A377875 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.

174, 232, 282, 325, 376, 438, 462, 474, 539, 584, 606, 616, 632, 654, 678, 798, 808, 872, 904, 906, 931, 966, 978, 1002, 1064, 1074, 1075, 1105, 1127, 1182, 1208, 1288, 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

Offset

1,1

Comments

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.

Links

Antti Karttunen, Table of n, a(n) for n = 1..15275

Formula

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

Example

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.
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.
A276085(4823) = 614889782588493750 =  2 * 3 * 5^5 * 13 * 2522624749081, thus 4823 is included.
A276085(1104299) = 11231250 = 2 * 3 * 5^5 * 599, thus 1104299 is included.

Prog

(PARI)
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
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); };
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))));
isA377875(n) = ((A083345(n)%2) && !A377868(n));

Crossrefs

Setwise difference A369003 \ A377869.

Setwise difference A377873 \ A369002.

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

Sequence in context: A351375 A160660 A025390 * A045257 A025381 A168349

Adjacent sequences: A377872 A377873 A377874 * A377876 A377877 A377878

Keyword

nonn,new

Author

Antti Karttunen, Nov 11 2024

Status

approved