298(A245821(A018252(n))).

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A245816 Permutation of natural numbers induced when A245822 is restricted to nonprime numbers: a(n) = A062298(A245822(A018252(n))).

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A191442 Dispersion of ([n*sqrt(3)+1/2]), where [ ]=floor, by antidiagonals.

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A191451 Dispersion of (3*n-2), for n>=2, by antidiagonals.

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A322027 Maximum order of primeness among the prime factors of n; a(1) = 0.

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A138947 Square array T[i+1,j] = prime(T[i,j]), T[1,j] = j-th nonprime = A018252(j); read by upward antidiagonals.

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A288469 a(n) = n if n is a nonprime, otherwise take the prime index of n and repeat until you get a nonprime which is then a(n).

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A322028 Number of distinct orders of primeness among the prime factors of n.

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