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A384658 Primitive terms in A384657: numbers k such that A384655(k) > k and A384655(d) <= d for every proper divisor d of k.

%I A384658 #7 Jun 07 2025 08:18:53
%S A384658 24,80,108,112,180,200,252,280,300,352,396,416,420,440,468,520,612,
%T A384658 660,684,780,828,1044,1088,1116,1216,1332,1472,1476,1548,1692,1856,
%U A384658 1908,1984,2124,2196,2288,2412,2556,2628,2844,2988,2992,3204,3344,3492,3536,3636,3708,3852,3924,3952
%N A384658 Primitive terms in A384657: numbers k such that A384655(k) > k and A384655(d) <= d for every proper divisor d of k.
%C A384658 If k is a term then every positive multiple of k is a term in A384657.
%H A384658 Amiram Eldar, <a href="/A384658/b384658.txt">Table of n, a(n) for n = 1..10000</a>
%e A384658 24 is a term since A384655(24) = 25 > 24, the proper divisors of 24 are {1, 2, 3, 4, 6, 8, 12}, A384655(1) = 0 < 1, A384655(2) = 1 < 2, A384655(3) = 1 < 3, A384655(4) = 3 < 4, A384655(6) = 4 < 6, A384655(8) = 7 < 8, and A384655(12) = 11 < 12.
%t A384658 f[p_, e_, k_] := p^e - If[e < k, 0, p^(e - k)]; q[n_] := Module[{fct = FactorInteger[n], emax, s}, emax = Max[fct[[;; , 2]]]; If[emax < 2, False, s = emax * n; Do[s -= Times @@ (f[#1, #2, k] & @@@ fct), {k, 1, emax}]; s > n]]; primQ[n_] := q[n] && AllTrue[Divisors[n], # == n || !q[#] &]; Select[Range[4000], primQ]
%o A384658 (PARI) is1(m) = {my(f = factor(m), p, e, emax, s); if(issquarefree(f), 0, p = f[,1]; e = f[,2]; emax = vecmax(e); s = emax*m; for(k = 1, emax, s -= prod(i = 1, #p, p[i]^e[i] - if(e[i] < k, 0, p[i]^(e[i]-k)))); s > m);}
%o A384658 isok(k) = if(!is1(k), 0, fordiv(k, d, if(d < k && is1(d), return(0))); 1);
%Y A384658 Subsequence of A013929 and A384657.
%Y A384658 Cf. A384655.
%K A384658 nonn
%O A384658 1,1
%A A384658 _Amiram Eldar_, Jun 06 2025