A113709 -id:A113709 - cp's OEIS frontend

A113710 a(n) = A113709(n)/(prime(n+1) - prime(n)).

2, 3, 2, 6, 4, 9, 5, 4, 15, 6, 10, 21, 11, 8, 9, 30, 11, 17, 36, 13, 20, 14, 12, 25, 51, 26, 54, 28, 9, 32, 22, 69, 14, 75, 26, 27, 41, 28, 29, 90, 19, 96, 49, 99, 17, 18, 56, 114, 58, 39, 120, 25, 42, 43, 44, 135, 46, 70, 141, 29, 21, 77, 156, 79, 23, 56, 34, 174, 88, 59, 45, 62

Offset: 2

Leroy Quet, Nov 06 2005

nonn

Records are in A040040. - Andres Cicuttin, Nov 26 2016

			Between the primes 67 and 71 is the composite 68 and 68 is divisible by (71-67)=4. So 68/(71-67)= 17 is in the sequence.

Cf. A113709.

a(n) = floor(p(n+1)/(p(n+1)-p(n))) = ceiling(p(n)/(p(n+1)-p(n))), where p(n) is the n-th prime. - Leroy Quet, Feb 03 2007

a(n) = A113709(n)/A001223(n). - Omar E. Pol, Nov 26 2016

More terms from R. J. Mathar, Aug 31 2007

A133555 Order of A113709(n) among composite positive integers.

Original entry on oeis.org

1, 2, 3, 6, 9, 10, 11, 14, 19, 24, 27, 28, 29, 32, 37, 42, 47, 48, 51, 56, 57, 60, 71, 74, 75, 76, 79, 82, 95, 96, 99, 104, 105, 114, 119, 124, 125, 128, 133, 138, 147, 148, 151, 152, 157, 168, 175, 178, 181, 182, 187, 196, 197, 202, 207, 212, 217, 220, 221, 228, 231

Offset: 2

Leroy Quet, Dec 25 2007

nonn

			The 10th prime - the 9th prime = 29-23 = 6. The integer between 23 and 29 that is divisible by 6 is 24. 24 is the 14th composite, so a(9) = 14.

Cf. A066246, A113709.

A113709 := proc(n) local d,a ; d := ithprime(n+1)-ithprime(n) ; for a from ithprime(n)+1 do if a mod d = 0 then RETURN(a) ; fi ; od: end: A066246 := proc(n) local a,i; if n = 1 or isprime(n) then 0 ; else a := 0 ; for i from 4 to n do if not isprime(i) then a := a+1 ; fi ; od: RETURN(a) ; fi ; end: A133555 := proc(n) A066246(A113709(n)) ; end: seq(A133555(n),n=2..80) ; # R. J. Mathar, Jan 12 2008

compositePi[n_] := n - PrimePi[n] - 1;
a[n_] := Module[{p1 = Prime[n], p2 = Prime[n+1], c}, c = SelectFirst[ Range[p1+1, p2-1], Divisible[#, p2-p1]&]; compositePi[c]];
Table[a[n], {n, 2, 62}] (* Jean-François Alcover, Apr 02 2024 *)

a(n) = A066246(A113709(n)). - R. J. Mathar, Jan 12 2008

More terms from R. J. Mathar, Jan 12 2008

A111379 Composite numbers n which are divisible by (nextprime(n) - prevprime(n)), but have fewer divisors than some number between those two primes.

Original entry on oeis.org

68, 126, 140, 162, 164, 174, 204, 258, 290, 294, 316, 322, 392, 410, 444, 456, 488, 496, 516, 550, 558, 624, 654, 676, 678, 688, 704, 710, 732, 772, 784, 790, 804, 820, 824, 830, 856, 868, 908, 920, 942, 948, 966, 978, 984, 1030, 1038, 1060, 1068, 1098

Offset: 1

Leroy Quet, Nov 07 2005

nonn

			68 is there because it is divisible by (71-67), but 70 has more divisors.

