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.

Previous Showing 11-19 of 19 results.

A179825 Multiples of 6 which are not the sum of a pair of twin primes.

Original entry on oeis.org

96, 402, 516, 786, 906, 1116, 1146, 1266, 1356, 3246, 4206
Offset: 1

Views

Author

M. B. Jones, Jul 28 2010

Keywords

Comments

Subset of A007534.

Crossrefs

Cf. A007534

A318202 Prime numbers p such that p-3 cannot be written as sum of two twin prime numbers (not necessarily forming a pair of twin primes).

Original entry on oeis.org

2, 3, 5, 7, 97, 101, 521, 787, 907, 911, 1117, 1151, 1361, 3251, 4211
Offset: 1

Views

Author

Dimitris Valianatos, Aug 21 2018

Keywords

Comments

Conjecture 1: All prime numbers >= 11 can be written as sum of 3 twin prime numbers.
Conjecture 2: All prime numbers aside from the 15 terms given here can be written as a sum of three twin prime numbers with (at least) one of them equal to 3.
If the sequence is finite then there are infinitely many twin primes.
For the terms in this sequence, the lexicographically least partitions into three twin primes are: 97=5+19+73, 101=11+17+73, 521=11+197+313, 787=5+139+643, 907=5+19+883, 911=11+17+883, 1117=5+19+1093, 1151=11+107+1033, 1361=11+29+1321, 3251=11+71+3169, 4211=11+41+4159.
a(16) > 10^6 if it exists. - Amiram Eldar, Dec 06 2018

Examples

			a(6) = 101 because 101 - 3 = 98 and (98 - 73 = 25, 98 - 71 = 27), (98 - 61 = 37, 98 - 59 = 39), ..., (98 - 5 = 93, 98 - 3 = 95) aren't twin primes.

Links

Crossrefs

Cf. A000040, A001097, A007534, A129363, A321221.

Programs

  • Mathematica
    p = Prime[Range[600]]; p2 = Select[p, PrimeQ[# - 2] || PrimeQ[# + 2] &]; Select[ p - 3, IntegerPartitions[#, {2}, p2] == {} &] + 3 (* Amiram Eldar, Nov 15 2018 *)
  • PARI
    {forprime(n=2,10^4,p=n-3;forprime(t1=2,n,forprime(t2=t1,n,t12=t1+t2; if((isprime(t1-2)||isprime(t1+2))&&(isprime(t2-2)||isprime(t2+2)), if(t12==p,break(2)))));if(t12==2*n,print1(n", ")))}
  • PARI
    isok(p) = {if (isprime(p), p -= 3; forprime(q = 2, p, if (isprime(r=p-q), if ((isprime(r+2) || isprime(r-2)) && (isprime(q-2) || isprime(q+2)), return (0)););); return (1));} \\ Michel Marcus, Dec 05 2018
  • PARI
    \\ See Corneth link \\ David A. Corneth, Dec 05 2018

Extensions

2,3,5,7 prepended by David A. Corneth, Dec 05 2018

A057702 Even numbers not the sum of a pair of twin lucky numbers.

Original entry on oeis.org

92, 94, 96, 110, 112, 114, 152, 154, 156, 188, 190, 192, 212, 214, 216, 230, 232, 234, 278, 280, 282, 344, 346, 348, 374, 376, 378, 380, 382, 384, 572, 574, 576, 698, 700, 702, 764, 766, 768, 824, 826, 828, 830, 832, 834, 842, 844, 846, 1742, 1744, 1746, 2168, 2170, 2172, 2546, 2548, 2550, 5948, 5950, 5952, 7034, 7036, 7038
Offset: 0

Views

Author

Naohiro Nomoto, Oct 23 2000

Keywords

Comments

Conjectured to be complete.

Crossrefs

Cf. A000959, A007534.

A133802 Even numbers which are not the sum of a pair of cousin primes.

