A392601 - OEIS

%I #17 Jan 25 2026 17:26:16

%S 1,2,10,15,28,42,88,132,176,198,220,416,484,624,832,936,1040,1053,

%T 1092,1144,1170,1248,1300,1404,1456,1560,1638,1664,1716,1755,1820,

%U 1872,1950,2028,2080,2106,2176,2184,2288,2340,2457,2548,2574,2704,2730,2860,2925,3042,3146,3264,3380,3808,4056,4080,4352,4459,4563

%N Numbers k such that A322582(k) <= A276085(k) <= A348507(k).

%C A322582 and A348507 give lower and upper bounds for arithmetic derivative, A003415. See comments in A369650.

%C Among the initial 100000 terms, there are 9968 odd numbers, 8711 terms that are also in A048103, and only 34 squarefree terms (given in A392607).

%H Antti Karttunen, <a href="/A392601/b392601.txt">Table of n, a(n) for n = 1..10000</a>

%H Antti Karttunen, <a href="/A392601/a392601.txt">The first 100000 terms of sequence; all terms <= 5677350912</a>

%o (PARI)

%o A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };

%o A322582(n) = (n-A003958(n));

%o A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };

%o A348507(n) = (A003959(n) - n);

%o is_A392601(n) = { my(f = factor(n), ul=A348507(n), pr=1, q=1, s=0); for(k=1, #f~, forprime(p = nextprime(1+q), f[k, 1]-1, pr *= p; if(pr > ul, return(0)); q = p); s += f[k, 2]*pr; if(s > ul, return(0))); (s >= A322582(n)); };

%Y Cf. A003415, A048103, A276085, A322582, A348507.

%Y Subsequences: A369650, A392607 (squarefree terms).

%Y Cf. also A392602.

%K nonn

%O 1,2

%A _Antti Karttunen_, Jan 20 2026