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.

User: Kok Seng Chua

Kok Seng Chua's wiki page.

Kok Seng Chua has authored 50 sequences. Here are the ten most recent ones:

A167605 Positive integers n such that the first n terms of A167604 form a permutation of the first n primes.

Original entry on oeis.org

1, 2, 3, 16, 18, 19, 26, 27, 34, 35, 36, 38, 39, 40, 48, 51, 52, 53, 54, 55, 63, 73, 75, 76, 77, 78, 83, 93, 94, 98, 99, 106, 113, 114, 121, 122, 123, 128, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 156, 157, 158, 159, 168, 173, 174
Offset: 1

Views

Author

Kok Seng Chua (chuakokseng(AT)hotmail.com), Nov 07 2009

Keywords

Comments

Is this sequence infinite?

Examples

			a(4)=16 since the first 16 terms of A167604 are 2,3,5,11,37,13,7,29,17,19,43,23,47,41,53,31 which is a permutation of the first 16 primes.

Crossrefs

A167604, A000945

Extensions

Edited and extended by Max Alekseyev, Nov 11 2009

A167604 A variant of Euclid-Mullin (A000945): a(1)=2, a(n+1) is the least prime dividing [Product_{i in I} a(i) + Product_{i not in I} a(i)], minimized over all subsets I of {1..n}.

Original entry on oeis.org

2, 3, 5, 11, 37, 13, 7, 29, 17, 19, 43, 23, 47, 41, 53, 31, 61, 59, 67, 79, 83, 73, 97, 71, 101, 89, 103, 127, 107, 113, 137, 131, 139, 109, 149, 151, 163, 157, 167, 173, 193, 211, 179, 191, 181, 223, 199, 197, 233, 227, 229, 239, 241, 251, 257, 307, 281, 269, 271, 293
Offset: 1

Views

Author

Kok Seng Chua (chuakokseng(AT)hotmail.com), Nov 07 2009

Keywords

Comments

By Euclid's argument, the a(i) are distinct.
One can ask whether all primes occur in this sequence.

Examples

			a(4)=11 which is the smallest prime dividing the 4 partitions 2+3*5=17, 3+2*5=13, 5+2*3=11, 1+2*3*5=31.

Links

Crossrefs

A167605 lists such n that the first n terms of a(n) is a permutation of the first n primes.
A000945 is the original Euclid-Mullin sequence (where I is restricted to the empty set).
Cf. A344020.

Programs

  • Maple
    with(numtheory):p:=proc(N) local S, d : S:=NULL:for d  in divisors(N) while d^2<=N  do S:=S,divisors(d+N/d)[2] od : return(min(S)) end:
a :=n->if n = 1 then 2 else p(mul(a(i),i = 1 .. n-1)) fi :
seq(a(n), n=1..15);
# Robert FERREOL, Oct 01 2019
  • Mathematica
    p[N_Integer] := Module[{S = {}, d, divisorsList},
For[d = 1, d^2 <= N, d++, If[Divisible[N, d], divisorsList = Divisors[d + N/d];
If[Length[divisorsList] >= 2, AppendTo[S, divisorsList[[2]]]];]]; Min[S]];
a[n_Integer] := If[n == 1, 2, p[Times @@ Table[a[i], {i, 1, n - 1}]]];
Table[a[n], {n, 1, 14}] (* Hilko Koning, Oct 30 2024 *)
  • PARI
    { A167604_list() = my(a,A,p,b,q,z,m); a = []; A=1; while(1, p=2; while( kronecker(-A,p)!=1, p=nextprime(p+1) ); b=lift(sqrt(-A+O(p))); z=znprimroot(p); m=nextprime(random(10^6)); q=lift(prod(i=1,#a, Mod(1+x^znlog(Mod(a[i],p),z,p-1),(1-x^(p-1))*Mod(1,m)) )); if( polcoeff(q,znlog(Mod(b,p),z,p-1),x)==0, error("conjecture failed mod",m)); a=concat(a,[p]); A*=p; print1(p,", ") ) } /* Max Alekseyev, May 20 2015 */