Original entry on oeis.org

2, 4, 8, 12, 68, 72, 76, 278, 282, 286, 638, 642, 644, 648, 652, 908, 912, 916, 1058, 1062, 1066, 1328, 1332, 1336, 1418, 1422, 2114, 2118, 2122, 2294, 2298, 2302, 2528, 2532, 2536, 2948, 2952
Offset: 1

Views

Author

Donovan Johnson, Jan 06 2008

Keywords

Comments

No other n < 10^9. Conjectured to be complete.

Examples

			The cousin primes < 100 are 3, 7, 11, 13, 17, 19, 23, 37, 41, 43, 47, 67, 71, 79, 83, 97. 76 is in the sequence because no combination of any two numbers from the set just enumerated can be summed to make 76.

References

  • D. Wells, Prime Numbers: The Most Mysterious Figures in Math. John Wiley & Sons, Inc., Hoboken, New Jersey, page 33.

Links

Crossrefs

Cf. A023200, A046132, A007534.

Programs

  • Mathematica
    Lim=10^4;cp1=Select[Range[Lim], PrimeQ[#] && PrimeQ[# + 4] &] ;cp=Union[Join[cp1,cp1+4]];sms=Total/@Tuples[cp,2];en=Range[2,Lim,2];Complement[en,sms] (* James C. McMahon, Mar 30 2025 *)

A179892 Numbers which are not the sum of three distinct members of twin primes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120
Offset: 1

Views

Author

Zak Seidov, Jul 30 2010

Keywords

Comments

All positive even numbers are terms of the sequence:
members of twin primes are odd and the sum of three odd primes is odd.
No odd numbers >=19 are in the sequence (conjecture?).

Examples

			15=3+5+7, 19=3+5+11 hence 15 and 19 are not in the sequence.

Crossrefs

Cf. A007534 Even numbers which are not the sum of a pair of twin primes.

Programs

  • Mathematica
    pr=Prime[Range[100]];se=Select[pr,PrimeQ[ #-2]||PrimeQ[ #+2]&];
se2=Select[Union[Total/@Subsets[se,{3}]],#<200&];Complement[Range[200],se2]

A231903 Even numbers which are the sum of two different primes > 3 in exactly one way.

Original entry on oeis.org

12, 16, 20, 22, 26, 32, 38, 62
Offset: 1

Views

Author

Pierre CAMI, Nov 15 2013

Keywords

Comments

Conjecture: there are no more numbers > 62 in the sequence.
Remark that the two primes are one of a twin primes pair and a pair of twin primes for 12=5+7.
The pairs of primes are : 12=5+7, 16=5+11, 20=7+13, 22=5+17, 26=7+19, 32=13+19, 38=7+31, 62=19+43.

Examples

			10=3+7=5+5 not in the sequence (not >3 or not different),
12=5+7, 12 first in the sequence,
14=3+11=7+7 not in the sequence.

Crossrefs

Cf. A007534

Programs

  • PARI
    is_A231903(n)={my(s=0);forprime(p=5,n\2-1,isprime(n-p)&&s++>1&&return);s} \\ - M. F. Hasler, Nov 22 2013

Extensions

Missing term a(7)=38 added by M. F. Hasler, Nov 22 2013

A273995 Even numbers with a unique representation as the difference of two primes, each of which is a member of a pair of twin primes, and one of which is smaller than the even number under consideration.

Original entry on oeis.org

4, 6, 20, 34, 46, 50, 74, 82, 86, 202, 206, 214, 218, 244, 248, 256, 260, 352, 356, 382, 386, 454, 472, 476, 524, 562, 604, 608, 664, 668, 724, 728, 772, 776, 982, 986, 1162, 1166, 1192, 1196, 1552, 1556, 1672, 1676, 2872, 2876, 3082, 3086, 6232, 6236, 6892, 6896
Offset: 1

Views

Author

Thomas Curtright, Jun 06 2016

Keywords

Comments

For the sequence to be infinite there must be an infinite number of twin prime pairs.
Can any even number n > 2 be so written (perhaps not uniquely) as the difference of two (unrelated) twins, one of which is smaller than n? (T. S. Van Kortryk conjectures there are, if any, only a finite number of even integers such that this is not true.)