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

Cf. A076082, A076083, A113709.

q:= 2: count:= 0: R:= NULL:
while count < 100 do
  p:= q; q:= nextprime(p);
  v:= q-p;
  m:= max({seq(numtheory:-tau(i),i=p+1 .. q-1)});
  S:= select(t -> numtheory:-tau(t) < m, [seq(i*v,i=ceil((p+1)/v) .. floor((q-1)/v))]);
  count:= count + nops(S);
  R:= R, op(S)
od:
R; # Robert Israel, Jun 03 2024

Edited by Don Reble, Nov 07 2005

A113728 a(n) is the integer between p(n) and p(n+2) which is divisible by (p(n+2)-p(n)), where p(n) is the n-th prime.

Original entry on oeis.org

3, 4, 6, 12, 12, 18, 18, 20, 24, 32, 40, 42, 42, 50, 48, 56, 64, 70, 72, 72, 80, 80, 84, 96, 102, 102, 108, 108, 126, 126, 130, 136, 144, 144, 152, 156, 160, 170, 168, 176, 180, 192, 192, 198, 210, 216, 224, 228, 228, 230, 240, 240, 256, 252, 264, 264, 272, 280

Offset: 1

Leroy Quet, Nov 08 2005

nonn

Exactly one integer exists between each p(n+2) and p(n) which is divisible by (p(n+2)-p(n)).

			Between the primes 19 and 29 is the composite 20 and 20 is divisible by (29-19)=10. So 20 is in the sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A113709, A113729.

For[n = 1, n < 50, n++, s := Prime[n] + 1; While[Floor[s/(Prime[n + 2] -Prime[n])] != s/(Prime[n + 2] - Prime[n]), s++ ]; Print[s]] (* Stefan Steinerberger, Feb 10 2006 *)
idp[n_]:=Module[{p1=Prime[n],p2=Prime[n+2]},Select[Range[p1+1,p2-1],Divisible[ #,p2-p1]&]]; Table[idp[n],{n,60}]//Flatten (* Harvey P. Dale, May 30 2021 *)

a(n) = A031131(n)*ceiling(A000040(n)/A031131(n)). - R. J. Mathar, Aug 31 2007

More terms from Stefan Steinerberger, Feb 10 2006

More terms from R. J. Mathar, Aug 31 2007

A113729 a(n) is the integer between p(n) and p(n+3) which is divisible by (p(n+3)-p(n)), where p(n) is the n-th prime.

Original entry on oeis.org

5, 8, 8, 10, 16, 20, 24, 24, 28, 36, 36, 40, 48, 48, 56, 56, 60, 72, 72, 72, 80, 90, 90, 98, 100, 104, 110, 120, 110, 120, 132, 144, 140, 144, 154, 160, 160, 176, 168, 180, 182, 192, 192, 198, 208, 224, 216, 230, 228, 240, 234, 252, 242, 252, 266, 266, 276, 276

Offset: 1

Leroy Quet, Nov 08 2005

nonn

Exactly one integer exists between each p(n+3) and p(n) which is divisible by (p(n+3)-p(n)).

			Between the primes 19 and 31 is the composite 24 and 24 is divisible by (31-19)=12. So 24 is in the sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A113709, A113728.

f[n_] := Block[{p = Prime[n], q = Prime[n + 3]}, q - Mod[p, q - p]]; Table[ f[n], {n, 58}] (* Robert G. Wilson v *)
id[{a_,b_,c_,d_}]:=Select[Range[a+1,d-1],Divisible[#,d-a]&]; Flatten[ id/@ Partition[Prime[Range[70]],4,1]] (* Harvey P. Dale, May 07 2015 *)

a(n)=p(n+3) - (p(n) (mod p(n+3)-p(n))).

More terms from Robert G. Wilson v, Nov 09 2005

A118335 a(n)= smallest multiple of (prime(n+1)-prime(n)) which is >= prime(n+1), where prime(m) is the m-th prime.

