A181177 Numbers k such that k^2 + 1 = p*q, p and q primes gives the sums p+q not unique.

22, 34, 46, 50, 28, 44, 76, 80, 100, 42, 114, 104, 136, 86, 254, 266, 274, 58, 106, 154, 194, 286, 334, 324, 456, 504, 516, 214, 374, 494, 526, 566, 140, 520, 78, 186, 354, 426, 530, 670, 366, 546, 726, 166, 614, 92, 220, 316, 620, 836, 844, 390, 414, 810

Offset

1,1

Comments

We obtain the subsets of numbers with the same sum of prime divisors: (22, 34, 46, 50), (28, 44, 76), (80, 100),... => sums 102, 162, 210, ...

Links

Table of n, a(n) for n=1..54.

Example

22^2 + 1 = 5*97 and 97 + 5 = 102;
34^2 + 1 = 13*89 and 13 + 89 = 102;
46^2 + 1 = 29*73 and 29 + 73 = 102;
50^2 + 1 = 41*61 and 41 + 61 = 102.

Maple

# storage of sums and indices
with(numtheory): nn:=350: T:=array(1..nn): U:=array(1..1000): B:=array(1..nn): V:=array(1..1000): k:=1: for p from 1 to 1000 do: n:=p^2+1: x:= factorset(n): n1:=nops(x): n2 :=bigomega(n): if n1=2 and n2 = 2 then zz:=x[1]+x[2]: T[k]:=zz: U[k]:=zz: V[k]:=p: k:=k+1: else fi: od:
# ascending sort
for i from 1 to k do: mini:=i :for j from i+1 to nn do: if T[j]< T[mini] then mini:=j: else fi: od: if mini<> i then x:=T[i]: T[i]:=T[mini]: T[mini]:=x: else fi: od:
# detection of identical sums
z:=1:for a from 1 to 52 do: if T[a]=T[a+1] or T[a]=T[a+1] or T[a]=T[a+2] or T[a]=T[a+3] or T[a]=T[a+4] or T[a]=T[a+5] then B[z]:=T[a]: z:=z+1: else fi: od: for d from 1 to 52 do: for e from 1 to 1000 do: if B[d]= U[e] then printf(`%d, `, V[e]): else fi: od: od:

Prog

(Sage)
def A181177_yield_vecs(max_sum):
    for s in [1..max_sum]:
        pqs = ((p, s-p) for p in prime_range(s//2) if is_prime(s-p) and (p != s-p))
        ss = list(sqrt(p*q-1) for p, q in pqs if is_square(p*q-1))
        if len(ss) > 1: yield ss
a = flatten(list(A181177_yield_vecs(2000))) # D. S. McNeil, Jan 27 2011

Crossrefs

Sequence in context: A103320 A306882 A125526 * A124317 A254691 A159518

Adjacent sequences: A181174 A181175 A181176 * A181178 A181179 A181180

Keyword

nonn

Author

Michel Lagneau, Jan 25 2011

Status

approved