A383760 -id:A383760 - cp's OEIS frontend

A383865 The number of divisors d of n having the property that for every prime p dividing n the p-adic valuation of d is either 0 or an infinitary divisor of the p-adic valuation of n.

1, 2, 2, 3, 2, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 3, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 3, 6, 2, 8, 2, 3, 4, 4, 4, 9, 2, 4, 4, 6, 2, 8, 2, 6, 6, 4, 2, 6, 3, 6, 4, 6, 2, 6, 4, 6, 4, 4, 2, 12, 2, 4, 6, 5, 4, 8, 2, 6, 4, 8, 2, 9, 2, 4, 6, 6, 4, 8, 2, 6, 3, 4, 2, 12, 4, 4, 4

Offset: 1

Amiram Eldar, May 12 2025

First differs from A383863 at n = 256.
The number of divisors d of n such that each is a unitary divisor of an exponential infinitary divisor of n (see A383760).	
Analogous to the number of (1+e)-divisors (A049599) as exponential infinitary divisors (A383760, A307848) are analogous to exponential divisors (A322791, A049419).
The sum of these divisors is A383866(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A036537, A037445, A049419, A049599, A307848, A322791, A383760, A383863, A383866.

f[p_, e_] := 2^DigitCount[e, 2, 1]; d[1] = 1; d[n_] := Times @@ f @@@ FactorInteger[n]; ff[p_, e_] := d[e] + 1; a[1] = 1; a[n_] := Times @@ ff @@@ FactorInteger[n]; Array[a, 100]

d(n) = vecprod(apply(x -> 2^hammingweight(x), factor(n)[, 2]));
a(n) = vecprod(apply(x -> 1 + d(x), factor(n)[, 2]));

Multiplicative with a(p^e) = 1 + A037445(e).

a(n) <= A049599(n), with equality if and only if all the exponents in the prime factorization of n are in A036537.

A383761 Irregular triangle read by rows in which the n-th row lists the exponential squarefree exponential divisors of n.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 7, 2, 8, 3, 9, 10, 11, 6, 12, 13, 14, 15, 2, 4, 17, 6, 18, 19, 10, 20, 21, 22, 23, 6, 24, 5, 25, 26, 3, 27, 14, 28, 29, 30, 31, 2, 32, 33, 34, 35, 6, 12, 18, 36, 37, 38, 39, 10, 40, 41, 42, 43, 22, 44, 15, 45, 46, 47, 6, 12, 7, 49, 10, 50

Offset: 1

Amiram Eldar, May 09 2025

Differs from A322791, A361255 and A383760 at rows 16, 48, 80, 81, 112, 144, 162, ... .

An exponential squarefree exponential divisor (or e-squarefree e-divisor) d of a number n is a divisor d of n such that for every prime divisor p of n, the p-adic valuation of d is a squarefree divisor of the p-adic valuation of n.

			The first 10 rows are:
  1
  2
  3
  2, 4
  5
  6
  7
  2, 8
  3, 9
  10

Amiram Eldar, Table of n, a(n) for n = 1..15331 (first 10000 rows, flattened)
László Tóth, On certain arithmetic functions involving exponential divisors, II, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 27 (2007), pp. 155-166; arXiv preprint, arXiv:0708.3557 [math.NT], 2007-2009.

Cf. A278908 (row lengths), A361174 (row sums).

Cf. A322791, A361255, A383760.

sqfDivQ[n_, d_] := SquareFreeQ[d] && Divisible[n, d];
expSqfDivQ[n_, d_] := Module[{f = FactorInteger[n]}, And @@ MapThread[sqfDivQ, {f[[;; , 2]], IntegerExponent[d, f[[;; , 1]]]}]]; expSqfDivs[1] = {1};
expSqfDivs[n_] := Module[{d = Rest[Divisors[n]]}, Select[d, expSqfDivQ[n, #] &]];
Table[expSqfDivs[n], {n, 1, 70}] // Flatten

