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.

A060328 Primes which are the sum of three consecutive composite numbers.

Original entry on oeis.org

23, 31, 41, 59, 67, 71, 109, 113, 131, 139, 157, 199, 211, 239, 251, 269, 293, 311, 337, 379, 383, 409, 419, 487, 491, 499, 503, 521, 571, 599, 631, 701, 751, 769, 773, 787, 829, 877, 881, 919, 941, 953, 991, 1009, 1013, 1039, 1049, 1061, 1103, 1117, 1151
Offset: 1

Views

Author

Robert G. Wilson v, Mar 30 2001

Keywords

Comments

"Consecutive" necessarily means consecutive in the list of composite numbers as opposed to consecutive in the integers, as the sum of any 3 consecutive integers is a multiple of 3. - Peter Munn, Aug 20 2023

Examples

			a(3) = 41 is equal to 12+14+15.

Crossrefs

Primes that are the sum of other numbers of consecutive composite numbers: A060254 (2), A060329 (4), A060330 (5), A060331 (6), A060332 (7), A060333 (8). See also A037174.
Cf. A034962.
Complement within A166039\{5, 11} of A151741.

Programs

  • Mathematica
    composite[ n_Integer ] := (k = n + PrimePi[ n ] + 1; While[ k - PrimePi[ k ] - 1 != n, k++ ]; k); b = {}; Do[ p = composite[ n ] + composite[ n + 1 ] + composite[ n + 2 ]; If[ PrimeQ[ p ], b = Append[ b, p ] ], {n, 1, 1000} ]; b

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

A161667 Smallest of 5 consecutive composite numbers which sum up to prime.

Original entry on oeis.org

4, 8, 10, 14, 16, 28, 56, 58, 68, 70, 98, 106, 134, 146, 148, 178, 188, 190, 194, 196, 236, 308, 310, 344, 346, 428, 520, 566, 568, 614, 638, 640, 658, 808, 824, 854, 856, 1018, 1028, 1030, 1058, 1226, 1276, 1318, 1448, 1480, 1484, 1616, 1784, 1876, 1946, 2024
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jun 15 2009

Keywords

Comments

There are at most 5n/log n members of this sequence up to n, since at least one of a(n), a(n) + 1, ..., a(n) + 4 is prime (else their sum is divisible by 5).

Examples

			4+6+8+9+10=37,.. p=37,53,67,83,97,157,..(A060330)

Crossrefs

Cf. A060329, A060330, A161666

Programs

  • Mathematica
    CompositeNext[n_]:=Module[{k=n+1},While[PrimeQ[k],k++ ];k]; lst={};Do[p=n+CompositeNext[n]+CompositeNext[CompositeNext[n]]+CompositeNext[CompositeNext[CompositeNext[n]]]+CompositeNext[CompositeNext[CompositeNext[CompositeNext[n]]]];If[ !PrimeQ[n]&&PrimeQ[p],AppendTo[lst,n]],{n,2,5*6!}];lst
Transpose[Select[Partition[Select[Range[2100],CompositeQ],5,1],PrimeQ[ Total[ #]]&]][[1]] (* Harvey P. Dale, Aug 14 2014 *)

Extensions

Comment from Charles R Greathouse IV, Nov 11 2009
Showing 1-5 of 5 results.

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