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.

A335369 Harmonic numbers k such that k*p is not a harmonic number for all the primes p that do not divide k.

Original entry on oeis.org

1, 6, 140, 496, 672, 2970, 27846, 105664, 173600, 237510, 539400, 695520, 726180, 753480, 1421280, 1539720, 2229500, 2290260, 8872200, 11981970, 14303520, 15495480, 33550336, 50401728, 71253000, 80832960, 90409410, 144963000, 221557248, 233103780, 287425800, 318177800
Offset: 1

Views

Author

Amiram Eldar, Jun 03 2020

Keywords

Comments

If k is a harmonic number (A001599) and p is a prime that does not divide k, then k*p is a harmonic number if and only if (p+1)/2 is a divisor of the harmonic mean of the divisors of k, h(k) = k*tau(k)/sigma(k) = k*A000005(k)/A000203(k). The terms of this sequence are harmonic numbers k such that for all the divisors d of h(k), 2*d - 1 is either a nonprime or a prime divisor of k.
The even perfect numbers, 2^(p-1)*(2^p - 1) where p is a Mersenne exponent (A000043), have harmonic mean of divisors p. Therefore, they are in this sequence if p = 2 or if 2*p - 1 is composite (i.e., not in A172461). Of the first 47 Mersenne exponents there are 37 such primes (p = 2, 5, 13, 17, ...), with the corresponding even perfect numbers 6, 496, 33550336, 8589869056, ...

Examples

			1 is a term since it is a harmonic number, and there is no prime p such that 1*p = p is a harmonic number (if p is a prime, h(p) = 2*p/(p+1) cannot be an integer).

Links

Crossrefs

Cf. A000005, A000043, A000203, A000396, A001599, A099377, A099378, A172461, A335368, A335370, A335371.

Programs

  • Mathematica
    harmNums = Cases[Import["https://oeis.org/A001599/b001599.txt", "Table"], {, }][[;; , 2]]; harMean[n_] := n * DivisorSigma[0, n]/DivisorSigma[1, n]; primeCountQ[n_] := Module[{d = Divisors[harMean[n]]}, Select[2*d - 1, PrimeQ[#] && ! Divisible[n, #] &] == {}]; Select[harmNums, primeCountQ]

A335368 Harmonic numbers k with a record number of primes p not dividing k such that k*p is also a harmonic number.

Original entry on oeis.org

1, 28, 1638, 30240, 2178540, 2457000, 32997888, 142990848, 1307124000, 71271827200, 547929930240, 2198278051200, 2567400675840, 54409216942080
Offset: 1

Views

Author

Amiram Eldar, Jun 03 2020

Keywords

Comments

The corresponding record values are 0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 17.
If k is a harmonic number (A001599) and p is a prime that does not divide k, then k*p is a harmonic number if and only if (p+1)/2 is a divisor of the harmonic mean of the divisors of k, k*tau(k)/sigma(k) = k*A000005(k)/A000203(k).

Examples

			1 is the first harmonic number, and it has 0 primes p such that 1*p = p is a harmonic number, since a prime number cannot be harmonic. The next harmonic number k with at least one prime p such that k*p is also a harmonic number is 28, since 28 * 5 = 140 is a harmonic number.

Links

Crossrefs

Cf. A000005, A000203, A001599, A099377, A099378, A335369, A335370, A335371.

Programs

  • Mathematica
    harmNums = Cases[Import["https://oeis.org/A001599/b001599.txt", "Table"], {, }][[;; , 2]]; harMean[n_] := n*DivisorSigma[0, n]/DivisorSigma[1, n]; primeCount[n_] := Module[{d = Divisors[harMean[n]]}, Length @ Select[2*d - 1, PrimeQ[#] && ! Divisible[n, #] &]]; primeCountMax = -1; seq = {}; Do[If[(pc = primeCount[harmNums[[k]]]) > primeCountMax, primeCountMax = pc; AppendTo[seq, harmNums[[k]]]], {k, 1, Length[harmNums]}]; seq

A335371 Harmonic numbers with a record number of harmonic numbers that can be generated from them using an iterative process of multiplying by primes (see Comments).

Original entry on oeis.org

1, 28, 1638, 6200, 950976, 2178540, 2457000, 4713984, 45532800, 142990848, 459818240, 1381161600, 10200236032, 57575890944, 109585986048, 513480135168, 1553357978368, 10881843388416, 43947421401888
Offset: 1

Views

Author

Amiram Eldar, Jun 03 2020

Keywords

Comments

If m is a harmonic number (A001599), then it is possible to generate a new harmonic number m*p if p is a prime number that does not divide m and (p+1)/2 is a divisor of the harmonic mean of the divisors of m, h(m) = m * tau(m)/sigma(m) = m * A000005(m)/A000203(m).
Given a harmonic number m, in the first iteration a finite set of new harmonic numbers, {m*p_1, m*p_2, ...} is being generated. In the second iteration, a new set of harmonic number is being generated from each of the harmonic numbers from the previous iteration, a union of these sets is be calculated (removing duplicates). The process is terminated when no more harmonic numbers can be generated. The total number of harmonic numbers from all the iterations is being counted. The terms of this sequence have a record count of new harmonic numbers.
The corresponding record values of k are 0, 1, 3, 5, 8, 12, 17, 36, 38, 40, 44, 62, 70, 82, 156, 226, 281, 335, 358, ...

Examples

			1638 is a term since a record number of 3 new harmonic numbers can be generated from it. In the first iteration 2 new harmonic numbers can be generated: 1638 * 5 = 8190, and 1638 * 17 = 27846. In the second iteration, a new harmonic number can be generated from 8190: 8190 * 29 = 237510.

Crossrefs

Cf. A000005, A000203, A001599, A099377, A099378, A335368, A335369, A335370.

Programs

  • Mathematica
    harmNums = Cases[Import["https://oeis.org/A001599/b001599.txt", "Table"], {, }][[;; , 2]]; harMean[n_] := n * DivisorSigma[0, n]/DivisorSigma[1, n]; harmGen[n_] := Module[{d = Divisors[harMean[n]]}, n * Select[2*d - 1, PrimeQ[#] && ! Divisible[n, #] &]]; harmGens[s_] := Union@Flatten[harmGen /@ s]; lenmax = -1; seq = {}; Do[len = Length @ Union @ Flatten @ FixedPointList[harmGens, {harmNums[[k]]}]; If[len > lenmax, lenmax = len; AppendTo[seq, harmNums[[k]]]], {k, 1, Length[harmNums]}]; seq
Showing 1-3 of 3 results.

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