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 1-4 of 4 results.

A348658 Numbers whose numerator and denominator of the harmonic mean of their divisors are both Fibonacci numbers.

Original entry on oeis.org

1, 3, 5, 6, 15, 21, 28, 140, 182, 496, 546, 672, 918, 1890, 2016, 4005, 4590, 24384, 52780, 55860, 68200, 84812, 90090, 105664, 145782, 186992, 204600, 381654, 728910, 907680, 1655400, 2302344, 2862405, 3828009, 3926832, 5959440, 21059220, 33550336, 33839988, 42325920
Offset: 1

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Author

Amiram Eldar, Oct 28 2021

Keywords

Comments

Terms that also Fibonacci numbers are 1, 3, 5, 21, and no more below Fibonacci(300).

Examples

			3 is a term since the harmonic mean of its divisors is 3/2 = Fibonacci(4)/Fibonacci(3).
15 is a term since the harmonic mean of its divisors is 5/2 = Fibonacci(5)/Fibonacci(3).

Crossrefs

Cf. A000045, A099377, A099378.
Similar sequences: A074266, A123193, A272412, A272440, A348659.

Programs

  • Mathematica
    fibQ[n_] := Or @@ IntegerQ /@ Sqrt[{5 n^2 - 4, 5 n^2 + 4}]; h[n_] := DivisorSigma[0, n]/DivisorSigma[-1, n]; q[n_] := fibQ[Numerator[(hn = h[n])]] && fibQ[Denominator[hn]]; Select[Range[1000], q]
  • Python
    from itertools import islice
from sympy import integer_nthroot, gcd, divisor_sigma
def A348658(): # generator of terms
    k = 1
    while True:
        a, b = divisor_sigma(k), divisor_sigma(k,0)*k
        c = gcd(a,b)
        n1, n2 = 5*(a//c)**2-4, 5*(b//c)**2-4
        if (integer_nthroot(n1,2)[1] or integer_nthroot(n1+8,2)[1]) and (integer_nthroot(n2,2)[1] or integer_nthroot(n2+8,2)[1]):
            yield k
        k += 1
A348658_list = list(islice(A348658(),10)) # Chai Wah Wu, Oct 28 2021

A348659 Numbers whose numerator and denominator of the harmonic mean of their divisors are both prime numbers.

Original entry on oeis.org

3, 5, 13, 14, 15, 37, 42, 61, 66, 73, 92, 114, 157, 182, 193, 258, 277, 308, 313, 397, 402, 421, 457, 476, 477, 541, 546, 570, 613, 661, 673, 733, 744, 757, 812, 877, 978, 997, 1093, 1148, 1153, 1201, 1213, 1237, 1266, 1278, 1321, 1381, 1428, 1453, 1621, 1657
Offset: 1

Views

Author

Amiram Eldar, Oct 28 2021

Keywords

Comments

The prime terms of this sequence are the primes p such that (p+1)/2 is also a prime (A005383).
If p is in A109835, then p*(2*p-1) is a semiprime term.

Examples

			3 is a term since the harmonic mean of its divisors is 3/2 and both 2 and 3 are primes.

Links

Crossrefs

Cf. A005383, A099377, A099378, A109835.
Similar sequences: A023194, A048968, A074266, A348659.

Programs

  • Mathematica
    q[n_] := Module[{h = DivisorSigma[0, n]/DivisorSigma[-1, n]}, And @@ PrimeQ[{Numerator[h], Denominator[h]}]]; Select[Range[2000], q]

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

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Nov 02 2021

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Amiram Eldar, Nov 02 2021

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    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]
Showing 1-4 of 4 results.

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