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 1-10 of 71 results. Next

A102540 Primes that are not Chen primes (see A109611).

Original entry on oeis.org

43, 61, 73, 79, 97, 103, 151, 163, 173, 193, 223, 229, 241, 271, 277, 283, 313, 331, 349, 367, 373, 383, 397, 421, 433, 439, 457, 463, 523, 547, 593, 601, 607, 613, 619, 643, 661, 673, 691, 709, 727, 733, 739, 757, 773, 823, 853, 859, 883, 907, 929, 967, 997
Offset: 1

Views

Author

Wilfredo Lopez (chakotay147138274(AT)yahoo.com), Aug 14 2005

Keywords

Links

Programs

  • PARI
    n=0; for(j=1, 1994, if(bigomega(prime(j)+2)>2, n++; write("b102540.txt", n " " prime(j)))) /* Donovan Johnson, Apr 29 2013 */

Extensions

More terms from Robert Happelberg (roberthappelberg(AT)yahoo.com), Aug 16 2005

A139690 a(n) = A109611(n) + 2.

Original entry on oeis.org

4, 5, 7, 9, 13, 15, 19, 21, 25, 31, 33, 39, 43, 49, 55, 61, 69, 73, 85, 91, 103, 109, 111, 115, 129, 133, 139, 141, 151, 159, 169, 181, 183, 193, 199, 201, 213, 229, 235, 241, 253, 259, 265, 271, 283, 295, 309, 313, 319, 339, 349, 355, 361, 381, 391, 403, 411
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 29 2008

Keywords

Links

Crossrefs

Intersection of A052147 and A037143; A006512 is a subsequence.
Cf. A109611.

Programs

  • Mathematica
    Cases[Import["https://oeis.org/A109611/b109611.txt", "Table"], {, }][[All, 2]] + 2 (* Robert Price, Apr 19 2025 *)
  • PARI
    list(lim)=my(v=List(),t); forprime(p=2,lim\2, forprime(q=2,min(p,lim\p), if(isprime(t=p*q-2), listput(v,t+2)))); t=2; forprime(p=3,lim, if(p-t==2, listput(v,p)); t=p); Set(v) \\ Charles R Greathouse IV, Jan 19 2017

Formula

A010051(a(n)) = A139689(n); A064911(a(n)) = 1 - A139689(n);
A001222(a(n)) = 2 - A139689(n).

A176012 Chen primes A109611(k) which have the same sum-of-digits as their index k.

Original entry on oeis.org

227, 827, 1201, 1621, 2179, 2333, 2441, 2711, 3041, 3251, 3329, 3541, 5147, 5167, 5701, 5711, 6131, 6661, 6833, 7321, 7331, 8501, 9209, 9239, 10271, 13807, 14251, 14449, 14629, 15731, 15761, 16007, 16139, 16619, 16741, 17291, 19421, 20231, 20441, 20507, 22441
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 06 2010

Keywords

Comments

The associated indices k are:
38, 98, 130, 163, 199, 209, 218, 236, 260, 272, 278, 292, 386, 388, 418, 419, 443, 469, 479, 508,...
The indices of A109611(k) in the primes A000040 are A000720(A109611(k)) =
49, 144, 197, 257, 327, 345, 362, 395, 436, 458, 469, 496, 686, 688, 751, 752, 799, 859, 880, 933, 934, 1060, ..
Some entries are also Honaker primes (A033548): 2441, 5701, 5711, 15761, 26119, 31517, 34471, 37019, 44221,...

Examples

			a(1) = 227 = A109611(38) where 2+2+7 = 11 = 3+8.
a(2) = 827 = A109611(98), where 8+2+7 =17= 9+8.

References

  • M. du Sautoy: Die Musik der Primzahlen: Auf den Spuren des groessten Raetsels der Mathematik, Beck, 4. Auflage, 2005

Links

Crossrefs

Cf. A033548, A109611, A174924.

Formula

{A109611(k): A007953(A109611(k)) = A007953(k) }.

Extensions

9241 replaced by 9239, and lists of examples reduced by R. J. Mathar, Jun 07 2010

A117244 Single (or isolated or non-twin) primes (A007510) that are not Chen primes (A109611).

