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 1-6 of 6 results.

A200072 Numbers k such that the sum of the prime distinct divisors of k^2+1 equals 2 times the largest prime divisor of k^2+1.

Original entry on oeis.org

1123, 1143, 6235, 8457, 11565, 21917, 22857, 33285, 41319, 58195, 119571, 124723, 128363, 173922, 178703, 188115, 243939, 280158, 308859, 309709, 409485, 430581, 565571, 703845, 961237, 1153362, 1170291, 1327998, 1409794, 1536651, 1586195, 1649395, 1665868
Offset: 1

Views

Author

Michel Lagneau, Nov 13 2011

Keywords

Examples

			1123 is in the sequence because the distinct prime divisors of 1123^2 + 1 are 2, 5, 13, 89, 109 and the sum 2 + 5 + 13 + 89 + 109 = 218 = 2*109.

Links

Crossrefs

Cf. A002522, A014442, A193462, A200071.

Programs

  • Mathematica
    Select[Range[1700000],Plus@@(pl=First/@FactorInteger[#^2+1])==2*pl[[-1]]&]

A212710 Smallest number k such that the difference between the greatest prime divisor of k^2+1 and the sum of the other prime distinct divisors equals n.

Original entry on oeis.org

411, 1, 3, 447, 2, 57, 212, 8, 307, 13, 5, 38, 319, 99, 3310, 70, 4, 242, 132, 50, 73, 17, 192, 12, 133, 3532, 41, 22231, 999, 43, 172, 68, 83, 11878, 294, 30, 6, 111, 9, 776, 2059, 922, 818, 46, 1183, 23, 216, 182, 557, 2010, 1818, 3323, 945, 512, 568, 76
Offset: 1

Views

Author

Michel Lagneau, May 24 2012

Keywords

Examples

			a(1) = 411 because 411^2+1 = 2 * 13 * 73 * 89  and 89 - (2 + 13 + 73) = 89 - 88 = 1.

Crossrefs

Cf. A182011, A014442, A193462.

Programs

  • Maple
    A212710 := proc(n)
    local fs,gpf,opf,k ;
    for k from 1 do
        fs := numtheory[factorset](k^2+1) ;
        gpf := max(op(fs)) ;
        opf := add( f,f=fs)-gpf ;
        if gpf-opf = n then
            return k;
        end if;
    end do:
end proc:
seq(A212710(n),n=1..50) ; # R. J. Mathar, Nov 14 2014
  • Mathematica
    lst={};Do[k=1;[While[!2*FactorInteger[k^2+1][[-1,1]]-Total[Transpose[FactorInteger[k^2+1]][[1]]]==n,k++]];AppendTo[lst,k],{n,0,60}];lst (* Michel Lagneau, Oct 28 2014 *)
  • PARI
    a(n) = {k = 1; ok = 0; while (!ok, f = factor(k^2+1); nbp = #f~; ok = (f[nbp, 1] - sum(i=1, nbp-1, f[i,1]) == n); if (!ok, k++);); k;} \\ Michel Marcus, Nov 09 2014

A216896 n - (sum of prime factors of n^2+1) is a positive square.

Original entry on oeis.org

38, 133, 172, 253, 460, 477, 580, 612, 717, 996, 1057, 1568, 1641, 2244, 2820, 3193, 3253, 3652, 3848, 4284, 4733, 4900, 4908, 5063, 5380, 6396, 7220, 8712, 9245, 9972, 10061, 10181, 10723, 11316, 11492, 12488, 12549, 12567, 13439, 14063, 14597, 15660, 15683
Offset: 1

Views

Author

Michel Lagneau, Sep 19 2012

Keywords

Examples

			38 is in the sequence because the prime divisors of 38^2 + 1 = 1445 are {5, 17}, and 38 - (5+17) = 16 is square.

Links

Crossrefs

Cf. A193462.

Programs

  • Maple
    with(numtheory): for n from 1 to 2500 do:x:=n^2+1:y:=factorset(x):n1:=nops(y): s:=sum('y[i] ', 'i'=1..n1):z:=n-s:if n> s and sqrt(z)=floor(sqrt(z)) then printf(`%d, `, n): else fi:od:
  • Mathematica
    aQ[n_] := (s = n - Plus @@ First @ Transpose @ FactorInteger[n^2+1]) > 0 && IntegerQ @ Sqrt @ s; Select[Range[16000], aQ] (* Amiram Eldar, Sep 09 2019 *)

A234646 Sum of the distinct prime divisors of n^3 + 1.

Original entry on oeis.org

0, 2, 3, 9, 18, 12, 38, 45, 22, 80, 31, 42, 39, 166, 69, 213, 258, 25, 326, 14, 137, 434, 486, 18, 91, 616, 41, 65, 786, 281, 111, 28, 345, 177, 1135, 402, 147, 95, 90, 1490, 271, 559, 1766, 165, 639, 315, 175, 115, 105, 201, 82, 2566, 439, 924, 432, 2980, 114
Offset: 0

Views

Author

Vincenzo Librandi, Jan 01 2014

Keywords

Examples

			a(3) = 9 because 3^3+1 = 28 and the sum of the 2 distinct prime divisors {2, 7} is 9.

Links

Crossrefs

Cf. A193462.

Programs

  • Mathematica
    Join[{0}, Table[Total[Transpose[FactorInteger[n^3 + 1]][[1]]], {n, 60}]]

A272175 Least number k such that (k^2+1) mod s = prime(n) where s is the sum of the distinct primes dividing k^2+1, or 0 if no such k exists.

Original entry on oeis.org

13, 3, 68, 182, 5, 32, 191, 333, 73, 70, 1068, 132, 507, 173, 774, 50, 11, 30, 1553, 3990, 338, 2307, 246, 2917, 1228, 80, 14369, 76, 114, 1590, 2529, 100, 28, 4952, 82, 659, 948, 7083, 2190, 8938, 19, 489, 11393, 1968, 2941, 21124, 3549, 1725, 64, 1382, 2540
Offset: 1

Views

Author

Michel Lagneau, Apr 28 2016

Keywords

Comments

Conjecture: a(n)> 0 for all n > 0.
The primes in the sequence are 3, 5, 11, 13, 19, 29, 73, 173, 191,...
The squares in the sequence are 25, 64, 100,...

Examples

			a(1)=13 because 13^2+1 = 170 = 2*5*17 => 170 mod(2+5+17) = 170 mod 24 = 2 = prime(1).

Crossrefs

Cf. A005574, A193462, A262965.

Programs

  • Mathematica
    Table[k=0;While[Mod[k^2+1,Plus@@First[Transpose[FactorInteger[k^2+1]]]]!=Prime[n],k++];k, {n,50}]
  • PARI
    a(n) = {k = 1; while ((m=k^2+1) && (lift(Mod(m, vecsum(factor(m)[,1]))) != prime(n)) , k++); k;} \\ Michel Marcus, Apr 29 2016

A358704 Numbers m such that the sum of the prime divisors and the sum of the nonprime divisors of m^2+1 are both prime.

Original entry on oeis.org

3, 9, 172, 309, 327, 392, 473, 483, 557, 578, 633, 657, 693, 699, 747, 767, 819, 820, 829, 909, 911, 1007, 1013, 1028, 1030, 1057, 1084, 1141, 1157, 1186, 1252, 1308, 1311, 1382, 1577, 1585, 1620, 1682, 1721, 1722, 1727, 1749, 1841, 1849, 1874, 1972, 2019, 2134
Offset: 1

Views

Author

Michel Lagneau, Nov 27 2022

Keywords

Comments

The primes of the sequence are 3, 557, 829, 911, 1013, 1721, ...
The corresponding pairs (p, q) = (sum of prime divisors, sum of nonprime divisors) are (7, 11), (43, 83), (163, 36293), ...
There is a subsequence {b(n)} = {3, 9, 309, 699, 819, ...} such that q/p < 2 (see the following table).
Conjecture: when b(n) tends to infinity, q/p tends to 2.
+------+----------+----------+-------------+
| b(n) | p | q | q/p |
+------+----------+----------+-------------+
| 3 | 7 | 11 | 1.571428571 |
| 9 | 43 | 83 | 1.930232558 |
| 309 | 47743 | 95483 | 1.999937164 |
| 699 | 244303 | 488603 | 1.999987720 |
| 819 | 335383 | 670763 | 1.999991055 |
| 909 | 413143 | 826283 | 1.999992739 |
| 1311 | 859363 | 1718723 | 1.999996509 |
| 1749 | 1529503 | 3059003 | 1.999998039 |
| 3201 | 5123203 | 10246403 | 1.999999414 |
| 4809 | 11563243 | 23126483 | 1.999999741 |
............................................

Examples

			3 is in the sequence because the divisors of 3^2 + 1 = 10 are {1, 2, 5, 10} and 1 + 10 = 11 and 2 + 5 = 7 are prime numbers.

Crossrefs

Cf. A193462, A194039, A194594.

Programs

  • Mathematica
    f[n_]:=Plus@@Select[Divisors[n^2+1], !PrimeQ[#]&]; g[n_]:=Plus@@First/@FactorInteger[n^2+1]; Select[Range[2200], PrimeQ[f[#]&&PrimeQ[g[#]]]&]
  • PARI
    isok(m) = my(f=factor(m^2+1), sp=vecsum(f[, 1])); isprime(sp) && isprime(sigma(f)-sp); \\ Michel Marcus, Nov 28 2022
Showing 1-6 of 6 results.

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