The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A295265 Numbers m such that sum of its i first divisors equals the sum of its j first non-divisors for some i, j. 1
4, 8, 10, 13, 14, 16, 19, 20, 21, 22, 26, 28, 30, 32, 34, 38, 39, 40, 43, 44, 46, 50, 52, 53, 56, 58, 60, 62, 63, 64, 68, 70, 72, 74, 76, 80, 82, 86, 88, 89, 90, 92, 94, 98, 99, 100, 103, 104, 106, 110, 111, 112, 116, 117, 118, 122, 124, 128, 130, 132, 134, 135 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Or numbers m such that Sum_{k=1..i} d(k) = Sum_{k=1..j} nd(k) for some i, j where d(k) are the i first divisors and nd(k) the j non-divisors of m.
The corresponding sums are 3, 3, 3, 14, 3, 3, 20, 3, 11, 3, 3, (3 or 14), 11, 3, 3, 3, 17, 3, 44, 3, 3, 3, 3, 54, 3, 3, 15, 3, 11, 3, 3, 3, 33, 3, 3, 3, ... containing the set of primes {3, 11, 17, 23, 29, 37, 41, 43, 53, 59, 61, 71, 79, ...}.
The equality Sum_{k=1..i} d(k) = Sum_{k=1..j} nd(k) is not always unique, for instance for a(12) = 28, we find 1 + 2 = 3 and 1 + 2 + 4 + 7 = 3 + 5 + 6 = 14.
The primes of the sequence are 13, 19, 43, 53, 89, 103, 151, 229, 251, 349, 433, ... (primes of the form k(k+1)/2 - 2; see A124199).
+-----+-----+-----+------+-----------------------------------------+
| n | i | j | a(n) | Sum_{k=1..i} d(k) = Sum_{k=1..j} nd(k) |
+-----+-----+-----+------+-----------------------------------------+
| 1 | 2 | 1 | 4 | 1 + 2 = 3 |
| 2 | 2 | 1 | 8 | 1 + 2 = 3 |
| 3 | 2 | 1 | 10 | 1 + 2 = 3 |
| 4 | 2 | 4 | 13 | 1 + 13 = 2 + 3 + 4 + 5 = 14 |
| 5 | 2 | 1 | 14 | 1 + 2 = 3 |
| 6 | 2 | 1 | 16 | 1 + 2 = 3 |
| 7 | 2 | 5 | 19 | 1 + 19 = 2 + 3 + 4 + 5 + 6 = 20 |
| 8 | 2 | 1 | 20 | 1 + 2 = 3 |
| 9 | 3 | 3 | 21 | 1 + 3 + 7 = 2 + 4 + 5 = 11 |
| 10 | 2 | 1 | 22 | 1 + 2 = 3 |
| 11 | 2 | 1 | 26 | 1 + 2 = 3 |
| 12 | 2 | 1 | 28 | 1 + 2 = 3 |
| | 4 | 3 | 28 | 1 + 2 + 4 + 7 = 3 + 5 + 6 = 14 |
| 13 | 4 | 2 | 30 | 1 + 2 + 3 + 5 = 4 + 7 = 11 |
| 14 | 2 | 1 | 32 | 1 + 2 = 3 | (End)
LINKS
EXAMPLE
30 is in the sequence because d(1) + d(2) + d(3) + d(4) = 1 + 2 + 3 + 5 = 11 and nd(1) + nd(2) = 4 + 7 = 11.
MAPLE
with(numtheory):nn:=300:
for n from 1 to nn do:
d:=divisors(n):n0:=nops(d):lst:={}:ii:=0:
for i from 1 to n do:
lst:=lst union {i}:
od:
lst:=lst minus d:n1:=nops(lst):
for m from 1 to n0 while(ii=0) do:
s1:=sum(‘d[i]’, ‘i’=1..m):
for j from 1 to n1 while(ii=0) do:
s2:=sum(‘lst[i]’, ‘i’=1..j):
if s1=s2
then
ii:=1:printf(`%d, `, n):
else
fi:
od:
od:
od:
MATHEMATICA
fQ[n_] := Block[{d = Divisors@ n}, nd = nd = Complement[Range@ n, d]; Intersection[Accumulate@ d, Accumulate@ nd] != {}]; Select[ Range@135, fQ] (* Robert G. Wilson v, Mar 06 2018 *)
PROG
(PARI) isok(n) = {d = divisors(n); psd = vector(#d, k, sum(j=1, k, d[j])); nd = setminus([1..n], d); psnd = vector(#nd, k, sum(j=1, k, nd[j])); #setintersect(psd, psnd) != 0; } \\ Michel Marcus, May 05 2018
CROSSREFS
Sequence in context: A238749 A310984 A346002 * A172153 A310985 A342734
KEYWORD
nonn
AUTHOR
Michel Lagneau, Feb 22 2018
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 18 13:40 EDT 2024. Contains 373481 sequences. (Running on oeis4.)