A377875 - OEIS

%I #23 Nov 13 2024 08:28:28

%S 174,232,282,325,376,438,462,474,539,584,606,616,632,654,678,798,808,

%T 872,904,906,931,966,978,1002,1064,1074,1075,1105,1127,1182,1208,1288,

%U 1302,1304,1336,1398,1432,1506,1519,1576,1626,1662,1736,1755,1842,1864,1866,2008,2168,2216,2226,2340,2425,2442,2456,2488,2514

%N Numbers k for which A276085(k) is not a multiple of 4 and has at least one divisor of the form p^p, with p an odd prime, where A276085 is fully additive with a(p) = p#/p.

%C Among the initial 15275 terms, in 110 cases A276085(k) is a multiple of 5^5, and for no cases a multiple of 7^7.

%H Antti Karttunen, <a href="/A377875/b377875.txt">Table of n, a(n) for n = 1..15275</a>

%F {k such that A377868(k) < A377874(k)}.

%e A276085(55) = 216 = 2^3 * 3^3, which although it has a divisor of the form p^p, with p an odd prime, it is also a multiple of 4, and therefore 55 is NOT included in this sequence.

%e A276085(174) = 223092873 = 3^3 * 3 * 1063 * 2591, which has a divisor of the form p^p, with p an odd prime, thus 174 is included in this sequence.

%e A276085(4823) = 614889782588493750 =  2 * 3 * 5^5 * 13 * 2522624749081, thus 4823 is included.

%e A276085(1104299) = 11231250 = 2 * 3 * 5^5 * 599, thus 1104299 is included.

%o (PARI)

%o A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };

%o A276085(n) = { my(f = factor(n), pr=1, i=1, s=0); for(k=1, #f~, while(i <= primepi(f[k, 1])-1, pr *= prime(i); i++); s += f[k, 2]*pr); (s); };

%o A377868(n) = if(isprime(n), 1, my(x=A276085(n),pp); forprime(p=2,, pp = p^p; if(!(x%pp), return(0)); if(pp > x, return(1))));

%o isA377875(n) = ((A083345(n)%2) && !A377868(n));

%Y Setwise difference A369003 \ A377869.

%Y Setwise difference A377873 \ A369002.

%Y Cf. A083345, A276085, A377868, A377872, A377874.

%K nonn,new

%O 1,1

%A _Antti Karttunen_, Nov 11 2024