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

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, 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, 22, 1, 23, 1, 2, 3, 6, 8, 24, 1, 25, 1, 2, 13, 26, 1, 3, 27, 1, 4, 7, 28

Offset: 1

Author

Amiram Eldar, Dec 14 2024

Keywords

• nonn
• easy
• tabf

Comments

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.

Examples

			The table starts:
  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;
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..11627 (first 2000 rows)
• Andrew V. Lelechenko, The Quest for the Generalized Perfect Numbers, Theoretical and Applied Aspects of Cybernetics, Proceedings, The 4th International Scientific Conference of Students and Young Scientists, Kyiv, 2014.
• David Moews, Perfect, amicable and sociable numbers.

Crossrefs

Cf. A379028 (row lengths), A241405 (row sums).

Similar tables: A027750 (all divisors), A077609 (infinitary), A077610 (unitary), A222266 (bi-unitary), A322791 (exponential), A361255 (exponential unitary).

Programs

Mathematica

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

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; }
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; }