Original entry on oeis.org

79, 97, 163, 173, 223, 277, 331, 367, 373, 383, 397, 439, 457, 547, 593, 607, 613, 673, 691, 709, 727, 733, 739, 757, 773, 853, 907, 929, 967, 997, 1013, 1069, 1087, 1103, 1123, 1129, 1171, 1181, 1213, 1223, 1237, 1249, 1307, 1373, 1423, 1433, 1447, 1493
Offset: 1

Views

Author

Jani Melik, Apr 22 2006

Keywords

Examples

			79 is single prime, but not Chen prime, since 79 -2 = 77 = 7*11 is composite, and 79 + 2 = 81 = 3^4 is neither prime nor semiprime.

Links

Crossrefs

Cf. A007510, A109611.

Programs

  • Maple
    isA001358 := proc(n) numtheory[bigomega](n) = 2 ; end proc: isA109611 := proc(n) if isprime(n) then isprime(n+2) or isA001358(n+2) ; else false; end if; end proc: isA007510 := proc(n) if isprime(n) then not isprime(n-2) and not isprime(n+2) ; else false; end if ; end proc: isA117244 := proc(n) isA007510(n) and not isA109611(n) ; end proc: for n from 1 to 4000 do if isA117244(n) then printf("%d,",n) ; fi; end do ; # R. J. Mathar, Dec 09 2009
  • Mathematica
    Select[Range[1500], PrimeQ[#] && !PrimeQ[#-2] && PrimeOmega[#+2] > 2 &] (* Amiram Eldar, Oct 19 2021 *)
  • PARI
    isok(p) = isprime(p) && !isprime(p-2) && !isprime(p+2) && (bigomega(p+2) > 2); \\ Michel Marcus, Oct 19 2021

Extensions

Terms beyond 397 from R. J. Mathar, Dec 09 2009
Offset corrected by Amiram Eldar, Oct 19 2021

A369385 The smallest number k that can be partitioned in n ways as the sum of two numbers from A109611.

Original entry on oeis.org

1, 4, 10, 22, 34, 60, 118, 112, 142, 198, 270, 298, 280, 364, 508, 460, 588, 490, 580, 658, 858, 670, 700, 994, 880, 1240, 1078, 1288, 910, 1120, 1428, 1510, 1300, 1330, 1930, 1960, 1750, 1540, 2128, 2170, 2140, 2470, 2560, 2380, 2590, 2770, 2728, 3838, 2968, 4000
Offset: 0

Views

Author

Marius A. Burtea, Jan 25 2024

Keywords

Examples

			a(0) = 1 because 1 cannot be written as the sum of two terms in A109611.
The numbers 2 and 3 cannot be written as the sum of two terms in A109611, and 4 = 2 + 2 = A109611(1) + A109611(1) is the only writing with terms in A109611, so a(1) = 4.
The numbers 5, 6, 7, 8, 9 can be written as the sum of two terms in A109611 in at most one way and 10 = 3 + 7 = A109611(2) + A109611(4) and 10 = 5 + 5 = A109611(3) + A109611(3), so a(2) = 10.

Crossrefs

Cf. A109611.

Programs

  • Magma
    IsSemiprime:=func;
ch:=func; b:=[n: n in [1..4000] |ch(n)]; a:=[]; for n in [0..47] do k:=1; while #RestrictedPartitions(k, 2, Set(b)) ne n do k:=k+1; end while; Append(~a,k); end for; a;

A063637 Primes p such that p+2 is a semiprime.

Original entry on oeis.org

2, 7, 13, 19, 23, 31, 37, 47, 53, 67, 83, 89, 109, 113, 127, 131, 139, 157, 167, 181, 199, 211, 233, 251, 257, 263, 293, 307, 317, 337, 353, 359, 379, 389, 401, 409, 443, 449, 467, 479, 487, 491, 499, 503, 509, 541, 557, 563, 571, 577, 587, 631, 647, 653, 677
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 21 2001

Keywords

Comments

Primes of the form p*q - 2, where p and q are primes.
Union of A049002 and A115093. - T. D. Noe, Mar 01 2006

Examples

			From _K. D. Bajpai_, Sep 06 2014: (Start)
a(3) = 13, which is prime, and 13 + 2 = 15 = 3 * 5, which is a semiprime.
a(4) = 19, which is prime, and 19 + 2 = 21 = 3 * 7, which is a semiprime.
(End)

References

  • J.-R. Chen, On the representation of a large even integer as the sum of a prime and a product of at most two primes, Sci. Sinica 16 (1973), 157-176.

