A018252 -id:A018252 - cp's OEIS frontend

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

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.

A083256 a(n) = A046523(n-th nonprime number) = A046523(A018252(n)).

Original entry on oeis.org

1, 4, 6, 8, 4, 6, 12, 6, 6, 16, 12, 12, 6, 6, 24, 4, 6, 8, 12, 30, 32, 6, 6, 6, 36, 6, 6, 24, 30, 12, 12, 6, 48, 4, 12, 6, 12, 24, 6, 24, 6, 6, 60, 6, 12, 64, 6, 30, 12, 6, 30, 72, 6, 12, 12, 6, 30, 48, 16, 6, 60, 6, 6, 6, 24, 60, 6, 12, 6, 6, 6, 96, 12, 12, 36, 30, 24, 30, 6, 72, 30, 6, 48

Offset: 1

Labos Elemer, May 09 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007395, A000040, A002808, A018252, A046523, A071364, A083257.

f[n_] := Module[{e = ReverseSort[FactorInteger[n][[;; , 2]]]}, Times @@ (Prime[Range[Length[e]]]^e)]; f[1] = 1; f /@ Select[Range[120], !PrimeQ[#] &] (* Amiram Eldar, Feb 22 2025 *)

Corrected by Ray Chandler, Aug 01 2004

A083257 a(n) = A071364(n-th nonprime number) = A071364(A018252(n)).

Original entry on oeis.org

1, 4, 6, 8, 4, 6, 12, 6, 6, 16, 18, 12, 6, 6, 24, 4, 6, 8, 12, 30, 32, 6, 6, 6, 36, 6, 6, 24, 30, 12, 12, 6, 48, 4, 18, 6, 12, 54, 6, 24, 6, 6, 60, 6, 12, 64, 6, 30, 12, 6, 30, 72, 6, 18, 12, 6, 30, 48, 16, 6, 60, 6, 6, 6, 24, 90, 6, 12, 6, 6, 6, 96, 18, 12, 36, 30, 24, 30, 6, 108, 30, 6

Offset: 1

Labos Elemer, May 09 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007395, A000040, A002808, A018252, A046523, A071364, A083256.

f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Times @@ (Prime[Range[Length[e]]]^e)]; f[1] = 1; f /@ Select[Range[120], !PrimeQ[#] &] (* Amiram Eldar, Feb 22 2025 *)

Corrected by Ray Chandler, Aug 01 2004

A141220 Write the n-th nonprime (A018252(n)) as a product of primes; increase one copy of the largest prime by 2 and decrease one copy of the smallest prime by 1, multiply the resulting numbers.

Original entry on oeis.org

1, 4, 5, 8, 10, 7, 10, 9, 14, 16, 15, 14, 18, 13, 20, 28, 15, 30, 18, 21, 32, 26, 19, 36, 30, 21, 30, 28, 27, 26, 42, 25, 40, 54, 35, 38, 30, 45, 52, 36, 42, 31, 42, 33, 54, 64, 60, 39, 38, 50, 45, 60, 39, 70, 42, 78, 45, 56, 90, 43, 54, 76, 45, 62, 52, 63, 90

Offset: 1

Juri-Stepan Gerasimov, Aug 07 2008

nonn

			1st nonprime = 1 (has no prime factors); a(1) = empty product = 1.
2nd nonprime = 4 = (p(max)=2)*(p(min)=2); a(2) = (2+2)*(2-1) = 4*1 = 4.
3rd nonprime = 6 = (p(max)=3)*(p(min)=2); a(3) = (3+2)*(2-1) = 5*1 = 5.
4th nonprime = 8 = (p(max)=2)*(p=2)*(p(min)=2); a(4) = (2+2)*2*(2-1) = 4*2*1 = 8.

