A052369 -id:A052369 - cp's OEIS frontend

A161850 Subsequence of A161986 consisting of all terms that are prime.

7, 11, 13, 17, 19, 23, 29, 31, 37, 37, 41, 43, 47, 47, 53, 53, 59, 61, 67, 71, 71, 73, 79, 83, 89, 89, 97, 97, 101, 101, 103, 107, 109, 113, 127, 131, 137, 137, 139, 149, 149, 151, 157, 163, 163, 167, 167, 173, 179, 179, 181, 193, 191, 193, 197, 199, 211, 223, 227

Offset: 1

Juri-Stepan Gerasimov, Jun 20 2009

nonn

A161986(n) = k+r where k is n-th composite and r is remainder of (largest prime divisor of k) divided by (smallest prime divisor k).

			A161986(1) to A161986(27) are 4, 7, 8, 9, 11, 13, 15, 17, 16, 19, 21, 22, 23, 25, 25, 27, 27, 29, 31, 32, 35, 35, 37, 37, 39, 40, 41. Hence a(1) to a(11) are the prime terms among them, namely 7, 11, 13, 17, 19, 23, 29, 31 ,37, 37, 41.

Cf. A161986 (A002808(n)+A161849(n)), A002808 (composite numbers), A161849 (A052369(n) mod A056608(n)), A052369 (largest prime factor of n-th composite), A056608 (smallest divisor of n-th composite).

[ p: n in [2..230] | not IsPrime(n) and IsPrime(p) where p is n+D[ #D] mod D[1] where D is PrimeDivisors(n) ];

Edited and corrected (a(19)=57 replaced by 67; a(38)=137, a(49)=179, a(50)=179 inserted) by Klaus Brockhaus, Jun 24 2009

A085434 Twice odd isolated primes.

Original entry on oeis.org

46, 74, 94, 106, 134, 158, 166, 178, 194, 226, 254, 262, 314, 326, 334, 346, 422, 446, 466, 502, 514, 526, 554, 586, 614, 634, 662, 674, 706, 718, 734, 746, 758, 766, 778, 794, 802, 818, 878, 886, 898, 914, 934, 958, 974, 982, 998, 1006, 1018, 1082, 1094

Offset: 6

Cino Hilliard, Aug 13 2003

Name was: n-th even number not a power of 2 whose largest and smallest factors do not add or subtract to a twin prime. - Robert Israel, Mar 11 2025

The density of these numbers approach 0 as n approaches oo.

Cf. A052369, A056608, A134797.

P:= select(isprime, {seq(i,i=3..1000,2)}):
A:= P minus (P +~ 2) minus (P -~ 2):
sort(convert(A,list)) *~ 2; # Robert Israel, Mar 11 2025

maxpmmintwin(n) = { c=0; forprime(p=3,n, if(!isprime(p-2) & !isprime(p+2),print1(p+p","); c++); ); print(); print(c" "c/n+.0) }

a(n) = 2 * A134797(n).

Definition corrected by Robert Israel, Mar 11 2025

A141553 Transformed nonprime products of prime factors of the composites, the largest prime decremented by 2 and the smallest prime incremented by 1.

Original entry on oeis.org

0, 0, 4, 9, 6, 15, 12, 0, 9, 18, 20, 27, 12, 18, 33, 12, 30, 27, 0, 36, 45, 30, 18, 51, 44, 36, 45, 54, 36, 63, 24, 40, 45, 60, 66, 27, 54, 60, 68, 81, 54, 87, 60, 0, 66, 81, 90, 84, 75, 36, 105, 60, 102, 72, 99, 72, 36, 117, 90, 90, 123, 108, 108, 81, 88, 126, 116, 135, 102, 48

Offset: 1

Juri-Stepan Gerasimov, Aug 14 2008

nonn

In the prime number decomposition of k=A002808(i), i=1,2,3,.., one instance of the largest prime, pmax=A052369(i), is replaced by pmax-2 and one instance of the smallest prime, pmin=A056608(i), is replaced by pmin+1. If the product of this modified list of factors, k*(pmax-2)*(pmin+1)/(pmin*pmax), is nonprime, it is added to the sequence.

			k(1)=(p(max)=2)*(p(min)=2), transformed (2-2)*(2+1)=0*3=0=a(1).
k(2)=(p(max)=3)*(p(min)=2), transformed (3-2)*(2+1)=1*3=3 (prime, skipped).
k(3)=(p(max)=2)*(p=2)*(p(min)=2), transformed (2-2)*2*(2+1)=0*2*3=0=a(2), etc.

