A085722 -id:A085722 - cp's OEIS frontend

A375390 Numbers k such that k^2 + 1, k^2 + 3 and k^2 + 5 are semiprimes.

44, 102, 104, 108, 152, 188, 226, 234, 296, 328, 426, 526, 586, 692, 720, 842, 846, 856, 926, 994, 1076, 1278, 1284, 1386, 1426, 1484, 1498, 1574, 1704, 1746, 1764, 1822, 1826, 1848, 1952, 2058, 2114, 2128, 2142, 2148, 2164, 2186, 2386, 2416, 2442, 2484, 2640, 2704, 2904, 2948, 3108, 3142, 3164

Offset: 1

Zak Seidov and Robert Israel, Aug 15 2024

nonn

All terms are even.

a(n)^2 + 3 or a(n)^2 + 5 is 3 times a prime. In the first case, a(n)/3 is in A111051.

			a(3) = 104 is a term because 104^2 + 1 = 10817 = 29 * 373, 104^2 + 3 = 10819 = 31 * 349 and 104^2 + 5 = 10821 = 3 * 3607 are all semiprimes.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001358, A111051. Intersection of A085722, A242331 and A242333.

select(t -> andmap(s -> numtheory:-bigomega(t^2+s)=2, [1,3,5]), 2*[$1..2000]);

Select[Range[3000], 2 == PrimeOmega[1 + #^2] == PrimeOmega[3 +
#^2] ==   PrimeOmega [5 + #^2] &]

A180507 Numbers k such that k^2 + 1 = p*q, p and q prime with p == q (mod k).

Original entry on oeis.org

3, 8, 12, 144, 1020, 8040, 13860, 34840, 729180, 1728240, 3232060, 17576520, 39279240, 85184880, 117649980, 778689840, 884737920, 1225045140, 1771563420, 3723878100, 3869896140, 4574299320, 7762395960, 12487172640, 14348911860, 14886940920, 21484957560, 24137574780

Offset: 1

Michel Lagneau, Jan 20 2011

nonn

q - p = k with k = 3, 8, 144.

The next terms with q - p = k are F(432) = 85738...5984 and F(570) where F(n) is the n-th Fibonacci number. All such terms are in A001906; the next such term, if one exists, has more than 25000 decimal digits. - Charles R Greathouse IV, Jan 21 2011

			a(3) = 12 because 12^2 + 1 = 5*29 and 29 - 5 = 2*12;
a(8) = 34840 because 34840^2 + 1 = 4289 * 283009 and 283009 - 4289 = 278720 = 8*34840.

Jinyuan Wang, Table of n, a(n) for n = 1..10000

Subset of A085722.

Cf. A001906, A002496, A005574, A027862, A134406, A134407.

with(numtheory):for k from 1 to 40000 do: x:=k^2+1:y:=factorset(x):yy:=bigomega(x):if
  yy=2 and irem(y[2],k) =y[1] then printf(`%d, `,k):else fi:od:

w(m, r) = Vec(x*(1-x)/(1-(m^2+2)*x+x^2) + O(x^r));
isok(s, t) = isprime(s) && isprime(s+t);
lista(nn) = {my(g, k, m=1, r, u=w(1, nn), v=List([])); for(i=2, r=#u, g=k=(u[i]+sqrtint(5*u[i]^2-4))/2; if(isok(u[i], k), listput(v, k))); while(r>2, u=w(m++, r); for(i=2, #u, k=(m*u[i]+sqrtint((m^2+4)*u[i]^2-4))/2; if(kJinyuan Wang, Mar 29 2020

More terms from Charles R Greathouse IV, Jan 24 2011

Missing terms inserted and more terms from Jinyuan Wang, Mar 30 2020

A186689 Numbers n such that n^4 + 1 is a semiprime.

Original entry on oeis.org

3, 5, 7, 8, 10, 11, 12, 13, 14, 17, 18, 21, 22, 23, 26, 29, 30, 32, 35, 36, 38, 39, 40, 42, 50, 52, 57, 58, 61, 62, 65, 68, 71, 72, 73, 78, 81, 84, 86, 92, 94, 98, 100, 102, 103, 105, 108, 112, 113, 114, 115, 116, 119, 120, 122, 124, 128, 129, 130, 138, 146, 148, 152, 153, 158

Offset: 1

Michel Lagneau, Feb 25 2011

nonn

Corresponding semiprimes n^4+1 are in A186688.

