cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A383863 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 a unitary divisor of the p-adic valuation of n.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, May 12 2025

Keywords

Comments

First differs from A073184 at n = 64.
First differs from A383865 at n = 256.
The number of divisors d of n such that each is a unitary divisor of an exponential unitary divisor of n (see A361255).
Analogous to the number of (1+e)-divisors (A049599) as exponential unitary divisors (A361255, A278908) are analogous to exponential divisors (A322791, A049419).
The sum of these divisors is A383864(n).
Also, 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 a squarefree divisor of the p-adic valuation of n. The sum of these divisors is A383867(n).

Links

Crossrefs

Cf. A001221, A034444, A049419, A049599, A073184, A209061, A278908, A322791, A361255, A383864, A383865, A383867.

Programs

  • Mathematica
    f[p_, e_] := 2^PrimeNu[e] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = vecprod(apply(x -> 1 + 1 << omega(x), factor(n)[, 2]));

Formula

Multiplicative with a(p^e) = 1 + 2^A001221(e) = 1 + A034444(e).
a(n) <= A049599(n), with equality if and only if n is an exponentially squarefree number (A209061).

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

Views

Author

Amiram Eldar, May 13 2025

Keywords

Comments

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).

Links

Crossrefs

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

Programs

  • Mathematica
    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]
  • PARI
    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]));}

Formula

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

Views

Author

Amiram Eldar, May 16 2025

Keywords

Comments

First differs from A238949 at n = 64.
First differs from A383959 at n = 256.

Links

Crossrefs

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

Programs

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

Formula

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... .
Showing 1-3 of 3 results.

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