Examples

			For even n with 4 <= n <= 100, all have at least one representation as the difference of two primes, each of which is a member of a pair of twin primes, but the following have only one such representation, and so belong to the sequence:
     4 =   7 -  3
     6 =  11 -  5
    20 =  31 - 11
    34 =  41 -  7
    46 =  59 - 13
    50 =  61 - 11
    74 = 103 - 29
    82 = 101 - 19
    86 = 103 - 17

Crossrefs

Cf. A007534.

Programs

  • PARI
    istwin(p) = isprime(p+2) || isprime(p-2);
isok(n) = {my(nb = 0); forprime(p=3, n, if (isprime(n+p) && istwin(p) && istwin(n+p), nb++);); if (nb == 1, return (1));}
lista(nn) = forstep(n=4, nn, 2, if (isok(n), print1(n, ", "))); \\ Michel Marcus, Jun 07 2016

Extensions

More terms from Michel Marcus, Jun 07 2016

A274001 Even numbers with a unique resolution as the sum of two primes, each of which has a twin.

Original entry on oeis.org

6, 8, 12, 28, 40, 52, 56, 68, 124, 128, 136, 172, 176, 188, 226, 262, 266, 304, 308, 394, 396, 398, 412, 416, 442, 446, 484, 488, 544, 548, 556, 560, 608, 634, 638, 668, 682, 686, 694, 696, 698, 724, 728, 736, 740, 754, 758, 772, 776, 802, 806, 874, 878, 934
Offset: 1

Views

Author

Thomas Curtright, Jun 06 2016

Keywords

Comments

The sequence is infinite only if the number of twin primes is infinite.
Note that not all even integers can be written as the sum of two twins (e.g. 94, 96, 98,...).

Examples

			6 = 3 + 3 is an element since (3,5) are twins, as is 8 = 5 + 3.
10 = 7 + 3 = 5 + 5 is not an element, since it is not uniquely resolved, even though the two resolutions both involve primes with twins.

Crossrefs

Cf. A129363, A007534.

Programs

  • Mathematica
    ok[n_] := 1 == Length@ IntegerPartitions[n, {2}, Select[Prime@ Range@ PrimePi@ n, Or @@ PrimeQ[# + {-2, 2}] &]]; Select[2 Range[500], ok] (* Giovanni Resta, Jun 06 2016 *)

Extensions

a(7)-a(54) from Giovanni Resta, Jun 06 2016

A376287 Index of first occurrence of n in A129363, or 0 if no such number exists.

Original entry on oeis.org

2, 6, 10, 22, 48, 120, 114, 298, 240, 540, 288, 1620, 210, 300, 702, 840, 660, 2312, 1290, 4284, 1332, 2580, 2070, 2100, 1890, 5100, 2340, 5580, 3720, 6660, 3612, 6240, 2310, 10288, 3540, 4680, 4788, 5460, 4410, 5940, 6120, 10200, 4200, 4620, 3570, 10560, 5700, 16588, 5250
Offset: 0

Views

Author

Robert G. Wilson v, Sep 19 2024

Keywords

Comments

Twin prime analogous to A023036.
Conjecture: a(n) > 0. Checked to 1010.
Conjectured last occurrence: 4208, 24536, 28916, 21278, 51806, 68078, 73538, 89216, 83978, ..., .
Conjecture number of terms for A129363(k) = n: 35, 115, 285, 327, 557, 537, 723, 652, 882, ..., .
A129363(n) = 0: A007534.

Crossrefs

Cf. A001097, A023036, A129363.

Programs

  • Mathematica
    tp = Select[Prime@Range@ 16340, PrimeQ[# -2] || PrimeQ[# +2] &]; f[n_] := Length@ IntegerPartitions[n, {2, 2}, tp]; t[_] := 0; k = 2; While[k < 10201, a = f@k; If[ t[a] == 0, t[a] = k]; k += 2]; t /@ Range[0, 75]
Previous Showing 11-19 of 19 results.

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