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 11-17 of 17 results.

A199924 Numbers k such that the sum of the largest and the smallest prime divisor of k^2 + 1 equals the sum of the other distinct prime divisors.

Original entry on oeis.org

948, 1560, 1772, 2153, 2697, 8487, 11293, 12553, 13236, 18065, 32247, 36984, 40452, 43999, 55945, 94536, 100512, 107607, 127224, 134223, 214641, 218783, 366937, 425808, 429855, 595471, 620865, 645327, 757382, 850416, 875784, 1241106, 1330849, 1363977, 1387689
Offset: 1

Views

Author

Michel Lagneau, Nov 12 2011

Keywords

Comments

Generalization of A192770 and A192771.

Examples

			2697 is in the sequence because 2697^2 + 1 = 7273810 has five distinct divisors  2, 5, 41, 113, 157 and 157 + 2 = 5 + 41 + 113 = 159.

Links

Crossrefs

Cf. A180278, A192771, A192770.

Programs

  • Mathematica
    Select[Range[1400000],Plus@@((pl=First/@FactorInteger[#^2+1])/2)==pl[[1]]+pl[[-1]]&](* program of Ray Chandler adapted for this sequence - see A199745 *)

A261529 Number k such that k^2 + 1 = p*q*r where p,q,r are distinct primes and the sum p+q+r is a perfect square.

Original entry on oeis.org

17, 37, 91, 235, 683, 1423, 1675, 2879, 8101, 9595, 13711, 18799, 19601, 21295, 25937, 30059, 32111, 36251, 39505, 41071, 49285, 60719, 79441, 90575, 93871, 94799, 103429, 112571, 132085, 136075, 144965, 180001, 180251, 188465, 189679
Offset: 1

Views

Author

Michel Lagneau, Aug 23 2015

Keywords

Comments

a(n) is odd. The prime numbers of the sequence are 17, 37, 683, 1423, 2879, 8101, 13711, 30059, 36251, 60719, 93871, 112571, 180001, ...

Examples

			17 is in the sequence because 17^2 + 1 = 2*5*29 and 2 + 5 + 29 = 6^2.

Crossrefs

Cf. A002522, A180278.

Programs

  • Maple
    with(numtheory):
for n from 1 to 200000 do:
  y:=factorset(n^2+1):n0:=nops(y):
   if n0=3 and bigomega(n^2+1)=3 and
   sqrt(y[1]+y[2]+y[3])=floor(sqrt(y[1]+y[2]+y[3]))
   then
   printf(`%d, `,n):
   else
   fi:
od:
  • PARI
    isok(n) = my(f = factor(n^2+1)); (#f~ == 3) && (vecmax(f[,2]) == 1) && issquare(vecsum(f[,1])); \\ Michel Marcus, Aug 24 2015

A138769 a(n) = least positive integer k such that k^2+3 is divisible by at least n distinct primes.

Original entry on oeis.org

1, 3, 9, 33, 201, 1125, 5259, 98481, 1176579, 4970985, 83471355, 607500315, 20298622815, 302065005093, 2979977447571, 46728566085441, 541457057096937, 13094093041014057, 231069516389617197, 5992213273680818217
Offset: 1

Views

Author

Emeric Deutsch, Apr 03 2008

Keywords

Comments

For n<=20, a(n)^2+3 happens to be divisible by exactly n distinct primes. - Max Alekseyev, Oct 10 2024

Examples

			a(3)=9 because 1^2+3=2*2, 2^2+3=7, 3^2+3=2*2*3, 4^2+3=19, 5^2+3=2*2*7, 6^2+3=3*13, 7^2+3=2*2*13, 8^2+3=67 have at most 2 distinct prime divisors, while 9^2+3=2*2*3*7 has 3 distinct prime divisors.

Links

  • Elgin H. Johnston, Problem 1792, Matematikolimpiyatokulu, Solution, 2009, p. 149. [broken link]
  • H. A. ShahAli, Problem 1792, Math. Magazine, vol. 81, No. 2, 2008, p. 155.

Crossrefs

Cf. A001221, A180278, A257366.

Programs

  • Maple
    n:=7: with(numtheory): for k while nops(factorset(k^2+3)) < n do end do: a[n]:=k; A[n]:=factorset(k^2+3); # yields a(7) as well as its 7 prime divisors; change the value of n to obtain other terms.
  • Mathematica
    a[n_] := Block[{k=1}, While[PrimeNu[k^2 + 3] != n, k++]; k]; Array[a, 8] (* Giovanni Resta, Nov 29 2019 *)

Extensions

a(11)-a(12) from Donovan Johnson, Aug 31 2008
a(13)-a(14) from Giovanni Resta, Nov 29 2019
a(15)-a(20) from Max Alekseyev, Oct 10 2024

A193164 a(1) = 1 ; for n > 1, a(n) is the smallest number such that a(n)^2 + 1 contains n distinct prime divisors dividing a(n+1)^2 + 1.

Original entry on oeis.org

1, 3, 13, 47, 463, 25683, 4187997
Offset: 1

Views

Author

Michel Lagneau, Jul 17 2011

Keywords

Comments

This sequence is not the same as A180278.

Examples

			a(1) = 1^1 + 1 = 2 ;
a(2) = 3^2 + 1 = 2*5 ;
a(3) = 13^2 + 1 = 2*5*17 ;
a(4) = 47^2 + 1 = 2*5*13*17 ;
a(5) = 463^2 + 1 = 2*5*13*17*97 ;
a(6) = 25683^2 + 1 = 2 * 5 * 13 * 17 ^ 2 * 97 * 181 ;
a(7) = 4187997^2 + 1 = 2 * 5 * 13 * 17 * 97 * 181 * 452033.

Crossrefs

Cf. A180278.

Programs

  • Maple
    with(numtheory):A:={2}:for n from 1 to 7 do:id:=0:for k from 1 to 4200000 (id=0) do:x:=k^2+1:y:=factorset(x):n1:=nops(y):if n1=n and A intersect y = A then A:=y:id:=1:printf ( "%d %d \n",n,k):else fi:od:od:

A356873 a(n) is the smallest number k such that 2^k+1 has at least n distinct prime factors.

Original entry on oeis.org

0, 5, 14, 18, 30, 42, 78, 78, 78, 90, 150, 150, 210, 210, 234, 234, 270, 390, 390, 390, 390, 450, 510, 630, 630, 630, 810, 810, 810, 966, 966, 1170, 1170, 1170, 1170, 1170, 1170, 1170
Offset: 1

Views

Author

Alex Ratushnyak, Sep 02 2022

Keywords

Comments

From Jon E. Schoenfield, Sep 04 2022: (Start)
a(39) <= a(40) <= a(41) <= 1530.
a(42) <= a(43) <= a(44) <= 1890.
a(45) <= a(46) <= 2070.
a(47) <= a(48) <= ... <= a(54) = 2730. (End)

Crossrefs

Cf. A046799, A071852.
Cf. A180278, A219108, A356872.

Programs

  • Mathematica
    a[n_] := Block[{k=0}, While[ Length@ FactorInteger[2^k + 1] < n, k++]; k]; Array[a, 12] (* Giovanni Resta, Oct 13 2022 *)
  • PARI
    a(n) = my(k=1); while (omega(2^k+1) < n, k++); k; \\ Michel Marcus, Sep 05 2022
  • Python
    from sympy import factorint, isprime
from itertools import count, islice
def f(n): return 1 if isprime(n) else len(factorint(n))
def agen():
    n = 1
    for k in count(0):
        v = f(2**k+1)
        while v >= n: yield k; n += 1
print(list(islice(agen(), 10))) # Michael S. Branicky, Sep 02 2022

Extensions

a(11)-a(38) from Michael S. Branicky, Sep 02 2022 using A071852

A365326 a(n) is the smallest positive number k such that k^2 - 1 and k^2 + 1 each have exactly n distinct prime divisors.

Original entry on oeis.org

2, 5, 13, 83, 463, 4217, 169333, 2273237, 23239523, 512974197, 5572561567
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Sep 01 2023

Keywords

Crossrefs

Cf. A001221, A180278, A219017.
Cf. A088075 (with k instead of k^2).

Programs

  • PARI
    isok(k, n) = (omega(k^2-1)==n) && (omega(k^2+1)==n);
a(n) = my(k=2); while (!isok(k, n), k++); k; \\ Michel Marcus, Sep 03 2023

Formula

a(n) >= max(A219017(n), A180278(n)). - Daniel Suteu, Sep 03 2023

Extensions

a(9)-a(11) from Amiram Eldar, Sep 03 2023

A380969 a(n) is the smallest k such that tau(k^2 + 1) is equal to 2^n, where tau = A000005 and a(n) = -1 if no such k exists.

Original entry on oeis.org

0, 1, 3, 13, 47, 307, 2163, 17557, 191807, 1413443, 16485763, 169053487
Offset: 0

Views

Author

Juri-Stepan Gerasimov, Feb 09 2025

Keywords

Crossrefs

Cf. A000005, A164511, A180278, A353008, A380798.

Programs

  • Mathematica
    a[n_]:=Module[{k=0}, While[DivisorSigma[0, k^2+1]!=2^n, k++]; k]; Array[a, 9,0] (* Stefano Spezia, Feb 10 2025 *)
  • PARI
    a(n) = my(k=0); while (numdiv(k^2+1) != 2^n, k++); k; \\ Michel Marcus, Feb 09 2025

Formula

a(n) = A353008(2^(n-1)) for n > 0.

Extensions

a(10)-a(11) from Stefano Spezia, Feb 12 2025
Previous Showing 11-17 of 17 results.

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