A377808 Prime numbers p such that the set of integers E = {1,2,3,...,p} can be divided into two subsets of consecutive integers E1 = {1,2,...,N} and E2 = {N+1,N+2,...,p} such that phi(1+2+...+N) = phi(N+1,N+2+...+p) for some N.

2, 3, 5, 13, 47, 67, 73, 83, 89, 103, 107, 137, 163, 179, 211, 239, 317, 331, 337, 347, 349, 359, 373, 401, 433, 491, 557, 599, 641, 659, 661, 683, 701, 743, 769, 787, 811, 827, 857, 983, 1069, 1093, 1109, 1117, 1123, 1181, 1217, 1259, 1279, 1289, 1303, 1361, 1429

Offset

1,1

Comments

Conjecture: the sequence is infinite.

Links

Robert Israel, Table of n, a(n) for n = 1..1000

Example

+----+-----------------------------------------------+---------+
|term|  subsets E1 and E2  | s1 = 1+2+...+N          | phi(s1) |
|    |                     | s2 = N+1,N+2+...+m      | phi(s2) |
+----+---------------------+-------------------------+---------+
|  3 | E1 = {1,2}          | s1 =  1+2 = 3           |    2    |
|    | E2 = {3}            | s2 =  3                 |    2    |
+----+---------------------+-------------------------+---------+
|  5 | E1 = {1,2,3,4}      | s1 = 1+2+3+4 = 10       |    4    |
|    | E2 = {5}            | s2 = 5 phi(5) = 4       |    4    |
+----+---------------------+-------------------------+---------+
| 13 | E1 = {1,2,3,...,11} | s1 = 1+2+...+11 = 66    |   20    |
|    | E2 = {12,13}        | s2 = 13+12 = 25         |   20    |
+----+---------------------+-------------------------+---------+
| 47 | E1 = {1,2,3,...,32} | s1 = 1+2+...+32 = 528   |  160    |
|    | E2 = {33,34,...,47} | s2 = 33+34+…+47 = 600   |  160    |
+----+---------------------+-------------------------+---------+
| 67 | E1 = {1,2,3,...,51} | s1 = 1+2+...+51 = 1326  |  384    |
|    | E2 = {52,53,...,67} | s2 = 52+53+...+67 =952  |  384    |
+----+---------------------+-------------------------+---------+
| 73 | E1 = {1,2,3,...,37} | s1 = 1+2+...+37 = 703   |  648    |
|    | E2 = {38,39,...,73} | s2 = 38+39+...+73 =1998 |  648    |
+----+---------------------+-------------------------+---------+

Maple

with(numtheory):for m from 1 to 240 do:
 ii:=0:p:=ithprime(m):
  for j from 1 to p while(ii=0) do:
    s1:=sum('i', 'i'=1..j):s2:=sum('i', 'i'=j+1..p):
     if phi(s1)=phi(s2) then ii:=i:printf(`%d, `, p):
     else fi:
  od:
od:

Prog

(PARI) isok(k) = if (isprime(k), for (i=2, k, if (eulerphi(sum(j=1, i-1, j)) == eulerphi(sum(j=i, k, j)), return(1)))); \\ Michel Marcus, Nov 08 2024

Crossrefs

Cf. A000010.

Sequence in context: A332060 A048634 A012899 * A215102 A110364 A111288

Adjacent sequences: A377805 A377806 A377807 * A377809 A377810 A377811

Keyword

nonn

Author

Michel Lagneau, Nov 08 2024

Status

approved