cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A037174 Primes which are not the sum of consecutive composite numbers.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 47, 61, 73, 107, 167, 179, 313, 347, 421, 479, 719, 863, 1153, 1213, 1283, 1307, 1523, 3467, 3733, 4007, 4621, 4787, 5087, 5113, 5413, 7523, 7703, 9817, 10333, 12347, 12539, 13381, 17027, 18553, 19717, 19813, 23399, 26003, 31873, 36097, 38833
Offset: 1

Views

Author

Naohiro Nomoto

Keywords

Comments

It seems reasonable that a(n)/A079149(n) has an asymptote that could be estimated. - Peter Munn, Aug 21 2023

Links

Crossrefs

Subsequence of A079149.
With {1}, the complement of A133576.
Primes that are the sum of specific numbers of consecutive composite numbers: A060254 (2), A060328 (3), A060329 (4), A060330 (5), A060331 (6), A060332 (7), A060333 (8).
Cf. A050940, A197227.

Programs

  • Maple
    N:= 5000:
primes,comps:= selectremove(isprime,{$2..N}):
M:= nops(comps):
X:= primes:
for n from 1 to floor(sqrt(2*N)) do
i:= 1;
T:= add(comps[k],k=1..n);
while T <= N do
X := X minus {T};
if i + n > M then break fi;
T := T + comps[i+n] - comps[i];
i := i+1;
od;
od:
X;
# Robert Israel, Jun 24 2008

Extensions

More terms from Jud McCranie, Jul 12 2000
Corrected by T. D. Noe, Aug 15 2008

A060333 Primes which are the sum of eight consecutive composite numbers.

Original entry on oeis.org

193, 277, 353, 433, 443, 613, 643, 653, 673, 683, 739, 881, 1109, 1129, 1237, 1511, 1531, 1609, 1619, 1697, 1873, 1999, 2017, 2027, 2113, 2207, 2239, 2281, 2371, 2447, 2621, 2657, 2677, 2687, 2749, 2801, 2833, 2843, 2909, 2927, 3023, 3083, 3121, 3167
Offset: 1

Views

Author

Robert G. Wilson v, Mar 30 2001

Keywords

Links

Crossrefs

Cf. A060254, A060328, A060329, A060330, A060331, A060332, A151738, A151739.

Programs

  • Maple
    comps:= remove(isprime, [$4..1000]):
S:= add(comps[i+1..i-8],i=0..7):
select(isprime,S); # Robert Israel, Dec 12 2019
  • Mathematica
    composite[ n_Integer ] := (k = n + PrimePi[ n ] + 1; While[ k - PrimePi[ k ] - 1 != n, k++ ]; k); a = {}; Do[ p = Sum[ composite[ n + k ], {k, 0, 7} ]; If[ PrimeQ[ p ], a = Append[ a, p ] ], {n, 1, 600} ]; a
Select[Total /@ Partition[ Select[ Range@ 500, CompositeQ], 8, 1], PrimeQ] (* Giovanni Resta, Dec 13 2019 *)

A151744 Primes which are the sum of two, three, four and five consecutive composite numbers.

Original entry on oeis.org

17783, 25057, 47303, 48383, 49297, 76343, 89783, 205703, 412343, 516457, 704183, 754417, 790703, 938183, 1105343, 1110743, 1279583, 1563503, 1632817, 1744583, 1890743, 1903103, 2062943, 2276303, 2714617, 2802383, 2812897, 2932703
Offset: 1

Views

Author

Claudio Meller, Jun 15 2009

Keywords

Comments

17783 is in the list because: 17783 = 8891 + 8892 (sum of two consecutive composite numbers)
17783 = 5926 + 5928 + 5929 (sum of three consecutive composite numbers)
17783 = 4444 + 4445 + 4446 + 4448 (sum of four consecutive composite numbers)
17783 = 3554 + 3555 + 3556 + 3558 + 3560 (sum of five consecutive composite numbers)

