login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Numbers that eventually reach zero when iterated with the arithmetic derivative (i.e., are in A099308), but some of their proper divisors will never reach it.
3

%I #11 Feb 11 2023 08:10:00

%S 30,70,78,105,126,130,138,150,165,174,182,222,238,246,255,258,266,273,

%T 282,285,286,306,310,315,318,333,338,342,345,350,357,366,369,370,375,

%U 385,390,399,402,414,426,430,442,455,465,474,483,490,494,495,498,510,518,530,546,549,550,555,561,570,574,575

%N Numbers that eventually reach zero when iterated with the arithmetic derivative (i.e., are in A099308), but some of their proper divisors will never reach it.

%C Numbers k for which A341999(k) is zero but A359542(k) is not zero.

%C Any such a nonreaching proper divisor must be one of the terms of A359547.

%e 30 = 2*3*5 is included in this sequence, as although it is in A099308, it is not included in A359544 because its proper divisor 15 is not in A099308. Note that 15 is a term of A359547.

%o (PARI)

%o A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i, 2]>=f[i, 1], return(0), s += f[i, 2]/f[i, 1])); (n*s));

%o A341999(n) = if(!n,n,while(n>1, n = A003415checked(n)); (!n));

%o A359542(n) = sumdiv(n,d,A341999(d));

%o isA359545(n) = ((0==A341999(n))&&(A359542(n)>0));

%Y Setwise difference A099308 \ A359544.

%Y Cf. A003415, A341999, A359542, A359547.

%K nonn

%O 1,1

%A _Antti Karttunen_, Jan 05 2023