A006530 := proc(n) if n = 1 then 1; else max(op(numtheory[factorset](n))) ; end if; end proc:
A020639 := proc(n) if n = 1 then 1; else min(op(numtheory[factorset](n))) ; end if; end proc:
A002808 := proc(n) if n = 1 then 4; else for a from procname(n-1)+1 do if not isprime(a) then return a; end if; end do; end if; end proc:
A052369 := proc(n) A006530(A002808(n)) ; end proc: A056608 := proc(n) A020639(A002808(n)) ; end proc:
A141220 := proc(n) if n = 0 then 1; else c := A002808(n) ; hi := A052369(n) ; lo := A056608(n) ; c*(hi+2)*(lo-1)/lo/hi ; end if; end proc:
printf("1,") ; for n from 1 to 400 do a := A141220(n) ; if not isprime(a) then printf("%d,",a) ; end if; end do: # R. J. Mathar, Mar 29 2010

nonp[1]=1;i=1;Do[If[n-PrimePi[n]>i,nonp[i+1]=n;i++],{n,2,lim}];f[k_]:=k*(FactorInteger[k][[1,1]]-1)/FactorInteger[k][[1,1]]*(FactorInteger[k][[-1,1]]+2)/FactorInteger[k][[-1,1]];Join[{1},f/@Array[nonp,66,2]] (* James C. McMahon, Jul 18 2025 *)

Entry revised by Jon E. Schoenfield, Mar 09 2014, following revision of A141218 by N. J. A. Sloane

A161570 Sum of all numbers from n up to A018252(n).

Original entry on oeis.org

1, 9, 18, 30, 35, 40, 57, 77, 84, 91, 116, 144, 153, 162, 195, 205, 215, 225, 235, 275, 318, 330, 342, 354, 366, 416, 429, 442, 497, 555, 570, 585, 648, 664, 680, 696, 712, 782, 799, 816, 833, 850, 927, 1007, 1026, 1045, 1064, 1083, 1170, 1190, 1210, 1302

Offset: 1

Juri-Stepan Gerasimov, Jun 14 2009

			a(2) = 2 + 3 + 4 = 9;
a(3) = 3 + 4 + 5 + 6 = 18;
a(4) = 4 + 5 + 6 + 7 + 8 = 30.

Cf. A018252, A000217.

A018252 := proc(n) option remember ; if n = 1 then 1; else for a from procname(n-1)+1 do if not isprime(a) then RETURN(a) ; fi; od: fi; end:
A000217 := proc(n) n*(n+1)/2 ; end:
A161570 := proc(n) A000217( A018252(n)) - A000217(n-1) ; end: seq(A161570(n),n=1..90) ; # R. J. Mathar, Aug 03 2009

print1(n=1);p=3;forprime(q=5,97,for(k=p+1,q-1,print1(", "k*(k+1)/2-n++*(n-1)/2));p=q) \\ Charles R Greathouse IV, Sep 03 2011

a(n) = A000217(A018252(n)) - A000217(n-1).

a(26) corrected and extended by R. J. Mathar, Aug 03 2009

Definition rephrased by R. J. Mathar, Sep 11 2009

A175251 Composites (A002808) with nonprime (A018252) subscripts.

Original entry on oeis.org

4, 9, 12, 15, 16, 18, 21, 24, 25, 26, 28, 32, 33, 34, 36, 38, 39, 40, 42, 45, 48, 49, 50, 51, 52, 55, 56, 57, 60, 63, 64, 65, 68, 69, 70, 72, 74, 76, 77, 78, 80, 81, 84, 86, 87, 88, 90, 91, 93, 94, 95, 98, 100

Offset: 1

Jaroslav Krizek, Mar 13 2010

nonn

a(n) = composite(nonprime(n)) = A002808(A018252(n)). a(n) U A065858(n) = A002808(n), a(n+1) U A022449(n) = A002808(n) for n >= 1. a(1) = 4, a(n) = A050435(n-1) = composites (A002808) with composite (A002808) subscripts for n >=2.

			a(5) = 16 because a(5) = c(b(5)) = c(9) = 16, c = composite, b = nonprime.

A203527 a(n) = Product_{1 <= i < j <= n} (A018252(i) + A018252(j)); A018252 = nonprime numbers.

Original entry on oeis.org

1, 5, 350, 529200, 17542980000, 14783258730240000, 511420331138811494400000, 871980665589501641034301440000000, 60150685659205753788492548338089984000000000, 182771197941564481989784945231570147139911680000000000000

Offset: 1

Clark Kimberling, Jan 03 2012

nonn

Each term divides its successor, as in A203528. It is conjectured that each term is divisible by the corresponding superfactorial, A000178(n); as in A203529. See A093883 for a guide to related sequences.