Links

Programs

  • Mathematica
    CompositeNext[n_]:=Module[{k=n+1},While[PrimeQ[k],k++ ];k]; q=9!; lst2={};Do[If[ !PrimeQ[n],c=CompositeNext[n];a2=n+c;If[PrimeQ[a2],AppendTo[lst2,a2]]],{n,q}];lst2; lst3={};Do[If[ !PrimeQ[n],c1=CompositeNext[n];c2=CompositeNext[c1];a3=n+c1+c2;If[PrimeQ[a3],AppendTo[lst3,a3]]],{n,q}];lst3; lst4={};Do[If[ !PrimeQ[n],c1=CompositeNext[n];c2=CompositeNext[c1];c3=CompositeNext[c2];a4=n+c1+c2+c3;If[PrimeQ[a4],AppendTo[lst4,a4]]],{n,q}];lst4; lst5={};Do[If[ !PrimeQ[n],c1=CompositeNext[n];c2=CompositeNext[c1];c3=CompositeNext[c2];c4=CompositeNext[c3];a5=n+c1+c2+c3+c4;If[PrimeQ[a5],AppendTo[lst5,a5]]],{n,q}];lst5; Intersection[lst2,lst3,lst4,lst5] (* Vladimir Joseph Stephan Orlovsky, Jun 17 2009 *)
Module[{comps=Select[Range[2*10^6],CompositeQ]},Intersection@@ Table[ Select[ Total/@ Partition[comps,n,1],PrimeQ],{n,2,5}]] (* Harvey P. Dale, Apr 16 2015 *)

Formula

Intersection of A060254, A060328, A060329 and A060330. - R. J. Mathar, Jun 17 2009

Extensions

Extended beyond a(7) by Klaus Brockhaus, Jun 16 2009

A128245 Smallest of three consecutive composite numbers whose sum is prime.

Original entry on oeis.org

6, 9, 12, 18, 21, 22, 35, 36, 42, 45, 51, 65, 69, 78, 82, 88, 96, 102, 111, 125, 126, 135, 138, 161, 162, 165, 166, 172, 189, 198, 209, 232, 249, 255, 256, 261, 275, 291, 292, 305, 312, 316, 329, 335, 336, 345, 348, 352, 366, 371, 382, 396, 399, 408, 429, 432
Offset: 1

Views

Author

Zak Seidov, May 03 2007

Keywords

Comments

If n is a member of this sequence, either n+1 or n+2 is prime. This suggests that the density of the sequence is roughly kn/log^2 n for some k. Counts up to 10^9 suggest k is about 5.26. - Charles R Greathouse IV, Sep 11 2009

Examples

			6 + 8 + 9 = 23 = A060328(1);
9 + 10 + 12 = 31 = A060328(2);
12 + 14 + 15 = 41 = A060328(3);
18 + 20 + 21 = 59 = A060328(4).

Links

Crossrefs

Cf. A060328.

Programs

  • Mathematica
    CompositeNext[n_]:=Module[{k=n+1},While[PrimeQ[k],k++ ];k]; lst={};Do[p=n+CompositeNext[n]+CompositeNext[CompositeNext[n]];If[ !PrimeQ[n]&&PrimeQ[p],AppendTo[lst,n]],{n,2,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 15 2009 *)
Select[Partition[Select[Range[500],CompositeQ],3,1],PrimeQ[Total[#]]&][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 24 2019 *)
  • PARI
    test(n)={my(b=a+1,c);b+=isprime(b);c=b+1;c+=isprime(c);isprime(a+b+c)};for(n=4,1e3,if(!isprime(n)&&test(n),print1(n","))) \\ Charles R Greathouse IV, Sep 11 2009

Formula

By Rosser's theorem, a(2n) > n log n. - Charles R Greathouse IV, Sep 11 2009

A133659 Primes that are the sum of three consecutive primes as well as the sum of three consecutive composite numbers.

Original entry on oeis.org

23, 31, 41, 59, 71, 109, 131, 199, 211, 251, 269, 311, 487, 503, 701, 829, 941, 1049, 1061, 1151, 1229, 1381, 1511, 1931, 2129, 2179, 2251, 2269, 2393, 2579, 2971, 3041, 3271, 3329, 3581, 3851, 3889, 3911, 4289, 4451, 4481, 4679, 4987, 4999
Offset: 1

Views

Author

Randy L. Ekl, Dec 28 2007

Keywords

Examples

			a(3) = 41 because 41 = 11+13+17 and 41 = 12+14+15.

Links

Crossrefs

Cf. A034962, A060328.

Programs

  • Mathematica
    a = {}; For[n = 2, n < 10000, n++, If[ ! PrimeQ[n], AppendTo[a, n + Select[Range[n + 1, n + 10], ! PrimeQ[ # ] &][[1]] + Select[Range[n + 1, n + 10], ! PrimeQ[ # ] &][[2]]]]]; b = Table[Prime[i] + Prime[i + 1] + Prime[i + 2], {i, 1, 10000}]; Select[Intersection[a, b], PrimeQ[ # ] &] (* Stefan Steinerberger, Dec 30 2007 *)

Formula

Equals A034962 INTERSECT A060328. - R. J. Mathar, Jan 11 2008

Extensions

More terms from Stefan Steinerberger, Dec 30 2007
Showing 1-5 of 5 results.

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