Formula

For any n, we have Legendre symbol (-a(1)*a(2)*...*a(n-1) / a(n)) = 1. If p is the smallest prime such that (-a(1)*a(2)*...*a(n-1) / p) = 1, then a(n) >= p. Conjecture: For all n, a(n) = p. Note that if b is such that b^2 == -a(1)*a(2)*...*a(n-1) (mod p) and for some I, b == prod_{i in I} a(i) (mod p), then a(n) = p. Heuristically, I must exist for large enough n, since the number of possible subsets I is much larger than p. - Max Alekseyev, Nov 11 2009, May 20 2015

Extensions

Edited and extended by Max Alekseyev, Nov 11 2009

A128276 a(n) is the least product of n odd primes m=p1*p2*...*pn, such that for all divisor d|2*m, d+2*m/d is prime.

Original entry on oeis.org

3, 15, 105, 93081
Offset: 1

Views

Author

Kok Seng Chua (chuakokseng(AT)hotmail.com), Feb 23 2007

Keywords

Comments

1. a(5) > 2*10^9 2. (C. Pomerance) The prime k-tuple conjecture implies a(n) exists for all n

Examples

			105=3*5*7 and 2*3*5*7+1, 3*5*7+2, 2*5*7+3, 2*3*7+5, 2*3*5+7, 2*3+5*7, 2*5+3*7, 2*7+3*5 are all primes and 105 is the smallest such integer which is the product of 3 odd primes, so a(3)=105

Extensions

Typo in a(4) corrected by T. D. Noe, Aug 04 2010

A128278 an=n-th smallest integer m=p1*p2*p3, product of 3 odd primes such that d+2m/d are all primes for d dividing 2m.

Original entry on oeis.org

105, 165, 231, 935, 2109, 2795, 3021, 3819, 6981, 7205, 11285, 12341, 13101, 16419, 17549, 19839, 21749, 21995, 26391, 31229, 31269, 46631, 62651, 63645, 65391, 76155, 77585, 100955, 110811, 113555, 118031, 136451, 148359, 150245, 154679
Offset: 1

Views

Author

Kok Seng Chua (chuakokseng(AT)hotmail.com), Mar 05 2007

Keywords

Examples

			E.g. 105=3.5.7, 2.3.5.7+1=211, 2+3.5.7=107, 3+2.5.7=73, 5+2.3.7=47, 7+2.3.5=37, 2.3+5.7=41, 2.5+3.7=31, 2.7+3.5=29 are all primes and 3.5.7 is the smallest such number, so a(1)=105.

Crossrefs

Cf. A128276, A128277, A128279, A045536.

A128279 an=n-th smallest integer of the form m=p1*p2 where pi are odd primes such that d+2m/d are all primes for d dividing 2m.

Original entry on oeis.org

15, 21, 35, 39, 51, 65, 95, 155, 221, 329, 371, 485, 519, 611, 905, 989, 1121, 1199, 1469, 1509, 1541, 1661, 1821, 3039, 3189, 3431, 3641, 3791, 4055, 4109, 4281, 4601, 4859, 5079, 5111, 5195, 5331, 5429, 5579, 5951, 5979, 6161, 6245, 6731, 6881, 7415
Offset: 1

Views

Author

Kok Seng Chua (chuakokseng(AT)hotmail.com), Mar 05 2007

Keywords

Examples

			E.g. 35=5.7, 2.5.7+1=71, 2.5+7=17, 2.7+5=19. 5.7+2=37 are all primes and this is the 3rd such number, so a(3)=35

Crossrefs

Cf. A128276, A128277, A128278, A045536.

A128281 a(n) is the least product of n distinct odd primes m=p_1*p_2*...*p_n, such that (d+m/d)/2 are all primes for each d dividing m.