Cf. A018252, A203415, A203528, A203529.

b:= proc(n) option remember; local k; if n=1 then 1
      else for k from 1+b(n-1) while isprime(k) do od; k fi
    end:
a:= n-> mul(mul(b(i)+b(j), i=1..j-1), j=2..n):
seq(a(n), n=1..10);  # Alois P. Heinz, Jul 23 2017

t = Table[If[PrimeQ[k], 0, k], {k, 1, 100}];
nonprime = Rest[Union[t]]              (* A018252 *)
f[j_] := nonprime[[j]]; z = 20;
v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
d[n_] := Product[(i - 1)!, {i, 1, n}]  (* A000178 *)
Table[v[n], {n, 1, z}]                 (* A203527 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]    (* A203528 *)
Table[v[n]/d[n], {n, 1, 20}]           (* A203529 *)

Name edited by Alois P. Heinz, Jul 23 2017

A245819 Permutation of natural numbers induced when A245703 is restricted to nonprime numbers: a(n) = 1+A091245(A245703(A018252(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 6, 12, 7, 9, 13, 26, 10, 14, 18, 11, 15, 48, 19, 20, 35, 16, 21, 32, 25, 17, 22, 63, 27, 56, 28, 138, 46, 23, 29, 43, 34, 38, 24, 30, 80, 60, 36, 88, 72, 37, 167, 42, 59, 31, 39, 55, 45, 62, 50, 33, 40, 100, 77, 320, 47, 92, 109, 90, 49, 201, 54, 98, 76, 41, 51

Offset: 1

Antti Karttunen, Aug 16 2014

nonn

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

Inverse: A245820.
Related permutations: A245703, A245814, A245815.
Cf. A008578, A018252, A091226, A091245.

\\ The rest of code can be found in A245703.
A245819(n) = if(1==n, 1, 1 + A245703(n-1));

(define (A245819 n) (if (<= n 1) n (+ 1 (A245703 (- n 1)))))

(define (A245819 n) (+ 1 (A091245 (A245703 (A018252 n)))))

(define (A245819 n) (+ 1 (A091226 (A245703 (A008578 n)))))

a(1) = 1, and for n > 1, a(n) = 1 + A245703(n-1).
a(n) = 1+A091245(A245703(A018252(n))). [Induced when A245703 is restricted to nonprime numbers].
a(n) = 1+A091226(A245703(A008578(n))). [Induced also when A245703 is restricted to noncomposite numbers].
As a composition of related permutations:
a(n) = A245814(A245815(n)).

A104655 Let c(i) = A018252(i) be the i-th nonprime; then the final entry in row n of A101513 is c(a(n)) (see A104656), for n >= 3.

Original entry on oeis.org

4, 8, 11, 17, 22, 30, 37, 46, 55, 66, 77, 90, 103, 117, 132, 148, 166, 183, 201, 222, 242, 264, 287, 310, 334, 360, 387, 413, 442, 470, 500, 533, 564, 598, 631, 665, 701, 738, 775, 813, 853, 893, 936, 979, 1022, 1065, 1110, 1154, 1202, 1251, 1299, 1349, 1401

Offset: 3

N. J. A. Sloane, Apr 22 2005

The entries up to 66 have been checked, but the remaining entries are based on my hypothetical formula (see A101513). (It would be easy to check them.)

I have checked that these are correct entries from the triangle, but not that they match the hypothetical formula (from A101513). - Joshua Zucker, May 20 2006

Cf. A101513, A104656, A073425.

More terms from Joshua Zucker, May 20 2006

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.

cp's OEIS Frontend

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

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A083256 a(n) = A046523(n-th nonprime number) = A046523(A018252(n)).

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A083257 a(n) = A071364(n-th nonprime number) = A071364(A018252(n)).

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A141220 Write the n-th nonprime (A018252(n)) as a product of primes; increase one copy of the largest prime by 2 and decrease one copy of the smallest prime by 1, multiply the resulting numbers.

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A161570 Sum of all numbers from n up to A018252(n).

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A175251 Composites (A002808) with nonprime (A018252) subscripts.

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A203527 a(n) = Product_{1 <= i < j <= n} (A018252(i) + A018252(j)); A018252 = nonprime numbers.

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A245819 Permutation of natural numbers induced when A245703 is restricted to nonprime numbers: a(n) = 1+A091245(A245703(A018252(n))).

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A104655 Let c(i) = A018252(i) be the i-th nonprime; then the final entry in row n of A101513 is c(a(n)) (see A104656), for n >= 3.

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