			3 is in the sequence because 3^4 + 1 = 82 = 2*41 is semiprime.

Robert Price, Table of n, a(n) for n = 1..1500

Cf. A001358, A006313, A085722, A103854, A104238, A104335, A105041, A105066, A105078, A105122, A105142, A105237, A104479, A104494, A104657, A105282, A186688.

SemiPrimeQ[ n_] := (n > 1) && (2 == Plus @@ (Transpose[FactorInteger[n]][[2]]));
  Select[Range[300], SemiPrimeQ[#^4 + 1] &]
Select[Range[200],PrimeOmega[#^4+1]==2&] (* Harvey P. Dale, Jan 27 2013 *)

A234693 Primes of the form n^2 + 1 such that (n - 1)^2 + 1 and (n + 1)^2 + 1 are semiprimes.

Original entry on oeis.org

17, 101, 28901, 324901, 608401, 902501, 2016401, 5664401, 7452901, 14822501, 16974401, 18490001, 34222501, 40449601, 41731601, 46240001, 48580901, 50410001, 52417601, 76038401, 92736901, 103022501, 111936401, 121220101, 124768901, 139948901, 151290001

Offset: 1

Michel Lagneau, Dec 29 2013

nonn

The corresponding n are 4, 10, 170, 570, 780, 950, 1420, 2380...

Property: n^2 + 1 = p + q - 1 and for a(n) > 17, a(n) == 1 mod 100.

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

Donovan Johnson and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from Johnson)

Cf. A002496, A144255, A085722.

with(numtheory):for n from 1 to 10^5 do:n1:=n^2+1:n2:=(n+1)^2+1:n3:=(n+2)^2+1: if type(n2,prime)=true and bigomega(n1)=2 and bigomega(n3)=2 then printf(`%d, `,n2):else fi:od:

forstep(n=4,1e5,2,if(isprime(n^2+1) && isprime(n^2/2-n+1) && isprime(n^2/2+n+1), print1(n^2+1", "))) \\ Charles R Greathouse IV, Dec 29 2013

A238947 Numbers k such that k^2 + 1 = p*q, p < q primes and q-p is a power of 2.

Original entry on oeis.org

8, 100, 3524, 5084, 36680, 77980, 21474824, 134201344, 148647496, 300741464, 73851531256, 153122539756, 778318386944, 6669171349484, 16526971109344, 596403262068016, 9376599920124524, 26698166963373164, 140144514160214876, 1613032378604451500

Offset: 1

Michel Lagneau, Mar 07 2014

Note that if n^2+1 = p*q, then p+q cannot be a power of 2. Proof by contradiction: There are two cases: p an odd prime and p=2. Case 1: suppose p and q are odd primes and q = 2^m-p. Then p(2^m-p) = n^2+1 for some even n. Rearranging terms, we obtain p*2^m-1 = p^2+n^2. Looking at this equation modulo 4, we obtain -1 = 1, a contradiction. Case 2: Let p=2. Then we obtain 2^(m+1)-n^2 = 5, which has no solutions in integers.

			8^2+1 = 65 = 5*13 and 13-5 = 2^3;
100^2+1 = 10001 = 73*137 and 137-73 = 2^6;
3524^2+1 = 12418577 = 3049*4073 and 4073-3049 = 2^10.

Giovanni Resta, Some terms greater than a(20)

Cf. A134406, A134407, A002522.

Subsequence of A085722.

with(numtheory):for a from 1 to 200000 do:p:=ithprime(a):for i from 1 to 50 do:q:=p+2^i:if type(q,prime)=true then x:=sqrt(p*q-1):if x=floor(x) then print(x):else fi:fi:od:od:

Select[Range[10^5],!PrimeQ[#^2+1]&&Plus@@Last/@FactorInteger[#^2+1]==2&&PrimeNu[#^2+1]==2&&IntegerQ[Log[2,FactorInteger[#^2+1][[2]][[1]]-FactorInteger[#^2+1][[1]][[1]]]]&]

isok(n) = (bigomega(n^2+1) == 2) && (f = factor(n^2+1)) && ((f[2, 1] - f[1, 1])== 2^(valuation(f[2, 1] - f[1, 1], 2))); \\ Michel Marcus, Mar 07 2014

a(7)-a(20) from Giovanni Resta, Mar 07 2014

A241975 Numbers n such that n^4 - n^3 - n - 1 is a semiprime.

Original entry on oeis.org

4, 6, 10, 14, 16, 20, 36, 40, 54, 56, 66, 84, 90, 94, 116, 126, 146, 150, 156, 160, 170, 204, 210, 260, 264, 306, 340, 350, 386, 396, 406, 420, 464, 474, 496, 570, 634, 674, 696, 700, 716, 740, 764, 780, 816, 826, 864, 890, 966, 1054, 1070, 1094, 1106, 1144

Offset: 1

Vincenzo Librandi, Aug 10 2014

Since n^4 - n^3 - n - 1 = (n^2 + 1)*(n^2 - n - 1), these are also numbers n such that n^2 + 1 and n^2 - n - 1 are both prime. Numbers in the intersection of A005574 and A002328. - Derek Orr, Aug 10 2014 [Sequence numbers corrected by Jens Kruse Andersen, Aug 11 2014]

Vincenzo Librandi, Table of n, a(n) for n = 1..400

Cf. A001358, A002327, A002496, A085722, A241670, A243436.

IsSemiprime:=func; [ n: n in [2..1500] | IsSemiprime(n^4 - n^3 - n - 1)];

Select[Range[2000], PrimeOmega[#^4 - #^3 - # - 1]==2 &]

for(n=1,10^4,if(isprime(n^2+1)&&isprime(n^2-n-1),print1(n,", "))) \\ Derek Orr, Aug 10 2014

A259036 Smallest divisor of n^2+1 >= sqrt(n^2+1).

Original entry on oeis.org

1, 2, 5, 5, 17, 13, 37, 10, 13, 41, 101, 61, 29, 17, 197, 113, 257, 29, 25, 181, 401, 26, 97, 53, 577, 313, 677, 73, 157, 421, 53, 37, 41, 109, 89, 613, 1297, 137, 85, 761, 1601, 58, 353, 50, 149, 1013, 73, 65, 461, 1201, 61, 1301, 541, 281, 2917, 89, 3137, 65

Offset: 0

Michel Lagneau, Jun 17 2015

Subsequence of A033677.
a(n) = n^2+1 if n^2+1 is prime (see A005574) or n=0. - Michel Marcus, Jul 01 2015
If n^2+1=p*q for primes p,q with pA085722), then a(n)=q. - Robert Israel, Dec 03 2019

			a(7) = 10 because 7^2+1 = 2*5*5 and 2*5 = 10 is the smallest divisor >=sqrt(7^2+1) = 7.0710678118...

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A002522, A005574, A033677, A085722.

[Min([d:d in Divisors(k^2+1)|d ge Sqrt(k^2+1) ]):k in [0..60]]; // Marius A. Burtea, Dec 03 2019

f:= proc(n) local m,k;
  m:= n^2+1;
  min(select(t -> t^2 >= m, numtheory:-divisors(m)))
end proc:
map(f, [$0..100]); # Robert Israel, Dec 03 2019

Table[Select[Divisors[n^2+1], # >= Sqrt[n^2+1] &, 1] // First, {n, 80}]

concat(1,vector(100,n,d=divisors(n^2+1);k=1;while(d[k]Derek Orr, Jun 27 2015

a(n) = A033677(A002522(n)).

A261546 Numbers k such that the five numbers k^2+1, (k+1)^2+1, ..., (k+4)^2+1 are all semiprime.

Original entry on oeis.org

48, 58, 1688, 2948, 28338, 36998, 38648, 96248, 100308, 133458, 136798, 187538, 207088, 224508, 253808, 309738, 375348, 545048, 598348, 607688, 659548, 672398, 793958, 1055648, 1055688, 1140008, 1270408, 1317808, 1388398, 1399098, 1529488, 1597008, 1655338

Offset: 1

Michel Lagneau, Aug 24 2015

a(n) == 8 (mod 10).

a(15017) > 10^10. - Hiroaki Yamanouchi, Oct 03 2015

			48 is in the sequence because of these five semiprimes:
48^2+1 = 2305 = 5*461;
49^2+1 = 2402 = 2*1201;
50^2+1 = 2501 = 41*61;
51^2+1 = 2602 = 2*1301;
52^2+1 = 2705 = 5*541.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..15016

Subsequence of A085722.

Cf. A001358, A144255.

IsSemiprime:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ n: n in [1..3*10^5] | IsSemiprime(n^2+1) and IsSemiprime(n^2+2*n+2)and IsSemiprime(n^2+4*n+5)and IsSemiprime(n^2+6*n+10)and IsSemiprime(n^2+8*n+17)]; // Vincenzo Librandi, Aug 24 2015

with(numtheory):
  n:=5:
  for k from 1 to 10^6 do:
    jj:=0:
    for m from 0 to n-1 do:
       x:=(k+m)^2+1:d0:=bigomega(x):
       if d0=2
       then
       jj:=jj+1:
       else
       fi:
     od:
        if jj=n
        then
        printf(`%d, `,k):
        else
        fi:
    od:

PrimeFactorExponentsAdded[n_]:=Plus@@Flatten[Table[#[[2]], {1}]&/@FactorInteger[n]]; Select[Range[2 10^5], PrimeFactorExponentsAdded[#^2+1] == PrimeFactorExponentsAdded[#^2 + 2 # + 2]== PrimeFactorExponentsAdded[#^2 + 4 # + 5]== PrimeFactorExponentsAdded[#^2 + 6 # + 10]== PrimeFactorExponentsAdded[#^2 + 8 # + 17] == 2 &] (* Vincenzo Librandi, Aug 24 2015 *)

has(n) = bigomega(n^2+1)==2;
isok(n) = has(n) && has(n+1) && has(n+2) && has(n+3) && has(n+4); \\ Michel Marcus, Aug 24 2015

a261546() = {
  nterm = 0;
  for (i = 0, 10^9,
    if (isprime(20*i*i + 32*i + 13) &&
      isprime(50*i*i + 90*i + 41) &&
      isprime(50*i*i + 110*i + 61) &&
      isprime(20*i*i + 48*i + 29) &&
      bigomega(100*i*i + 200*i + 101) == 2,
      nterm += 1;
      print(nterm, " ", 10 * i + 8);
    );
  );
} \\ - Hiroaki Yamanouchi, Oct 03 2015

issemi(n)=forprime(p=2,97, if(n%p==0, return(isprime(n/p)))); bigomega(n)==2
list(lim)=my(v=List()); forstep(k=48,lim,[10,30,10], if(issemi(k^2+1) && issemi((k+1)^2+1) && issemi((k+3)^2+1) && issemi((k+4)^2+1) && issemi((k+2)^2+1), listput(v,k))); Vec(v) \\ Charles R Greathouse IV, Jul 06 2017

a(18)-a(33) from Hiroaki Yamanouchi, Oct 03 2015

A261803 a(n) is the smallest number satisfying a(n)^2+1 = p(n)*q(n), p(n) < q(n) both prime, such that q(n+1)/p(n+1) < q(n)/p(n) with the initial condition q(1)/p(1) < 3/2.

Original entry on oeis.org

50, 334, 516, 670, 844, 1164, 1250, 1800, 2450, 9800, 14450, 20000, 24200, 101250, 105800, 135200, 162450, 168200, 204800, 304200, 336200, 451250, 480200, 490050, 530450, 696200, 924800, 966050, 1008200, 1125000, 1155200, 1428050, 1805000, 2332800, 2420000

Offset: 1

Michel Lagneau, Sep 01 2015

The sequence is probably infinite. [This follows from Schinzel's hypothesis H, for example. - Charles R Greathouse IV, Aug 31 2021]

A majority of numbers in the sequence are of the form 2*q^2 with q = 5, 25, 30, 35, 70, 85, 100, 110, 225, 230, 260, 285, 290, 320, 390, 410, ... So, it seems that {a(n)} = {334, 516, 670, 844, 1164} union {2*A109306(n)^2} where A109306 are the numbers k such that k^2 + (k-1)^2 and k^2 + (k+1)^2 are both primes.

			a(1) = 50 because 50^2+1 = 41*61 => 61/41 = 1.4878... < 1.5
a(2) = 334 because 334^2+1 = 281*397 => 397/281 = 1.4128... < 1.4878...
a(3) = 516 because 516^2+1 = 449*593 => 593/449 = 1.3207... < 1.4128...
a(4) = 670 because 670^2+1 = 593*757 => 757/593 = 1.2765... < 1.3207...