Cf. A141218, A141219, A141220, A141284.

Edited and corrected by R. J. Mathar, Aug 18 2008

A161003 A list of the composite numbers divided by their largest prime factors.

Original entry on oeis.org

2, 2, 4, 3, 2, 4, 2, 3, 8, 6, 4, 3, 2, 8, 5, 2, 9, 4, 6, 16, 3, 2, 5, 12, 2, 3, 8, 6, 4, 9, 2, 16, 7, 10, 3, 4, 18, 5, 8, 3, 2, 12, 2, 9, 32, 5, 6, 4, 3, 10, 24, 2, 15, 4, 7, 6, 16, 27, 2, 12, 5, 2, 3, 8, 18, 7, 4, 3, 2, 5, 32, 14, 9, 20, 6, 8, 15, 2, 36, 10, 3, 16, 6, 5, 4, 9, 2, 7, 24, 11, 2, 3, 4

Offset: 1

Trevor Cassiliano (casstjc(AT)gmail.com), Jun 01 2009

nonn

a(A120389(n)) = A000040(n). - Gionata Neri, May 07 2015

For n >= 2, a(x) = n where x = A066246(n*A006530(n)). - Robert Israel, May 07 2015

			n=1 4/2; n=2 6/3; n=3 8/2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A002808, A006530, A052369, A066246, A120389, A161004, A160180.

with(numtheory): a := proc (n) if isprime(n) = false then n/factorset(n)[nops(factorset(n))] else end if end proc: seq(a(n), n = 2 .. 130); # Emeric Deutsch, Jun 27 2009

With[{cmps=Select[Range[200],CompositeQ]},#/FactorInteger[#][[-1,1]]&/@ cmps] (* Harvey P. Dale, Mar 29 2017 *)

a(n) = A002808(n)/A052369(n). - Robert Israel, May 07 2015

Extended by Emeric Deutsch, Jun 27 2009

A161990 Composites which have the same largest prime factor as their index.

Original entry on oeis.org

10, 12, 14, 25, 36, 39, 42, 45, 77, 124, 132, 140, 147, 224, 234, 266, 345, 365, 370, 375, 380, 385, 390, 494, 621, 638, 660, 671, 682, 782, 899, 945, 1001, 1086, 1140, 1377, 1558, 1577, 1628, 1696, 1728, 1760, 1798, 1885, 2046, 2145, 2484, 2550, 2970, 3101, 3122, 3477

Offset: 1

Juri-Stepan Gerasimov, Jun 24 2009

nonn

If A052369(k) = A006530(k), we add the associated composite A002808(k) to the sequence.

			The 6th composite is 12=2^2*3 with largest prime factor 3, and the largest prime factor of the index 6=2*3 is also 3, which adds 12 to the sequence.
The 7th composite is 14=2*7 with largest prime factor 7, and the largest prime factor of the index 7 is also 7, which adds 14 to the sequence.

Robert Israel, Table of n, a(n) for n = 1..4262

Cf. A002808, A050696.

A006530 := proc(n) sort(convert(numtheory[factorset](n),list)); op(-1,%) ; end:
A002808 := proc(n) if n = 1 then 4; else for a from procname(n-1)+1 do if not isprime(a) then RETURN(a) ; fi; od: fi; end:
A052369 := proc(n) A006530(A002808(n)) ; end:
for n from 1 to 10000 do if A052369(n) = A006530(n) then printf("%d,",A002808(n)) ; fi; od: # R. J. Mathar, Aug 14 2009
# More efficient alternative:
N:= 10000: # to get terms <= N
Lpf:= [seq(max(numtheory:-factorset(n)),n=1..N)]:
comps:= select(n -> Lpf[n]Robert Israel, Mar 05 2018

lpf[n_] := FactorInteger[n ][[-1, 1]];
cc = Select[Range[10000], CompositeQ];
Select[{Range[Length[cc]], cc} // Transpose, lpf[#[[1]]] == lpf[#[[2]]]&][[All, 2]] (* Jean-François Alcover, Aug 19 2020 *)

Corrected and extended by R. J. Mathar, Aug 14 2009

A085428 Sum of the smallest and largest prime divisors of the n-th composite number.

