A358936 Numbers k such that for some r we have phi(1) + ... + phi(k - 1) = phi(k + 1) + ... + phi(k + r), where phi(i) = A000010(i).

3, 4, 6, 38, 40, 88, 244, 578, 581, 602, 1663, 2196, 10327, 17358, 28133, 36163, 42299, 123556, 149788, 234900, 350210, 366321, 620478, 694950, 869880, 905807, 934286, 1907010, 2005592, 5026297, 7675637, 11492764, 12844691, 14400214, 15444216, 18798939, 20300872

Offset: 1

Author

Ctibor O. Zizka, Dec 07 2022

Keywords

• nonn

Comments

These numbers might be called "Euler totient function sequence balancing numbers", after the Behera and Panda link.

Numbers k such that A002088(k-1) + A002088(k) is a term of A002088.

Examples

			k = 3:
phi(1) + phi(2) = phi(4) = 2.
Thus the balancing number k = 3 is a term. The balancer r is 1.
k = 4:
phi(1) + phi(2) + phi(3) = phi(5) = 4.
Thus the balancing number k = 4 is a term. The balancer r is 1.
phi(i) = A000010(i).
		

Links

• Ctibor O. Zizka, Table of k < 10^9, Terms 38..45 from David A. Corneth.
• A. Behera and G. K. Panda, On the square roots of triangular numbers, The Fibonacci Quarterly, 37.2 (1999), 98-105.

Crossrefs

Cf. A000010, A001109, A002088, A064018.

Programs

Mathematica

With[{m = 30000},phi = EulerPhi[Range[m]]; s = Accumulate[phi]; Select[Range[2, m], MemberQ[s, 2*s[[#]] - phi[[#]]] &]] (* Amiram Eldar, Dec 07 2022 *)

PARI

upto(n) = {my(res = List(), lefttotal = 1, righttotal = 2, k = 2, nplusr = 3, sumf = 1, oldfk = 1); for(i = 1, n,  while(lefttotal > righttotal, nplusr++; righttotal+=f(nplusr) ); if(lefttotal == righttotal, listput(res, k)); lefttotal+=oldfk; k++; fk = f(k); righttotal-=fk; oldfk = fk ); res }
f(k) = eulerphi(k) \\ David A. Corneth, Dec 07 2022

Python

from sympy import totient as phi
from itertools import count, islice
def f(n): # function we wish to "balance"
    return phi(n)
def agen(): # generator of terms
    s, sset, i = [0, f(1), f(1)+f(2)], set(), 3
    for k in count(2):
        target = s[k-1] + s[k]
        while s[-1] < target:
            fi = f(i); nexts = s[-1] + fi; i += 1
            s.append(nexts); sset.add(nexts)
        if target in sset: yield k
print(list(islice(agen(), 17))) # Michael S. Branicky, Dec 07 2022

Extensions

a(8)-a(15) from Amiram Eldar, Dec 07 2022

a(16)-a(37) from Michael S. Branicky, Dec 07 2022