Links

Crossrefs

Cf. A005383, A001358, A063638.
Cf. A109611 (Chen primes).

Programs

  • Haskell
    a063637 n = a063637_list !!(n-1)
a063637_list = filter ((== 1) . a064911 . (+ 2)) a000040_list
-- Reinhard Zumkeller, Nov 15 2011
  • Maple
    select(t -> isprime(t) and numtheory:-bigomega(t+2)=2, [2, seq(2*i+1,i=1..500)]); # Robert Israel, Sep 07 2014
  • Mathematica
    f[n_] := Plus @@ Flatten[ Table[ # [[2]], {1}] & /@ FactorInteger[ n]]; Select[ Prime[ Range[ 123]], f[ # + 2] == 2 &] (* Robert G. Wilson v, Apr 30 2005 *)
Select[Prime[Range[500]],PrimeOmega[#+2]==2&]  (* K. D. Bajpai, Sep 06 2014 *)
  • PARI
    { n=0; for (m=1, 10^9, p=prime(m); if (bigomega(p + 2) == 2, write("b063637.txt", n++, " ", p); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 26 2009

Formula

a(n) = A062721(n) - 2.
A010051(a(n)) * A064911(a(n) + 2) = 1. - Reinhard Zumkeller, Nov 15 2011

A063638 Primes p such that p-2 is a semiprime.

Original entry on oeis.org

11, 17, 23, 37, 41, 53, 59, 67, 71, 79, 89, 97, 113, 131, 157, 163, 179, 211, 223, 239, 251, 269, 293, 307, 311, 331, 337, 367, 373, 379, 383, 397, 409, 419, 439, 449, 487, 491, 499, 503, 521, 547, 593, 599, 613, 631, 673, 683, 691, 701, 709, 719, 733, 739
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 21 2001

Keywords

Comments

Primes of form p*q + 2, where p and q are primes.
11 is the only prime of this form where p=q. For prime p>3, 3 divides p^2+2. - T. D. Noe, Mar 01 2006
The asymptotic growth of this sequence is relevant for A204142. We have a(10^k) = (11, 79, 1571, 27961, 407741, 5647823, ...). - M. F. Hasler, Feb 13 2012

Links

Crossrefs

Cf. A005385, A001358, A063637, A109611 (Chen primes), A204142, A064911, A040976, A241809.

Programs

  • Haskell
    a063638 n = a063638_list !! (n-1)
a063638_list = map (+ 2) $ filter ((== 1) . a064911) a040976_list
-- Reinhard Zumkeller, Feb 22 2012
  • Mathematica
    Take[Select[ # + 2 & /@ Union[Flatten[Outer[Times, Prime[Range[100]], Prime[Range[100]]]]], PrimeQ], 60]
Select[Prime[Range[200]],PrimeOmega[#-2]==2&] (* Paolo Xausa, Oct 30 2023 *)
  • PARI
    n=0; for (m=2, 10^9, p=prime(m); if (bigomega(p - 2) == 2, write("b063638.txt", n++, " ", p); if (n==1000, break))) \\ Harry J. Smith, Aug 26 2009
  • PARI
    forprime(p=3,9999, bigomega(p-2)==2 & print1(p","))
  • PARI
    p=2; for(n=1,1e4, until(bigomega(-2+p=nextprime(p+1))==2,); write("b063638.txt", n" "p)) \\ M. F. Hasler, Feb 13 2012
  • PARI
    list(lim)=my(v=List(), t); forprime(p=3, (lim-2)\3, forprime(q=3, min((lim-2)\p, p), t=p*q+2; if(isprime(t), listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Aug 05 2016

Formula

a(n) = A241809(n) + 2. - Hugo Pfoertner, Oct 30 2023

A063643 Primes with 2 representations: p*q - 2 = u*v + 2 where p, q, u and v are primes.

Original entry on oeis.org

23, 37, 53, 67, 89, 113, 131, 157, 211, 251, 293, 307, 337, 379, 409, 449, 487, 491, 499, 503, 631, 683, 701, 719, 751, 769, 787, 919, 941, 953, 991, 1009, 1039, 1117, 1193, 1201, 1259, 1381, 1399, 1439, 1459, 1471, 1499, 1511, 1567, 1709, 1733, 1759, 1801
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 21 2001

Keywords

Comments

Or, primes p such that p+/-2 are semiprimes. - Zak Seidov, Mar 08 2006

Examples

			A063643(25) = 751: 751 = A063637(60)= 753 - 2 = 3*251 - 2, 751 = A063638(55)= 749 + 2 = 7*107 + 2.

