A348658 -id:A348658 - cp's OEIS frontend

A348867 Numbers whose numerator and denominator of the harmonic mean of their divisors are both 3-smooth numbers.

1, 2, 3, 6, 28, 40, 84, 120, 135, 224, 270, 672, 819, 1638, 3780, 10880, 13392, 30240, 32640, 32760, 167400, 950976, 1303533, 2178540, 2607066, 3138345, 4713984, 6276690, 8910720, 14705145, 17428320, 29410290, 45532800, 52141320, 179734464, 301953024, 311323824

Offset: 1

Amiram Eldar, Nov 02 2021

nonn

The terms that are also harmonic numbers (A001599) are those whose harmonic mean of divisors (A001600) is a 3-smooth number. Of the 937 harmonic numbers below 10^14, 38 are terms in this sequence.
If a term is not a harmonic number, then its numerator and denominator of the harmonic mean of its divisors are powers of 2 and 3, or vice versa.
If k1 and k2 are coprime terms, then k1*k2 is also a term. In particular, if k is an odd term, then 2*k is also a term.

			2 is a term since the harmonic mean of its divisors is 4/3 = 2^2/3.
3 is a term since the harmonic mean of its divisors is 3/2.
40 is a term since the harmonic mean of its divisors is 32/9 = 2^5/3^2.

Amiram Eldar, Table of n, a(n) for n = 1..47

Cf. A000079, A000244, A001599, A001600, A003586, A099377, A099378.
Subsequence of A348868.
Similar sequences: A074266, A122254, A348658, A348659.

smQ[n_] := n == 2^IntegerExponent[n, 2] * 3^IntegerExponent[n, 3]; h[n_] := DivisorSigma[0, n]/DivisorSigma[-1, n]; q[n_] := smQ[Numerator[(hn = h[n])]] && smQ[Denominator[hn]]; Select[Range[10^5], q]

A348868 Numbers whose numerator and denominator of the harmonic mean of their divisors are both 5-smooth numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 10, 15, 24, 27, 28, 30, 40, 54, 84, 120, 135, 140, 216, 224, 270, 420, 496, 672, 756, 775, 819, 1080, 1120, 1488, 1550, 1638, 2176, 2325, 2480, 3360, 3780, 4095, 4650, 6048, 6200, 6528, 6552, 7440, 8190, 10880, 11375, 13392, 18600, 20925, 21700

Offset: 1

Amiram Eldar, Nov 02 2021

nonn

The terms that are also harmonic numbers (A001599) are those whose harmonic mean of divisors (A001600) is a 5-smooth number. Of the 937 harmonic numbers below 10^14, 83 are terms in this sequence.

If k1 and k2 are coprime terms, then k1*k2 is also a term. In particular, if k is an odd term, then 2*k is also a term.

			8 is a term since the harmonic mean of its divisors is 32/15 and both 32 = 2^5 and 15 = 3*5 are 5-smooth numbers.

Amiram Eldar, Table of n, a(n) for n = 1..234

Cf. A001599, A001600, A051037, A099377, A099378.
A348867 is a subsequence.
Similar sequences: A074266, A348658, A348659.

smQ[n_] := n == 2^IntegerExponent[n, 2] * 3^IntegerExponent[n, 3] * 5^IntegerExponent[n, 5]; h[n_] := DivisorSigma[0, n]/DivisorSigma[-1, n]; q[n_] := smQ[Numerator[(hn = h[n])]] && smQ[Denominator[hn]]; Select[Range[22000], q]

A349687 Numbers whose numerator and denominator of their abundancy index are both Fibonacci numbers.

Original entry on oeis.org

1, 2, 6, 15, 24, 26, 28, 84, 90, 96, 120, 270, 330, 496, 672, 1335, 1488, 1540, 1638, 8128, 24384, 27280, 44109, 68200, 131040, 447040, 523776, 18506880, 22256640, 33550336, 36197280, 38257095, 65688320, 91963648, 95472000, 100651008, 102136320, 176432256, 197308800

Offset: 1

Amiram Eldar, Nov 25 2021

nonn

This sequence includes all the perfect numbers (A000396), 3-perfect numbers (A005820) and 5-perfect numbers (A046060).

The deficient terms, 1, 2, 15, 26, 1335, 44109, 38257095, ..., have an abundancy index which is a ratio of two consecutive Fibonacci numbers, 1/1, 3/2, 8/5, 21/13, 144/89, 610/377, 46368/28657, ..., which approaches the golden ratio phi = 1.618... (A001622) as the numerators and denominators get larger.

			2 is a term since sigma(2)/2 = 3/2 = Fibonacci(4)/Fibonacci(3).
15 is a term since sigma(15)/15 = 8/5 = Fibonacci(6)/Fibonacci(5).

Michel Marcus, 85 terms (some terms might be missing in this list).

Cf. A000045, A010056, A000203, A001622, A017665, A017666.
Subsequences: A000396, A005820, A046060.
Similar sequences: A069070, A216780, A247086, A348658.

fibQ[n_] := Or @@ IntegerQ /@ Sqrt[{5 n^2 - 4, 5 n^2 + 4}]; ai[n_] := DivisorSigma[1, n]/n; q[n_] := fibQ[Numerator[(ain = ai[n])]] && fibQ[Denominator[ain]]; Select[Range[10^6], q]

isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8));
isok(n) = my(q=sigma(n)/n); isfib(numerator(q)) && isfib(denominator(q)); \\ Michel Marcus, Nov 25 2021

cp's OEIS Frontend

A348867 Numbers whose numerator and denominator of the harmonic mean of their divisors are both 3-smooth numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A348868 Numbers whose numerator and denominator of the harmonic mean of their divisors are both 5-smooth numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A349687 Numbers whose numerator and denominator of their abundancy index are both Fibonacci numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI