A200070 -id:A200070 - cp's OEIS frontend

A200071 Numbers n such that the sum of the prime distinct divisors of n^2+1 equals 2 times the difference between the largest and the smallest prime divisor.

447, 2042, 4942, 8673, 17232, 18321, 38232, 52953, 54468, 54974, 55174, 57229, 66567, 71132, 83071, 101499, 113667, 121206, 133047, 173932, 297907, 325286, 430353, 447131, 656079, 702969, 842151, 937313, 1061846, 1173886, 1613346, 1721094, 1754679, 1759310

Offset: 1

Michel Lagneau, Nov 13 2011

nonn

			447 is a term because the distinct prime divisors of 447^2 + 1 are 2, 5, 13, 29, 53 and their sum, 2 + 5 + 13 + 29 + 53 = 102, equals 2*(53 - 2).

Amiram Eldar, Table of n, a(n) for n = 1..500

Cf. A200070.

Select[Range[1800000],Plus@@(pl=First/@FactorInteger[#^2+1])/2==pl[[-1]]-pl[[1]]&]
spddQ[n_]:=Module[{fi=FactorInteger[n^2+1][[All,1]]},Total[fi] == 2*(Last[ fi]-First[fi])]; Select[Range[176*10^4],spddQ] (* Harvey P. Dale, Jan 12 2019 *)

A200090 Numbers k such that the sum of the distinct prime divisors of k equals three times the largest prime divisor of k.

Original entry on oeis.org

15015, 45045, 51051, 62985, 72930, 74613, 75075, 105105, 106590, 135135, 145860, 153153, 156009, 165165, 187473, 188955, 190190, 195195, 213180, 218790, 222870, 223839, 225225, 279565, 291720, 314925, 315315, 319770, 335478, 357357, 364650, 375375, 380380

Offset: 1

Michel Lagneau, Nov 13 2011

nonn

			15015 is a term because its distinct prime divisors are 3, 5, 7, 11, 13 and their sum 3 + 5 + 7 + 11 + 13 = 39 equals 3*13.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006530 (gpf), A008472 (sopf), A200070.

Select[Range[400000],Plus@@(pl=First/@FactorInteger[#])==3*pl[[-1]]&]

A324210 Squarefree numbers k such that the sum of the distinct prime factors of k is twice the difference between the largest and the smallest prime factors of k.

Original entry on oeis.org

110, 182, 374, 494, 782, 1334, 2294, 3182, 3854, 4982, 6254, 7905, 7917, 8174, 9782, 11534, 12765, 14774, 15810, 15834, 18705, 19982, 20757, 21614, 22330, 22454, 24182, 25530, 27265, 28210, 30381, 30597, 32637, 35894, 37410, 40205, 41181, 41514, 43005, 47414, 49210

Offset: 1

David A. Corneth, Apr 09 2019

nonn

This sequence is a primitive subsequence of A200070. If p|a(n) for some prime p then p*a(n) is in A200070.
From Robert Israel, Apr 09 2019: (Start)
All terms have at least three prime factors.
The number of prime factors is odd if and only if the term is even.
The terms with three prime factors are 2*A111192. (End)

			110 = 2 * 5 * 11 is squarefree. The minimal and maximal prime divisors of 110 are 2 and 11 respectively. Twice their difference is 2 * (11-2) = 18 which is also the sum of the distinct prime divisors of 110; 2 + 5 + 11 = 18.

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

Cf. A200070, A111192.

filter:= proc(n) local P;
if not numtheory:-issqrfree(n) then return false fi;
P:= numtheory:-factorset(n);
  convert(P, `+`) = 2*(max(P)-min(P))
end proc:
select(filter, [$1..50000]);# Robert Israel, Apr 09 2019

Select[Select[Range[2, 5*10^4], SquareFreeQ], Total@ # == 2 (Last@ # - First@ #) &@ FactorInteger[#][[All, 1]] &] (* Michael De Vlieger, Apr 11 2019 *)

is(n) = if(!issquarefree(n), return(0)); my(f=factor(n)[, 1]~); sum(i=1, #f, f[i])==2*(f[#f]-f[1])
forcomposite(c=1, 50000, if(is(c), print1(c, ", "))) \\ Felix Fröhlich, Apr 11 2019

cp's OEIS Frontend

A200071 Numbers n such that the sum of the prime distinct divisors of n^2+1 equals 2 times the difference between the largest and the smallest prime divisor.

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A200090 Numbers k such that the sum of the distinct prime divisors of k equals three times the largest prime divisor of k.

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Mathematica

A324210 Squarefree numbers k such that the sum of the distinct prime factors of k is twice the difference between the largest and the smallest prime factors of k.

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