Jean-François Alcover, Table of n, a(n) for n = 1..500

Cf. A002496, A109306, A206328.

Subsequence of A085722.

with(numtheory):nn:=100:d:=1.5:
for n from 1 to nn do:
  x:=factorset(n^2+1):n0:=bigomega(n^2+1):
   if n0=2
   then
   q:=evalf(x[2]/x[1]):
   if q

(* Assumption: n>7 ==> a(n)=0 mod 50 *) a[n_] := a[n] = For[k = Which[n==1, 0, n <= 7, a[n-1]+1, True, a[n-1] + 50], True, k = Which[n <= 7, k+1, k == a[n-1]+1, k+49, True, k+50], f = FactorInteger[k^2+1]; If[Length[f] == 2, If[f[[All, 2]] == {1, 1}, {p1, q1} = f[[All, 1]]; If[q1/p1 < If[n == 1, 3/2, q[n-1]/p[n-1]], p[n] = p1; q[n] = q1; Return[k]]]]]; Table[Print["a(", n, ") = ", a[n], "  p = ", p[n], "  q = ", q[n],  "  q/p = ", N[q[n]/p[n], 10], "  q-p = ", q[n]-p[n]]; a[n], {n, 1, 100}] (* Jean-François Alcover, Sep 28 2015 *)

A333635 Numbers m such that m^2 + 1 has at most 2 prime factors.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 19, 20, 22, 24, 25, 26, 28, 29, 30, 34, 35, 36, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 56, 58, 59, 60, 61, 62, 64, 65, 66, 69, 71, 74, 76, 78, 79, 80, 84, 85, 86, 88, 90, 92, 94, 95, 96, 100

Offset: 1

Bernard Schott, Mar 30 2020

nonn

Equivalently, numbers m such that m^2 + 1 is prime or semiprime.
Henryk Iwaniec proved in 1978 that this sequence is infinite (see link). By contrast, it is not known whether there are infinitely many primes of the form m^2 + 1 (or infinitely many semiprimes of that form).
The integers that have at most 2 prime factors counted with multiplicity are called almost-primes of order 2 and they are in A037143. Here, as m^2 + 1 is not a square for m > 0, all the semiprimes of this form have two distinct prime factors (A144255), and with the primes of the form m^2 + 1 (A002496), they constitute A248742.

			10^2 + 1 = 101, which is prime, so 10 is in the sequence.
11^2 + 1 = 122 = 2 * 61, so 11 is in the sequence.
12^2 + 1 = 145 = 5 * 29, so 12 is in the sequence.
13^2 + 1 = 170 = 2 * 5 * 17, so 13 is not in the sequence.

R. K. Guy, Unsolved Problems in Number Theory, A1.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Henryk Iwaniec, Almost-primes represented by quadratic polynomials, Inventiones Mathematicae 47 (2) (1978), pp. 171-188.

Union of A005574 and A085722.
Cf. A002496 (m^2 + 1 is prime), A005574 (corresponding m).
Cf. A144255 (m^2 + 1 is semiprime), A085722 (corresponding m).
Cf. A248742 (m^2 + 1 is prime or semiprime), this sequence (corresponding m).
Cf. A037143 (numbers with at most 2 prime factors counted with multiplicity).

Select[Range[100], PrimeQ[(k = #^2 + 1)] || PrimeOmega[k] == 2 &]  (* Amiram Eldar, Mar 30 2020 *)
Select[Range[100],PrimeOmega[#^2+1]<3&] (* Harvey P. Dale, Aug 08 2025 *)

cp's OEIS Frontend

A375390 Numbers k such that k^2 + 1, k^2 + 3 and k^2 + 5 are semiprimes.

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A180507 Numbers k such that k^2 + 1 = p*q, p and q prime with p == q (mod k).

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A186689 Numbers n such that n^4 + 1 is a semiprime.

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A234693 Primes of the form n^2 + 1 such that (n - 1)^2 + 1 and (n + 1)^2 + 1 are semiprimes.

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A238947 Numbers k such that k^2 + 1 = p*q, p < q primes and q-p is a power of 2.

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A241975 Numbers n such that n^4 - n^3 - n - 1 is a semiprime.

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A259036 Smallest divisor of n^2+1 >= sqrt(n^2+1).

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