Original entry on oeis.org

3, 6, 8, 12, 14, 20, 20, 24, 30, 32, 42, 44, 44, 48, 54, 60, 62, 72, 72, 74, 84, 84, 90, 104, 104, 104, 108, 110, 116, 140, 132, 138, 140, 150, 152, 162, 168, 168, 174, 180, 182, 200, 194, 200, 200, 216, 228, 228, 230, 236, 240, 242, 260, 258, 264, 270, 272, 282

Offset: 1

Leroy Quet, Apr 25 2006

nonn

a(n) < a(n-1) for n: 31,43,54,83,98,100,116,122,140,142,155,163,169,173,178,..., . - Robert G. Wilson v, Apr 27 2006

Cf. A113709, A118336.

f[n_] := Block[{m = Prime[n + 1] - Prime[n]}, k = Mod[m, Prime[n + 1]]; While[k < Prime[n + 1], k += m]; k]; Array[f, 58] (* Robert G. Wilson v, Apr 27 2006 *)

a(n) - A118336(n) = 2 *(prime(n+1) - prime(n)).

a(n) - A113709(n) = prime(n+1) - prime(n), for n>= 2.

More terms from Robert G. Wilson v, Apr 27 2006

A118336 a(n) = greatest multiple of (p(n+1) - p(n)) which is < p(n), where p(m) is the m-th prime.

Original entry on oeis.org

1, 2, 4, 4, 10, 12, 16, 16, 18, 28, 30, 36, 40, 40, 42, 48, 58, 60, 64, 70, 72, 76, 78, 88, 96, 100, 100, 106, 108, 112, 124, 126, 136, 130, 148, 150, 156, 160, 162, 168, 178, 180, 190, 192, 196, 192, 204, 220, 226, 228, 228, 238, 240, 246, 252, 258, 268, 270, 276

Offset: 1

Leroy Quet, Apr 25 2006

nonn

A118335(n) - a(n) = 2*(p(n+1) - p(n)). A113709(n) - a(n) = p(n+1) - p(n), for n >= 2.

Cf. A113709, A118335.

a:=n->(ithprime(n+1)-ithprime(n))*floor(ithprime(n)/(ithprime(n+1)-ithprime(n))): 1,seq(a(n),n=2..64); # Emeric Deutsch, Apr 27 2006

Join[{1},Floor[First[#]/(Last[#]-First[#])](Last[#]-First[#])&/@Partition[Prime[ Range[ 2,60]],2,1]] (* Harvey P. Dale, Feb 25 2013 *)

a(1)=1; a(n) = (p(n+1) - p(n))*floor(p(n)/(p(n+1)-p(n))) for n >= 2. - Emeric Deutsch, Apr 27 2006

More terms from Emeric Deutsch, Apr 27 2006

A140785 a(n) = the single integer k, where p(n) <= k <= p(n+1), that is divisible by (p(n+1)-p(n)+1), where p(n) is the n-th prime.

Original entry on oeis.org

2, 3, 6, 10, 12, 15, 18, 20, 28, 30, 35, 40, 42, 45, 49, 56, 60, 63, 70, 72, 77, 80, 84, 90, 100, 102, 105, 108, 110, 120, 130, 133, 138, 143, 150, 154, 161, 165, 168, 175, 180, 187, 192, 195, 198, 208, 221, 225, 228, 230, 238, 240, 242, 252, 259, 266, 270, 273

Offset: 1

Leroy Quet, Jul 13 2008

nonn

Diana Mecum, Table of n, a(n) for n = 1..1000

Cf. A076368, A113709.

(#[[2]]-#[[1]]+1)Floor[#[[2]]/(#[[2]]-#[[1]]+1)]&/@Partition[ Prime[ Range[ 60]],2,1] (* Harvey P. Dale, Apr 07 2018 *)

a(n) = (p(n+1)-p(n)+1) * floor(p(n+1)/(p(n+1)-p(n)+1)), where p(n) is the n-th prime.