Original entry on oeis.org

3, 21, 105, 1365, 884037
Offset: 1

Views

Author

Kok Seng Chua (chuakokseng(AT)hotmail.com), Mar 05 2007

Keywords

Comments

From Iain Fox, Aug 26 2020: (Start)
a(6) > 10^9 if it exists.
All terms are members of A076274 since the definition requires that (1+m)/2 be prime.
The number of prime factors of m congruent to 3 (mod 4) must be even except for n=1.
(End)
a(6) > 2*10^11 if it exists. - David A. Corneth, Aug 27 2020
a(n) >= A070826(n+1) by definition of the sequence. - Iain Fox, Aug 28 2020

Examples

			105=3*5*7, (3*5*7+1)/2=53, (3+5*7)/2=19, (5+3*7)/2=13, (7+3*5)/2=11 are all primes and 105 is the least such number which is the product of 3 primes, so a(3)=3.

Crossrefs

Subsequence of A076274.
Lower bound: A070826.
Cf. A002145, A128276, A128282, A128283, A128284, A128285, A128286, A168352.

Programs

  • PARI
    a(n)=if(n==1, return(3)); my(p=prod(k=1, n, prime(k+1))); forstep(m=p+if(p%4-1, 2), +oo, 4, if(bigomega(m)==n && omega(m)==n, fordiv(m, d, if(!isprime((d+m/d)/2), next(2))); return(m))) \\ Iain Fox, Aug 27 2020

Extensions

Definition corrected by Iain Fox, Aug 25 2020

A128285 Numbers of the form m = p1 * p2 * p3 * p4 where for each d|m we have (d+m/d)/2 prime and p1 < p2 < p3 < p4 each prime.

Original entry on oeis.org

1365, 4305, 10465, 11685, 15873, 27105, 31845, 35245, 50065, 54033, 58765, 74965, 84513, 91977, 95557, 95613, 96033, 104377, 113997, 114405, 117957, 118105, 126357, 127605, 136437, 170905, 197985, 209605, 215373, 226185, 248385, 277797
Offset: 1

Views

Author

Kok Seng Chua (chuakokseng(AT)hotmail.com), Mar 05 2007

Keywords

Examples

			1365=3 * 5 * 7 * 13 and (3 * 5 * 7 * 13+1)/2, (3+5 * 7 * 13)/2, (5+3 * 7 * 13)/2, (7+3 * 5 * 13)/2, (13+3 * 5 * 7)/2, (3 * 5+7.13)/2, (3 * 7+5 * 13)/2, (3 * 13+5 * 7)/2 are all primes and 1365 is the smallest such integer which is the product of 4 primes, so 1365 is in the sequence.

Links

Crossrefs

Cf. A128281, A005383, A128283, A128284, A128286.
Subsequence of A046390.

Extensions

New name from David A. Corneth, Jan 09 2021

A128284 Numbers of the form m = p1 * p2 * p3 where for each d|m we have (d+m/d)/2 prime and p1 < p2 < p3 each prime.

Original entry on oeis.org

105, 165, 273, 345, 357, 385, 777, 897, 1045, 1173, 1353, 1653, 1677, 1705, 2193, 2233, 2373, 2905, 3157, 3237, 3333, 3417, 3445, 3553, 3565, 3945, 4053, 4585, 4953, 5665, 5817, 6097, 6513, 6693, 7077, 7833, 8437, 8565, 8845, 10153, 11005, 11433
Offset: 1

Views

Author

Kok Seng Chua (chuakokseng(AT)hotmail.com), Mar 05 2007

Keywords

Comments

The symmetric representation of sigma (cf. A237593), SRS(a(n)), of any number in this sequence has between 4 and 8 regions with 3 regions impossible because p1 < p2 < p3 implies 2*p3 < p1*p2. When there are 8 regions they all have width 1 and their areas are the prime numbers (d+a(n)/d)/2 for the 4 respective pairs of divisors of a(n). In general, the areas of the regions in SRS(a(n)) need not be prime, except for the two symmetric outer regions (n+1)/2. - Hartmut F. W. Hoft, Jan 09 2021