A383866 The sum of divisors d of n having the property that for every prime p dividing n the p-adic valuation of d is either 0 or an infinitary divisor of the p-adic valuation of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 11, 13, 18, 12, 28, 14, 24, 24, 19, 18, 39, 20, 42, 32, 36, 24, 44, 31, 42, 31, 56, 30, 72, 32, 35, 48, 54, 48, 91, 38, 60, 56, 66, 42, 96, 44, 84, 78, 72, 48, 76, 57, 93, 72, 98, 54, 93, 72, 88, 80, 90, 60, 168, 62, 96, 104, 79, 84, 144

Offset: 1

Amiram Eldar, May 13 2025

First differs from A383864 at n = 256.
The sum of divisors d of n such that each is a unitary divisor of an exponential infinitary divisor of n (see A383760).	
Analogous to the sum of (1+e)-divisors (A051378) as exponential infinitary divisors (A383760, A361175) are analogous to exponential divisors (A322791, A051377).
The number of these divisors is A383865(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A036537, A051377, A051378, A077609, A322791, A361175, A383760, A383864, A383865.

infdivs[n_] := If[n == 1, {1}, Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]];  (* Michael De Vlieger at A077609 *)
f[p_, e_] := 1 + Total[p^infdivs[e]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); }
infdivs(n) = {d = divisors(n); f = factor(n); idiv = []; for (k=1, #d, if (isidiv(d[k], f), idiv = concat(idiv, d[k])); ); idiv; } \\ Michel Marcus at A077609
a(n) = {my(f = factor(n), d); prod(i = 1, #f~, d = infdivs(f[i, 2]); 1 + sum(j = 1, #d, f[i, 1]^d[j]));}

Multiplicative with a(p^e) = 1 + Sum_{d infinitary divisor of e} p^d.
a(n) <= A051378(n), with equality if and only if all the exponents in the prime factorization of n are in A036537.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} f(1/p) = 1.52187097260174705015..., and f(x) = (1-x) * (1 + Sum_{k>=1} (1 + Sum{d infinitary divisor of k} x^(2*k-d))).

A383960 The number of prime powers p^e having the property that e is an infinitary divisor of the p-adic valuation of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 3, 2, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 4, 1, 2, 2, 3, 1, 3, 1, 3, 3, 2, 1, 3, 2, 3, 2, 3, 1, 3, 2, 3, 2, 2, 1, 4, 1, 2, 3, 4, 2, 3, 1, 3, 2, 3, 1, 4, 1, 2, 3, 3, 2, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2

Offset: 1

Amiram Eldar, May 16 2025

First differs from A238949 at n = 64.

First differs from A383959 at n = 256.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A037445, A085548, A238949, A383760, A383865, A383959.

f[p_, e_] := 2^DigitCount[e, 2, 1]; d[1] = 1; d[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 0; ff[p_, e_] := d[e]; a[n_] := Plus @@ ff @@@ FactorInteger[n]; Array[a, 100]

d(n) = vecprod(apply(x -> 1 << hammingweight(x), factor(n)[, 2]));
a(n) = vecsum(apply(x -> d(x), factor(n)[, 2]));

Additive with a(p^e) = A037445(e).

Sum_{k=1..n} a(k) ~ n*(log(log(n)) + B - C + D), where B is Mertens's constant (A077761), C = Sum_{p prime} 1/p^2 (A085548), and D = Sum_{p prime, e>=2} (1-1/p)*A037445(e)/p^e = 0.92752481299257205938... .

cp's OEIS Frontend

A383865 The number of divisors d of n having the property that for every prime p dividing n the p-adic valuation of d is either 0 or an infinitary divisor of the p-adic valuation of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A383761 Irregular triangle read by rows in which the n-th row lists the exponential squarefree exponential divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A383866 The sum of divisors d of n having the property that for every prime p dividing n the p-adic valuation of d is either 0 or an infinitary divisor of the p-adic valuation of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A383960 The number of prime powers p^e having the property that e is an infinitary divisor of the p-adic valuation of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula