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.

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A105237 Positive integers n such that n^13 + 1 is semiprime.

Original entry on oeis.org

2, 22, 108, 126, 180, 256, 336, 490, 630, 652, 660, 682, 708, 760, 828, 862, 882, 1030, 1038, 1128, 1162, 1216, 1318, 1450, 1612, 1930, 1950, 2010, 2236, 2268, 2380, 2436, 2658, 2752, 2800, 2962, 2998, 3036, 3048, 3318, 3672, 3922, 4152, 4396, 4506, 4816
Offset: 1

Views

Author

Jonathan Vos Post, Apr 12 2005

Keywords

Comments

We have the polynomial factorization: n^13+1 = (n+1) * (n^12 - n^11 + n^10 - n^9 + n^8 - n^7 + n^6 - n^5 + n^4 - n^3 + n^2 - n + 1) Hence after the initial n=1 prime, the binomial can never be prime. It can be semiprime iff n+1 is prime and n^12 - n^11 + n^10 - n^9 + n^8 - n^7 + n^6 - n^5 + n^4 - n^3 + n^2 - n + 1 is prime.

Examples

			2^13+1 = 8193 = 3 * 2731,
22^13+1 = 282810057883082753 = 23 * 12296089473177511,
1030^13+1 = 1468533713451564313811276230000000000001 = 1031 * 1424377995588326201562828545101842871.

Links

Crossrefs

Cf. A001358, A085722, A096173, A186669, A104238, A103854, A105041, A105066, A105078, A105122, A105142, A105237, A104335, A104479, A104494, A104657, A105282.

Programs

  • Magma
    IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [1..1600]|IsSemiprime(n^13+1)] // Vincenzo Librandi, Dec 21 2010
  • Mathematica
    Select[Range[0, 300000], PrimeQ[# + 1] && PrimeQ[(#^13 + 1)/(# + 1)] &] (* Robert Price, Mar 11 2015 *)

Extensions

a(19)-a(24) from Vincenzo Librandi, Dec 21 2010

A242331 Numbers k such that k^2 + 3 is a semiprime.

Original entry on oeis.org

1, 6, 16, 18, 20, 24, 26, 32, 34, 36, 40, 44, 46, 48, 56, 60, 66, 68, 78, 80, 88, 98, 100, 102, 104, 108, 116, 118, 120, 128, 136, 148, 152, 164, 170, 174, 176, 182, 188, 190, 192, 196, 200, 204, 212, 220, 226, 232, 234, 238, 246, 250, 252, 258, 260, 262, 266
Offset: 1

Views

Author

Vincenzo Librandi, May 14 2014

Keywords

Comments

The semiprimes of this form are: 4, 39, 259, 327, 403, 579, 679, 1027, 1159, 1299, 1603, 1939, 2119, 2307, 3139, 3603, 4359, 4627, ...

Links

Crossrefs

Cf. A049422, A049423, A085722, A242330, A242332, A242333.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [0..300] | IsSemiprime(s) where s is n^2+3];
  • Mathematica
    Select[Range[300], PrimeOmega[#^2 + 3] == 2 &]

A242330 Numbers k such that k^2 + 2 is a semiprime.

Original entry on oeis.org

2, 6, 7, 11, 12, 17, 18, 27, 29, 35, 37, 42, 43, 48, 51, 53, 54, 55, 60, 65, 66, 69, 72, 73, 75, 79, 83, 84, 87, 90, 93, 97, 115, 119, 125, 132, 133, 135, 137, 141, 144, 150, 153, 155, 159, 161, 165, 169, 174, 183, 186, 187, 189, 191, 192, 195, 198
Offset: 1

Views

Author

Vincenzo Librandi, May 14 2014

Keywords

Comments

The semiprimes of this form are: 6, 38, 51, 123, 146, 291, 326, 731, 843, 1227, 1371, 1766, 1851, 2306, 2603, 2811, 2918, 3027, 3602, ....
There are no four consecutive terms in this sequence, that is, a(n) > a(n-3) + 3 (check mod 6). Probably sieve theory can show that this sequence has density 0. - Charles R Greathouse IV, Feb 24 2023

Links

Crossrefs

Cf. A056899, A067201, A085722, A242331, A242332, A242333.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [2..200] | IsSemiprime(s) where s is n^2+2];
  • Mathematica
    Select[Range[300], PrimeOmega[#^2 + 2] == 2 &]
  • PARI
    issemi(n)=forprime(p=2,997,if(n%p==0, return(isprime(n/p)))); bigomega(n)==2
is(n)=issemi(n^2+2) \\ Charles R Greathouse IV, Feb 24 2023

Formula

a(n) > 2n for n > 1. - Charles R Greathouse IV, Feb 24 2023

A242333 Numbers k such that k^2 + 5 is a semiprime.

Original entry on oeis.org

1, 2, 3, 4, 8, 9, 14, 18, 21, 22, 24, 26, 27, 28, 30, 33, 42, 44, 51, 54, 57, 58, 62, 63, 64, 68, 69, 82, 84, 86, 90, 93, 98, 99, 102, 104, 108, 111, 118, 132, 134, 138, 144, 152, 154, 156, 166, 174, 177, 180, 183, 184, 186, 188, 189, 194, 208, 210, 212, 216
Offset: 1

Views

Author

Vincenzo Librandi, May 14 2014

Keywords

Comments

The semiprimes of this form are: 6, 9, 14, 21, 69, 86, 201, 329, 446, 489, 581, 681, 734, 789, 905, 1094, 1769, 1941, 2606, 2921, 3254, ...

Links

Crossrefs

Cf. A056905, A078402, A085722, A242330, A242331, A242332.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [0..300] | IsSemiprime(s) where s is n^2+5];
  • Mathematica
    Select[Range[0, 300], PrimeOmega[#^2 + 5] == 2 &]

A242332 Numbers k such that k^2 + 4 is a semiprime.

Original entry on oeis.org

0, 9, 19, 21, 23, 25, 31, 41, 43, 51, 53, 55, 63, 69, 71, 75, 77, 79, 83, 91, 93, 105, 107, 109, 113, 119, 123, 129, 131, 133, 143, 145, 149, 151, 153, 157, 165, 171, 173, 175, 181, 185, 187, 191, 195, 197, 201, 209, 221, 223, 225, 227, 241, 249, 251, 257, 259
Offset: 1

Views

Author

Vincenzo Librandi, May 14 2014

Keywords

Comments

The semiprimes of this form are: 4, 85, 365, 445, 533, 629, 965, 1685, 1853, 2605, 2813, 3029, 3973, 4765, 5045, 5629, 5933, 6245, ...

Links

Crossrefs

Cf. A005473, A007591, A085722, A242330, A242331, A242333.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [0..300] | IsSemiprime(s) where s is n^2+4];
  • Mathematica
    Select[Range[0, 300], PrimeOmega[#^2 + 4] == 2 &]

A278162 Least number with the prime signature of n^2 + 1.

Original entry on oeis.org

1, 2, 2, 6, 2, 6, 2, 12, 6, 6, 2, 6, 6, 30, 2, 6, 2, 30, 12, 6, 2, 30, 6, 30, 2, 6, 2, 30, 6, 6, 6, 30, 12, 30, 6, 6, 2, 30, 12, 6, 2, 12, 6, 60, 6, 6, 6, 210, 6, 6, 6, 6, 6, 30, 2, 30, 2, 120, 6, 6, 6, 6, 6, 30, 6, 6, 2, 30, 24, 6, 12, 6, 30, 210, 2, 30, 6, 30, 6, 6, 6, 30, 12, 210, 2, 6, 6, 30, 6, 30, 2, 30, 6, 60, 2, 6, 6, 30, 30, 60, 6, 6, 6, 30, 6, 30, 6
Offset: 0

Views

Author

Antti Karttunen, Nov 19 2016

Keywords

Links

Crossrefs

Bisection of A278260.
Cf. A002522, A046523.
Cf. A005574 (positions of 2's), A085722 (of 6's).
Cf. also A278160, A278244, A278254.

Programs

  • Mathematica
    Table[Times @@ MapIndexed[(Prime@ First@ #2)^#1 &, #] &@ If[Length@ # == 1 && #[[1, 1]] == 1, {0}, Reverse@ Sort@ #[[All, -1]]] &@ FactorInteger[ n^2 + 1], {n, 0, 120}] (* Michael De Vlieger, Nov 21 2016 *)
  • PARI
    A046523(n) = my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]) \\ From Charles R Greathouse IV, Aug 17 2011
A278162(n) = A046523((n^2)+1);
for(n=0, 10000, write("b278162.txt", n, " ", A278162(n)));
  • Scheme
    (define (A278162 n) (A046523 (+ 1 (* n n))))

Formula

a(n) = A046523(A002522(n)) = A046523((n^2)+1).
a(n) = A278260(2*n).

A209877 a(n) = A209874(n)/2: Least m > 0 such that 4*m^2 = -1 modulo the Pythagorean prime A002144(n).

Original entry on oeis.org

1, 4, 2, 6, 3, 16, 15, 25, 23, 17, 11, 5, 38, 49, 50, 22, 14, 40, 81, 56, 7, 61, 72, 32, 8, 41, 30, 114, 69, 144, 57, 74, 68, 21, 52, 137, 167, 10, 133, 196, 127, 191, 174, 24, 104, 143, 26, 59, 43, 12, 258, 238, 289, 97, 77, 252, 53, 29, 13, 283, 48, 190, 335, 361, 31, 228, 291, 159, 263, 123, 260, 325, 363, 247, 162
Offset: 1

Views

Author

M. F. Hasler, Mar 14 2012

Keywords

Comments

Also: Square root of -1/4 in Z/pZ, for Pythagorean primes p=A002144(n).
Also: Least m>0 such that the Pythagorean prime p=A002144(n) divides 4(kp +/- m)^2+1 for all k>=0.
In practice these can also be determined by searching the least N^2+1 whose least prime factor is p=A002144(n): For given p, all of these N will have a(n) or p-a(n) as remainder mod 2p.

Examples

			a(1)=1 since A002144(1)=5 and 4*1^2+1 is divisible by 5; as a consequence 4*(5k+/-1)^2+1 = 100k^2 +/- 40k + 5 is divisible by 5 for all k.
a(2)=4 since A002144(2)=13 and 4*4^2+1 = 65 is divisible by 13, while 4*1^1+1=5, 4*2^2+1=17 and 4*3^2+1=37 are not. As a consequence, 4*(13k+/-4)^2+1 = 13(...)+4*4^1+1 is divisible by 13 for all k.

Links

Crossrefs

Cf. A209874.
Cf. A002496, A014442, A085722, A144255, A089120, A193432.
Cf. A002314, A177979.

Programs

  • Maple
    f:= proc(p) local m;
   if not isprime(p) then return NULL fi;
   m:= numtheory:-msqrt(-1/4, p);
   min(m,p-m);
end proc:
map(f, [seq(i,i=5..1000,4)]); # Robert Israel, Mar 13 2018
  • Mathematica
    f[p_] := Module[{r}, r /. Solve[4 r^2 == -1, r, Modulus -> p] // Min];
f /@ Select[4 Range[300] + 1, PrimeQ] (* Jean-François Alcover, Jul 27 2020 *)
  • PARI
    apply(p->lift(sqrt(Mod(-1,p)/4)), A002144)

A247340 Numbers n such that each prime divisor of the semiprime n^2+1 is also a divisor of a^2+1 and b^2+1 respectively for some a, b < n.

Original entry on oeis.org

3, 8, 30, 46, 50, 76, 100, 144, 254, 266, 274, 286, 334, 380, 456, 494, 504, 516, 520, 526, 566, 664, 670, 726, 756, 810, 836, 844, 874, 1040, 1064, 1086, 1130, 1164, 1216, 1250, 1300, 1476, 1714, 1740, 1800, 1826, 1834, 1946, 1950, 2014, 2194, 2200, 2220, 2324
Offset: 1

Views

Author

Michel Lagneau, Sep 14 2014

Keywords

Comments

Or numbers n such that n^2+1 = p*q, p and q primes => p | a^2+1 and q | b^2+1 for some a,b < n.
Subsequence of A085722 and except the first term, a(n) is even.
The squares of the sequence are 100, 144, 3364, 6084, 7396, 10404, 24964, 45796, 47524, 68644, 71824, 93636,...
Observation : a(n) = p*q => there exists a and b such that a^2+1 = m*p and b^2+1 = m*q. (see the examples).

Examples

			3^2+1 = 2*5 => 1^1+1 = 2 and 2^2+1 = 5 ;
8^2+1 = 5*13 => 3^2+1 = 2*5 and 5^2+1 = 2*13 ;
30^2+1 = 17*53 => 13^2+1=2*5*17 and 23^2+1 = 2*5*53 ;
46^2+1 = 29*73 => 17^2+1 = 2*5*29 and 27^2+1=2*5*73 ;
50^2+1 = 41*61 => 9^2+1 = 2*41 and 11^2+1 = 2*61 ;
76^2+1 = 53*109 => 23^2+1 = 2*5*53 and 33^2+1 = 2*5*109 ;
100^2+1 = 73*137 => 27^2+1=2*5*73 and 37^2+1 = 2*5*137 ;
144^2+1 = 89*233 => 55^2+1 = 2*17*89 and 89^2+1 = 2*17*233 ;
254^2+1 = 149*433 => 105^2+1 = 2*37*149 and 179^2+1 = 2*37*433 ;
266^2+1 = 173*409 => 93^2+1 = 2*5^2*173 and 143^2+1 = 2*5^2*409.

Links

Crossrefs

Cf. A085722, A144255.

Programs

  • Maple
    with(numtheory):lst:={}:
for n from 1 to 3000 do:
   x:=factorset(n^2+1):n0:=nops(x):
     for i from 1 to n0 do:
      lst:=lst union {x[i]}:
     od:
      lst1:={}:nn:=n+1:xx:=factorset(nn^2+1):nn0:=nops(xx):
        for j from 1 to nn0 do:
         lst1:=lst1 union {xx[j]}:
        od:
        if
         nn0=2
         and bigomega(nn^2+1)=2
         and {xx[1],xx[2]} intersect lst = {xx[1],xx[2]}
         then
         printf(`%d, `,n+1):
         else
        fi:
        lst:=lst union lst1:
  od:

A263876 Numbers n such that n^2 + 1 has two distinct prime divisors less than n.

Original entry on oeis.org

7, 18, 38, 41, 68, 70, 182, 239, 500, 682, 776, 800, 1068, 1710, 1744, 4030, 4060, 5604, 5744, 8119, 12156, 15006, 16610, 17684, 21490, 25294, 26884, 27590, 32060, 32150, 37416, 37520, 45630, 47321, 58724, 71264, 84906, 88526, 98864, 109054, 109610, 128766
Offset: 1

Views

Author

Michel Lagneau, Oct 28 2015

Keywords

Comments

Subsequence of A256011.
The numbers n such that n^2 + 1 = p*q are semiprimes (A085722) are not in the sequence. According to this property, the corresponding sequence of the number of prime divisors with multiplicity is 3, 3, 3, 3, 4, 3, 5, 5, 3, 5, 3, 3, 7, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 6, ...

Examples

			7 is in the sequence because 7^2 + 1 = 2*5^2 => 2 and 5 are less than 7.

Links

Crossrefs

Cf. A085722, A144255, A256011, A263877.

Programs

  • Mathematica
    Select[Range[150000], PrimeNu[#^2+1] == 2&&FactorInteger[#^2+1][[1,1]]<# &&FactorInteger[#^2+1][[2,1]]<#&]
  • PARI
    for(n=1, 1e5, t=n^2+1; if ((omega(t) == 2) && (factor(t)[, 1][2] < n), print1(n, ", "))); \\ Altug Alkan, Oct 28 2015

A321795 Numbers m such that m^2+1 is prime with (m-1)^2+1 and (m+1)^2+1 semiprimes.

Original entry on oeis.org

4, 10, 170, 570, 780, 950, 1420, 2380, 2730, 3850, 4120, 4300, 5850, 6360, 6460, 6800, 6970, 7100, 7240, 8720, 9630, 10150, 10580, 11010, 11170, 11830, 12300, 14290, 16330, 17670, 17810, 17850, 17860, 18940, 19030, 20500, 21930, 23960, 24490, 25830, 26050
Offset: 1

Views

Author

Michel Lagneau, Nov 19 2018

Keywords

Comments

Subsequence of A005574.
For n>1, a(n) == 0 (mod 10).
The corresponding pairs of semiprimes ((m-1)^2+1, (m+1)^2+1) are of the form (2p, 2q) with p, q primes == 1 (mod 10). So, a(n) = (q - p)/2 and a(n)^2 + 1 = p + q - 1.

Examples

			10 is in the sequence because 10^2 + 1 = 101 is prime, and 9^2 + 1 = 2*41, 11^2 + 1 = 2*61 are semiprimes.

Links

Crossrefs

Cf. A005574, A085722.

Programs

  • Mathematica
    Select[Range[50000],PrimeOmega[(#-1)^2+1]==2&&PrimeQ[#^2+1]&&PrimeOmega[(#+1)^2+1]==2&]
Mean/@SequencePosition[Table[Which[PrimeQ[m^2+1],1,PrimeOmega[m^2+1]==2,2,True,0],{m,30000}],{2,1,2}] (* Requires Mathematica version 10 or later *)  (* Harvey P. Dale, Sep 04 2019 *)
  • PARI
    isok(m) = isprime(m^2+1) && (bigomega((m-1)^2+1) == 2) && (bigomega((m+1)^2+1) == 2); \\ Michel Marcus, Nov 20 2018
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