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A379027 Irregular table read by rows in which the n-th row lists the modified exponential divisors of n.

%I A379027 #10 Dec 16 2024 02:18:43
%S A379027 1,1,2,1,3,1,4,1,5,1,2,3,6,1,7,1,2,8,1,9,1,2,5,10,1,11,1,3,4,12,1,13,
%T A379027 1,2,7,14,1,3,5,15,1,16,1,17,1,2,9,18,1,19,1,4,5,20,1,3,7,21,1,2,11,
%U A379027 22,1,23,1,2,3,6,8,24,1,25,1,2,13,26,1,3,27,1,4,7,28
%N A379027 Irregular table read by rows in which the n-th row lists the modified exponential divisors of n.
%C A379027 If the prime factorization of n is Product_{i} p_i^e_i, then the modified exponential divisors of n are all the divisors of n that are of the form Product_{i} p_i^b_i such that 1 + b_i | 1 + e_i for all i.
%H A379027 Amiram Eldar, <a href="/A379027/b379027.txt">Table of n, a(n) for n = 1..11627</a> (first 2000 rows)
%H A379027 Andrew V. Lelechenko, <a href="http://taac.org.ua/files/a2014/proceedings/UA-2-Andrew%20Lelechenko-440.pdf">The Quest for the Generalized Perfect Numbers</a>, Theoretical and Applied Aspects of Cybernetics, Proceedings, The 4th International Scientific Conference of Students and Young Scientists, Kyiv, 2014.
%H A379027 David Moews, <a href="http://djm.cc/amicable.html">Perfect, amicable and sociable numbers</a>.
%e A379027 The table starts:
%e A379027   1;
%e A379027   1, 2;
%e A379027   1, 3;
%e A379027   1, 4;
%e A379027   1, 5;
%e A379027   1, 2, 3, 6;
%e A379027   1, 7;
%e A379027   1, 2, 8;
%e A379027   1, 9;
%e A379027   1, 2, 5, 10;
%e A379027   1, 11;
%e A379027   1, 3, 4, 12;
%t A379027 modexpDivQ[n_, d_] := Module[{f = FactorInteger[n]}, And @@ MapThread[Divisible, {f[[;; , 2]] + 1, IntegerExponent[d, f[[;; , 1]]] + 1}]]; row[1] = {1}; row[n_] := Select[Divisors[n], modexpDivQ[n, #] &]; Table[row[n], {n, 1, 28}] // Flatten
%o A379027 (PARI) ismodexpdiv(f, d) = {my(e); for(i=1, #f~, e = valuation(d, f[i, 1]); if((f[i, 2]+1) % (e+1), return(0))); 1; }
%o A379027 row(n) = {my(f = factor(n), d = divisors(f), mediv = [1]); if(n == 1, return(mediv)); for(i=2, #d, if(ismodexpdiv(f, d[i]), mediv = concat(mediv, d[i]))); mediv; }
%Y A379027 Cf. A379028 (row lengths), A241405 (row sums).
%Y A379027 Similar tables: A027750 (all divisors), A077609 (infinitary), A077610 (unitary), A222266 (bi-unitary), A322791 (exponential), A361255 (exponential unitary).
%K A379027 nonn,easy,tabf
%O A379027 1,3
%A A379027 _Amiram Eldar_, Dec 14 2024