Original entry on oeis.org

4, 5, 4, 6, 7, 5, 9, 8, 4, 5, 7, 10, 13, 5, 10, 15, 6, 9, 7, 4, 14, 19, 12, 5, 21, 16, 7, 9, 13, 8, 25, 5, 14, 7, 20, 15, 5, 16, 9, 22, 31, 7, 33, 10, 4, 18, 13, 19, 26, 9, 5, 39, 8, 21, 18, 15, 7, 6, 43, 9, 22, 45, 32, 13, 7, 20, 25, 34, 49, 24, 5, 9, 14, 7

Offset: 0

Cino Hilliard, Aug 13 2003

Equals A056608(n) + A052369(n).

maxpmindivc(n) = { for(x=4,n, s=0; forstep(p=2,x-1,1, if(x%p==0 & isprime(p),s=p; break); ); forstep(p=x-1,2,-1, if(x%p==0 & isprime(p),print1(s+p,","); break); ) ) }

A109709 Triangle in which n-th row gives the prime factors of the n-th composite number with repetition.

Original entry on oeis.org

2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 5, 2, 2, 3, 2, 7, 3, 5, 2, 2, 2, 2, 2, 3, 3, 2, 2, 5, 3, 7, 2, 11, 2, 2, 2, 3, 5, 5, 2, 13, 3, 3, 3, 2, 2, 7, 2, 3, 5, 2, 2, 2, 2, 2, 3, 11, 2, 17, 5, 7, 2, 2, 3, 3, 2, 19, 3, 13, 2, 2, 2, 5, 2, 3, 7, 2, 2, 11, 3, 3, 5, 2, 23, 2, 2, 2, 2, 3, 7, 7, 2, 5, 5, 3, 17, 2, 2, 13, 2, 3, 3, 3

Offset: 1

Lior Manor, Aug 08 2005

			Irregular triangle begins:
  2, 2;
  2, 3;
  2, 2, 2;
  3, 3;
  2, 5;
  ...

Column k=1 gives A056608.
Last terms in rows give A052369.
Row lengths give A062502(n+1).
Cf. A002808, A027746.

fn[{a_,b_}]:=Table[a,b];Flatten[Table[fn/@FactorInteger[ ResourceFunction["Composite"][n]],{n,37}]] (* James C. McMahon, Mar 29 2024 *)

A141462 Transformed nonprime products of prime factors of the composites, the largest prime decremented by 2, the smallest by 1.

Original entry on oeis.org

0, 1, 0, 6, 0, 6, 10, 9, 4, 12, 6, 10, 9, 0, 18, 15, 20, 6, 22, 12, 15, 18, 18, 21, 8, 30, 15, 30, 22, 9, 36, 20, 34, 27, 18, 30, 0, 44, 27, 30, 42, 25, 12, 35, 30, 34, 54, 33, 24, 18, 39, 30, 60, 54, 36, 27, 66, 42, 58, 45, 68, 16, 35, 54, 30, 45, 44, 50, 51, 18, 45, 70, 40, 51, 84

Offset: 1

Juri-Stepan Gerasimov, Aug 08 2008

nonn

In the prime factorization of k=A002808(i), i=1,2,3,..., one instance of the largest prime, pmax=A052369(i), is replaced by pmax-2 and one instance of the smallest prime, pmin=A056608(i), is replaced by pmin-1. If the product of this modified set of factors, k*(pmax-2)*(pmin-1)/(pmin*pmax), is nonprime, it is a term of the sequence.

			  composite k     transformed product
  -----------  -------------------------
   4 = 2*2     (2-1)*(2-2)   = 1*0   = 0  = a(1)
   6 = 2*3     (2-1)*(3-2)   = 1*1   = 1  = a(2)
   8 = 2*2*2   (2-1)*2*(2-2) = 1*2*0 = 0  = a(3)
   9 = 3*3     (3-1)*(3-2)   = 2*1   = 2  (prime)
  10 = 2*5     (2-1)*(5-2)   = 1*3   = 3  (prime)
  12 = 2*2*3   (2-1)*2*(3-2) = 1*2*1 = 2  (prime)
  14 = 2*7     (2-1)*(7-2)   = 1*5   = 5  (prime)
  15 = 3*5     (3-1)*(5-2)   = 2*3   = 6  = a(4)

Definition rephrased by R. J. Mathar, Aug 14 2008

Example section edited by Jon E. Schoenfield, Feb 20 2021

A141465 Prime transformed products of prime factors of the composites, the largest prime decremented by 2, the smallest by 1.

