cp's OEIS Frontend

A364559 a(n) = A005941(n) - n.

%I A364559 #21 Aug 07 2023 18:26:57
%S A364559 0,0,0,0,0,0,2,0,-2,0,6,0,20,4,-4,0,48,-4,110,0,-2,12,234,0,-12,40,
%T A364559 -12,8,484,-8,994,0,2,96,-14,-8,2012,220,28,0,4056,-4,8150,24,-22,468,
%U A364559 16338,0,-24,-24,80,80,32716,-24,-18,16,202,968,65478,-16,131012,1988,-24,0,4,4,262078,192,446,-28,524218,-16
%N A364559 a(n) = A005941(n) - n.
%H A364559 Antti Karttunen, <a href="/A364559/b364559.txt">Table of n, a(n) for n = 1..10000</a>
%F A364559 a(n) = -A364499(A005941(n)).
%F A364559 a(n) = Sum_{d|n} A364558(d).
%e A364559 a(528581) = -4 as A005941(528581) = 528577 = 528581-4. Notably, 528581 = 17^2 * 31 * 59, with divisors [1, 17, 31, 59, 289, 527, 1003, 1829, 8959, 17051, 31093, 528581]. Applying A364557 to these divisors gives [1, 64, 1024, 65536, 128, 1024, 65536, 65536, 2048, 131072, 65536, 131072], while applying Euler totient phi (A000010) to them gives [1, 16, 30, 58, 272, 480, 928, 1740, 8160, 15776, 27840, 473280], their differences being [0, 48, 994, 65478, -144, 544, 64608, 63796, -6112, 115296, 37696, -342208], whose sum is -4.
%o A364559 (PARI)
%o A364559 A005941(n) = { my(f=factor(n), p, p2=1, res=0); for(i=1, #f~, p = 1 << (primepi(f[i, 1])-1); res += (p * p2 * (2^(f[i, 2])-1)); p2 <<= f[i, 2]); (1+res) }; \\ (After _David A. Corneth_'s program for A156552)
%o A364559 A364559(n) = (A005941(n)-n);
%o A364559 (Python)
%o A364559 from sympy import factorint, primepi
%o A364559 def A364559(n): return sum(1<<primepi(p)+i for i, p in enumerate(factorint(n, multiple=True),-1))+1-n # _Chai Wah Wu_, Jul 29 2023
%Y A364559 Cf. A005941, A364499, A364557, A364558 (Möbius transform).
%Y A364559 Cf. A029747 (known positions of 0's), A364560 (of terms <= 0), A364562 (of terms > 0), A364576.
%Y A364559 Cf. also A364288.
%K A364559 sign
%O A364559 1,7
%A A364559 _Antti Karttunen_, Jul 28 2023