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 A257996 Let s0 and s1 be the sums of the reciprocals of the even and odd divisors of n, respectively. The sequence lists the numbers n such that 3*s0 - 2*s1 = 1. 0
 120, 1456, 121024, 2198352216064, 576458003527499776 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Let D0 = {d0(i)}, i = 1..p, the set of the p even divisors of a number n and D1 = {d1(n)}, j = 1..q the set of the q odd divisors of n. Then a(n) is the number such that 3*Sum_{i=1..p} 1/d0(i)- 2*Sum_{j=1..q} 1/d1(j) = 1. Property of the sequence: We observe that a(n) = 2^(k+1)*(2^k-1)*(2^(k+1) - 3) = (2*A000668(m) + 2)*A000668(m)*(2*A000668(m) - 1) where A000668(m) = 2^k - 1 is a Mersenne prime and (2*A000668(m)-1) = 2^(k+1)- 3 is also a prime number. The corresponding values of k are 2, 3, 5, 13, 19, ... and the corresponding values of m are 1, 2, 3, 5, 7, ... Generalization: It is possible to introduce general sequences of numbers such that a*s0 + b*s1 = c with very interesting properties for some integers a, b, c. Example 1: with (a, b, c) = (2, -1, 1) we find the sequence A064591 = 24, 112, 1984, 32512, ... (non-unitary perfect numbers). Example 2: with (a, b, c) = (2, -1, 0) we find the sequence A016825(n) = 2, 6, 10, 14, 18, 22, ... Example 3: with (a, b, c) = (1, 1, 2) we find the sequence A000396(n) = 6, 28, 496, 8128,... (perfect numbers). Example 4: with (a, b, c) = (4, -3, 1) we find the sequence 48, 224, 3968, 65024, ... = 2*A064591(n) = A000668(n)*2^p for some p where A000668 lists the Mersenne primes. Example 5: with (a, b, c) = (6, -5, 1) we find the sequence 240, 2912, 242048, ... which equals twice the sequence obtained with (a, b, c) = (3, -2, 1). Example 6: with (a, b, c) = (7, -6, 1) we find the sequence 2150, 13104, 24800, ... LINKS EXAMPLE 120 = 2^3*3*5 = (2*A000668(1)+2)* A000668(1)*(2*A000668(1)-1); 1456 = 2^4*7*13 = (2*A000668(2)+2)* A000668(2)*(2*A000668(2)-1); 121024 = 2^6*31*61 =(2*A000668(3)+2)* A000668(3)*(2*A000668(3)-1); 2198352216064 = 2^14*8191*16381= (2*A000668(5)+2)*A000668(5)*(2*A000668(5)-1); 576458003527499776 = 2^20*524287*1048573 = (2*A000668(7)+2)* A000668(7)*(2*A000668(7)-1). MAPLE with(numtheory):nn:=100000: for n from 2 by 2 to nn do :    x:=divisors(n):n0:=nops(x):s:=sum('x[i]', 'i'=1..n0):     s0:=0:s1:=0:     for k from 1 to n0 do:      if irem(x[k], 2)=0      then      s0:=s0+1/x[k]      else      s1:=s1+1/x[k]:      fi:     od:     if 3*s0-2*s1=1 then print(n):else fi:od: MATHEMATICA Do[s0=0; s1=0; Do[d=Divisors[n][[i]]; If[Mod[d, 2]==0, s0=s0+1/d, s1=s1+1/d], {i, 1, Length[Divisors[n]]}]; If[3*s0-2*s1==1, Print[n]], {n, 2, 10^9, 2}] PROG (PARI) siod(n) = sumdiv(n, d, (d%2)/d); seod(n) = sumdiv(n, d, (1-d%2)/d); isok(n) = 3*seod(n)-2*siod(n) == 1; \\ Michel Marcus, May 16 2015 CROSSREFS Cf. A000668. Sequence in context: A052776 A052770 A175112 * A183597 A027795 A223427 Adjacent sequences:  A257993 A257994 A257995 * A257997 A257998 A257999 KEYWORD nonn AUTHOR Michel Lagneau, May 16 2015 STATUS approved

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Last modified June 14 21:27 EDT 2021. Contains 345041 sequences. (Running on oeis4.)