A358936 - OEIS

%I #31 Jan 05 2025 19:51:42

%S 3,4,6,38,40,88,244,578,581,602,1663,2196,10327,17358,28133,36163,

%T 42299,123556,149788,234900,350210,366321,620478,694950,869880,905807,

%U 934286,1907010,2005592,5026297,7675637,11492764,12844691,14400214,15444216,18798939,20300872

%N 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).

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

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

%H Ctibor O. Zizka, <a href="/A358936/b358936.txt">Table of k < 10^9</a>, Terms 38..45 from David A. Corneth.

%H A. Behera and G. K. Panda, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/37-2/behera.pdf">On the square roots of triangular numbers</a>, The Fibonacci Quarterly, 37.2 (1999), 98-105.

%e k = 3:

%e phi(1) + phi(2) = phi(4) = 2.

%e Thus the balancing number k = 3 is a term. The balancer r is 1.

%e k = 4:

%e phi(1) + phi(2) + phi(3) = phi(5) = 4.

%e Thus the balancing number k = 4 is a term. The balancer r is 1.

%e phi(i) = A000010(i).

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

%o (Python)

%o from sympy import totient as phi

%o from itertools import count, islice

%o def f(n): # function we wish to "balance"

%o     return phi(n)

%o def agen(): # generator of terms

%o     s, sset, i = [0, f(1), f(1)+f(2)], set(), 3

%o     for k in count(2):

%o         target = s[k-1] + s[k]

%o         while s[-1] < target:

%o             fi = f(i); nexts = s[-1] + fi; i += 1

%o             s.append(nexts); sset.add(nexts)

%o         if target in sset: yield k

%o print(list(islice(agen(), 17))) # _Michael S. Branicky_, Dec 07 2022

%o (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 }

%o f(k) = eulerphi(k) \\ _David A. Corneth_, Dec 07 2022

%Y Cf. A000010, A001109, A002088, A064018.

%K nonn

%O 1,1

%A _Ctibor O. Zizka_, Dec 07 2022

%E a(8)-a(15) from _Amiram Eldar_, Dec 07 2022

%E a(16)-a(37) from _Michael S. Branicky_, Dec 07 2022