Original entry on oeis.org

2, 3, 2, 5, 3, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451, 1481

Offset: 1

Juri-Stepan Gerasimov, Aug 08 2008

nonn

In the prime factorization of k=A002808(i), i=1,2,3,..., one instance of the largest prime, pmax=A052369(i), is replaced by pmax-2 and one instance of the smallest prime, pmin=A056608(i), is replaced by pmin-1. If the product of this modified set of factors, k*(pmax-2)*(pmin-1)/(pmin*pmax), is prime, it is appended to the sequence.

			  composite k     transformed product
  -----------  -------------------------
   4 = 2*2     (2-1)*(2-2)   = 1*0   = 0  (nonprime)
   6 = 2*3     (2-1)*(3-2)   = 1*1   = 1  (nonprime)
   8 = 2*2*2   (2-1)*2*(2-2) = 1*2*0 = 0  (nonprime)
   9 = 3*3     (3-1)*(3-2)   = 2*1   = 2  = a(1)
  10 = 2*5     (2-1)*(5-2)   = 1*3   = 3  = a(2)
  12 = 2*2*3   (2-1)*2*(3-2) = 1*2*1 = 2  = a(3)
  14 = 2*7     (2-1)*(7-2)   = 1*5   = 5  = a(4)

Edited by Jon E. Schoenfield, Feb 20 2021

A141466 Nonprime transformed products of prime factors of the composites, the largest and smallest prime decremented by 1.

Original entry on oeis.org

1, 4, 4, 4, 6, 8, 4, 6, 8, 12, 10, 8, 16, 12, 12, 12, 12, 8, 20, 16, 24, 12, 18, 24, 16, 18, 20, 24, 22, 16, 36, 20, 32, 24, 18, 40, 24, 36, 28, 24, 30, 36, 16, 48, 30, 32, 44, 30, 24, 36, 40, 36, 60, 36, 32, 36, 40, 36, 64, 42, 56, 40, 36, 72, 44, 60, 46, 72, 32, 42, 60, 40, 48, 48, 60, 52

Offset: 1

Juri-Stepan Gerasimov, Aug 08 2008

nonn

In the prime factorization of k=A002808(i), i=1,2,3,..., one instance of the largest prime, pmax=A052369(i), is replaced by pmax-1 and one instance of the smallest prime, pmin=A056608(i), is replaced by pmin-1. If the product of this modified set of factors, k*(pmax-1)*(pmin-1)/(pmin*pmax), is nonprime, it is appended to the sequence.

			  composite k     transformed product
  -----------  -------------------------
   4 = 2*2     (2-1)*(2-1)   = 1*1   = 1  = a(1)
   6 = 2*3     (2-1)*(3-1)   = 1*2   = 2  (prime)
   8 = 2*2*2   (2-1)*2*(2-1) = 1*2*1 = 2  (prime)
   9 = 3*3     (3-1)*(3-1)   = 2*2   = 4  = a(2)
  10 = 2*5     (2-1)*(5-1)   = 1*4   = 4  = a(3)
  12 = 2*2*3   (2-1)*2*(3-1) = 1*2*2 = 4  = a(4)
  14 = 2*7     (2-1)*(7-1)   = 1*6   = 6  = a(5)

Definition rephrased by R. J. Mathar, Aug 14 2008

Edited by Jon E. Schoenfield, Feb 20 2021

cp's OEIS Frontend

A161850 Subsequence of A161986 consisting of all terms that are prime.

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A085434 Twice odd isolated primes.

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A141553 Transformed nonprime products of prime factors of the composites, the largest prime decremented by 2 and the smallest prime incremented by 1.

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A161003 A list of the composite numbers divided by their largest prime factors.

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A161990 Composites which have the same largest prime factor as their index.

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A085428 Sum of the smallest and largest prime divisors of the n-th composite number.

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PARI

A109709 Triangle in which n-th row gives the prime factors of the n-th composite number with repetition.

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Mathematica

A141462 Transformed nonprime products of prime factors of the composites, the largest prime decremented by 2, the smallest by 1.

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A141465 Prime transformed products of prime factors of the composites, the largest prime decremented by 2, the smallest by 1.

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