More terms from Diana L. Mecum, Jul 21 2008

A191612 Image of A008578 (the noncomposite numbers) under the "forming" transformation.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 16, 18, 20, 24, 30, 36, 40, 42, 44, 48, 54, 60, 66, 68, 72, 78, 80, 84, 96, 100, 102, 104, 108, 112, 126, 128, 132, 138, 140, 150, 156, 162, 164, 168, 174, 180, 190, 192, 196, 198, 204, 216, 224, 228

Offset: 1

Jaroslav Krizek, Oct 16 2011

We define a transformation T_f [b(n)] = [c(n)] - the index f means "forming" - of an increasing sequence b(n) of integers b(1), b(2), b(3), ..., b(k) which produces an increasing sequence c(n) of the same length, c(1), c(2), c(3), ..., c(k) such that c(1) = b(1), and for j>1, c(j) is the only integer b(j-1) < c(j) <= b(j), with (b(j)-b(j-1)) | c(j). We say b(n) is forming c(n).
An increasing sequence c(n) is called formed from the increasing sequence b(n) by T_f [b(n)] when there is an increasing sequence b(n) such that b(1) = c(1), for j > 1, b(j) is an integer c(j) <= b(j) < c(j+1) such that difference b(j) - b(j-1) divides c(j).
This transformation T_invf [c(n)] is an inverse of T_f [b(n)], but this inversion of c(n) back to b(n) may not be unique, and there are also increasing sequences c(n) which do not have an image T_invf [c(n)]. We call the latter sequences c(n) "unformed."
Each increasing sequences b(n) can be transform by transformation T_f [b(n)] but this does not apply to transformation T_invf [b(n)]. An increasing sequence c(n) is called totally formed if c(n) = T_f [c(n)] = T_invf [c(n)]. Each totally formed sequence is formed.
There are infinitely many formed, totally formed and unformed increasing sequences.
Examples of totally formed sequences: A047229, A004277, A002808, A000079, A000027.
Examples of formed, but not totally formed, sequences: A000225, A000295, A018252.
Examples of unformed sequences: A000040, A008578, A005117, A005408.

			a(10) = 20 because 20 is the only integer such that 19 = A008578(9) < 20 <= A008578(10) = 23 and simultaneously is multiple of difference A008578(10) - A008578(9) = 4.

Tf := proc(L)
        local a,j,c ;
        a := [op(1,L)] ;
        while nops(a) < nops(L)-1 do
                j := nops(a)+1 ;
                for c from op(j-1,L)+1 to op(j,L) do
                        if (c mod ( op(j,L)-op(j-1,L) )) = 0 then
                                a := [op(a),c] ;
                                break;
                        end if;
                end do:
        end do:
        a ;
end proc:
nonc := [seq(A008578(n),n=1..80)] ;
Tf(nonc) ; # R. J. Mathar, Oct 27 2011

For n > 3, a(n) = A113709(n-2).

cp's OEIS Frontend

A113710 a(n) = A113709(n)/(prime(n+1) - prime(n)).

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A133555 Order of A113709(n) among composite positive integers.

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A111379 Composite numbers n which are divisible by (nextprime(n) - prevprime(n)), but have fewer divisors than some number between those two primes.

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A113728 a(n) is the integer between p(n) and p(n+2) which is divisible by (p(n+2)-p(n)), where p(n) is the n-th prime.

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A113729 a(n) is the integer between p(n) and p(n+3) which is divisible by (p(n+3)-p(n)), where p(n) is the n-th prime.

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A118335 a(n)= smallest multiple of (prime(n+1)-prime(n)) which is >= prime(n+1), where prime(m) is the m-th prime.

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A118336 a(n) = greatest multiple of (p(n+1) - p(n)) which is < p(n), where p(m) is the m-th prime.

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A140785 a(n) = the single integer k, where p(n) <= k <= p(n+1), that is divisible by (p(n+1)-p(n)+1), where p(n) is the n-th prime.

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