Examples

			165=3*5*11 and (3*5*11+1)/2=83, (3+5*7)/2=19, (5+3*7)/2=13, (7+3*5)/2=11 are all primes, so 165 is a term.
From _Hartmut F. W. Hoft_, Jan 09 2021: (Start)
a(1) = 105 = 3*5*7 and SRS(a(1)) consists of four regions with areas ( 53, 43, 43, 53 ); the center areas have maximum width 2 and represent the sum of primes (3+35)/2 + (5+21)/2 + (7+15)/2 = 43.
a(17) = 2373 = 3*7*17 is the first number in the sequence whose symmetric representation of sigma consists of 8 regions, all of width 1 and the respective symmetric regions have areas:  (2373 + 1)/2 = 1187, (791 + 3)/2 = 397, (339 + 7)/2 = 173, (21 + 113)/2 = 67. (End)

Links

Crossrefs

Cf. A128281, A005383, A128283, A128285, A128286.
Cf. A237048, A237591, A237593, A249223, A262045.
Subsequence of A046389.

Programs

  • Mathematica
    (* function goodL[] is defined in A128283 *)
a128284[n_] := goodL[{1, n}, 3]
a128284[11433] (* Hartmut F. W. Hoft, Jan 09 2021 *)

A128286 a(n) is the n-th smallest product of 5 odd primes m = p1*p2*p3*p4*p5 such that (d+m/d)/2 are all primes for each d dividing m.

Original entry on oeis.org

884037, 1137565, 2398377, 123156993, 681714273, 2347722213, 7283144845, 7794246057, 8953447917, 10287992785, 13749228493, 38108016453, 38901676405, 70918253385, 71809744693, 120418624965, 148282565865, 150721729873
Offset: 1

Views

Author

Kok Seng Chua (chuakokseng(AT)hotmail.com), Mar 05 2007

Keywords

Comments

a(6) > 10^9.

Examples

			884037 = 3*7*11*43*89 and (1 + 884037)/2, (3 + 7*11*43*89)/2,
(7 + 3*11*43*89)/2, (11 + 3*7*43*89)/2, (43 + 3*7*11*89)/2, (89 + 3*7*11*43)/2,
(3*7 + 11*43*89)/2, (3*11 + 7*43*89)/2, (3*43 + 7*11*89)/2,(3*89 + 7*11*43)/2,
(7*11 + 3*43*89)/2, (7*43 + 3*7*89)/2, (7*89 + 3*7*43)/2, (11*43 + 3*7*89)/2,
(11*89 + 3*7*43)/2, (43*89 + 3*7*11)/2 are all primes and 884037 is the smallest such integer, so a(1) = 884037.

Crossrefs

Cf. A128281, A005383, A128283, A128284, A128285.

Extensions

a(6)-a(18) from Donovan Johnson, Oct 12 2008

A128277 a(n) is the n-th smallest integer m which is the product of 4 odd primes m=p1*p2*p3*p4 such that d+2*m/d are all primes for each d dividing 2*m.

Original entry on oeis.org

93081, 449985, 1523705, 301921991, 899343761, 1581262341, 7290929465, 12102153569, 25404516309, 27482957831, 38661868781, 49656488021, 240305617889, 305000299185, 341656377581, 377737353491
Offset: 1

Views

Author

Kok Seng Chua (chuakokseng(AT)hotmail.com), Feb 23 2007

Keywords

Comments

1. a(6) > 2*10^9
2. (C. Pomerance) The prime k-tuple conjecture implies the sequence is infinite.
a(17) > 4*10^11. - Donovan Johnson, Sep 06 2010

Examples

			93081 = 3*19*23*71 and 2*93081+1, 2+3*19*23*71, 3+2*19*23*71, 19+2*3*23*71, 71+2*3*19*23, 2*3+19*23*71, 2*19+3*23*71, 2*23+3*19*23*71, 2*71+3*19*23, 3*19+2*23*71, 3*23+2*19*71, 3*71+2*19*23, 19*23+2*3*71, 19*71+2*3*23, 23*71+2*3*19 are all primes and 93081 is smallest such integer, so a(1)=93081.

Crossrefs

Cf. A128276.

Extensions

Changed every occurrence of 93801 to 93081. - T. D. Noe, Aug 05 2010
Added missing term 899343761 and a(7)-a(16) from Donovan Johnson, Sep 06 2010

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