Links

Crossrefs

Cf. A063637, A063638, A001358.
Cf. A109611 (Chen primes).

Programs

  • Maple
    q:= p-> isprime(p) and map(numtheory[bigomega], {p-2, p+2})={2}:
select(q, [$2..2000])[];  # Alois P. Heinz, Apr 01 2024
  • Mathematica
    Select[Prime[Range[300]], PrimeOmega[#+2] == PrimeOmega[#-2] == 2&] (* Jean-François Alcover, Mar 02 2019 *)
  • PARI
    { n=0; for (m=2, 10^9, p=prime(m); if (bigomega(p + 2) == 2 && bigomega(p - 2) == 2, write("b063643.txt", n++, " ", p); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 26 2009

Formula

Intersection of A063637 and A063638. - Zak Seidov, Mar 14 2011

A086005 Semiprimes sandwiched between semiprimes.

Original entry on oeis.org

34, 86, 94, 122, 142, 202, 214, 218, 302, 394, 446, 634, 698, 842, 922, 1042, 1138, 1262, 1346, 1402, 1642, 1762, 1838, 1894, 1942, 1982, 2102, 2182, 2218, 2306, 2362, 2434, 2462, 2518, 2642, 2722, 2734, 3098, 3386, 3602, 3694, 3866, 3902, 3958, 4286, 4414
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 07 2003

Keywords

Comments

These are some of the balanced semiprimes (see A213025). - Alonso del Arte, Jun 04 2012

Examples

			94 = 47*2: 94 - 1 = 3*31 and 94 + 1 = 5*19, therefore 94 is in the sequence.

Links

Crossrefs

Cf. A001358, A056809, A064911, A100484, A123255.

Programs

  • Haskell
    a086005 n = a086005_list !! (n-1)
a086005_list = filter
   (\x -> a064911 (x - 1) == 1 && a064911 (x + 1) == 1) a100484_list
-- Reinhard Zumkeller, Aug 08 2013, Jun 10 2012
  • Mathematica
    u[n_]:=Plus@@Last/@FactorInteger[n]==2;lst={};Do[If[u[n],sp=n;If[u[sp-1]&&u[sp+1],AppendTo[lst,sp]]],{n,8!}];lst  (* Vladimir Joseph Stephan Orlovsky, Nov 16 2009 *)
(* First run program for A109611 to define semiPrimeQ *) Select[Range[4000], Union[{semiPrimeQ[# - 1], semiPrimeQ[#], semiPrimeQ[# + 1]}] == {True} &] (* Alonso del Arte, Jun 03 2012 *)
Select[Partition[Range@ 4000, 3, 1], Union@ PrimeOmega@ # == {2} &][[All, 2]] (* Michael De Vlieger, Jun 14 2017 *)
  • Python
    from itertools import count, islice
from sympy import factorint, isprime
def agen(): # generator of terms
    nxt = 0
    for k in count(2, 2):
        prv, nxt = nxt, sum(factorint(k+1).values())
        if prv == nxt == 2 and isprime(k//2): yield k
print(list(islice(agen(), 46))) # Michael S. Branicky, Nov 26 2022

Formula

a(n) = 2*A086006(n).
a(n) = A056809(n)+1. - Zak Seidov, Sep 30 2012

A002267 The 15 supersingular primes: primes dividing order of Monster simple group.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

The supersingular primes are a subset of the Chen primes (A109611). - Paul Muljadi, Oct 12 2005
PROD(a(k): 1<=k<=15) = 1618964990108856390 = A174848(26). - Reinhard Zumkeller, Apr 02 2010

References

  • J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].
  • A. P. Ogg, Modular functions, in The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), pp. 521-532, Proc. Sympos. Pure Math., 37, Amer. Math. Soc., Providence, R.I., 1980.

Links

Crossrefs

Cf. A003131, A001379, A051161, A109611.

Programs

  • Mathematica
    FactorInteger[GroupOrder[MonsterGroupM[]]][[All, 1]] (* Jean-François Alcover, Oct 03 2016 *)
  • PARI
    A002267=vecextract(primes(20),612351) \\ bitmask 2^20-1-213<<11: remove primes # 12, 14, 16, 18 and 19. - M. F. Hasler, Nov 10 2017
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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