A335371 -id:A335371 - cp's OEIS frontend

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

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

Amiram Eldar, Jun 03 2020

nonn

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, ...

			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).

Amiram Eldar, Table of n, a(n) for n = 1..246 (terms below 10^14)
Mariano Garcia, On numbers with integral harmonic mean, The American Mathematical Monthly, Vol. 61, No. 2 (1954), pp. 89-96. See page 95.

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

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

Amiram Eldar, Jun 03 2020

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).

			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.

Mariano Garcia, On numbers with integral harmonic mean, The American Mathematical Monthly, Vol. 61, No. 2 (1954), pp. 89-96. See page 95.

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

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

A335370 Harmonic numbers m with a record number k of distinct prime numbers p_i (i = 1..k) that do not divide m such that mp_1, mp_1p_2, ... , mp_1...p_k are all harmonic numbers.

Original entry on oeis.org

1, 28, 1638, 6200, 2457000, 4713984, 1381161600, 10200236032

Offset: 1

Amiram Eldar, Jun 03 2020

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).
The terms of this sequence begin a chain of harmonic numbers of a record length. In each chain, each member, except the first, is generated from its predecessor by multiplying it by a prime that does not divide it.
The corresponding record values of k are 0, 1, 2, 3, 4, 6, 7, 8, ...
The list of primes or their order may not be unique.

			28 is the least harmonic number with one prime, p = 5, such that 28*p = 140 is also a harmonic number.
1638 is the least harmonic number with 2 primes, 5 and 29, such that 1638*5 = 8190 and 1638*5*29 = 237510 are also harmonic numbers.
.
n  a(n)         k   primes p_i, i = 1..k                 number of permutations
-------------------------------------------------------------------------------
1  1            0   -                                         -
2  28           1   5                                         1
3  1638         2   5, 29                                     1
4  6200         3   19, 37, 73                                1
5  2457000      4   11, 19, 37, 73                            4
6  4713984      6   5, 7, 13, 19, 37, 73                      15
                    5, 7, 19, 37, 73, 1021                    5
7  1381161600   7   11, 19, 37, 43, 73, 6277, 12553           10
                    11, 19, 37, 43, 3181, 6361, 12721         6
8  10200236032  8   3, 5, 79, 157, 313, 1877, 7507, 15013     5

Cf. A000005, A000203, A001599, A099377, A099378, A335368, A335369, A335371.

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 = -3 + Length @ FixedPointList[harmGens, {harmNums[[k]]}]; If[len > lenmax, lenmax = len; AppendTo[seq, harmNums[[k]]]], {k, 1, Length[harmNums]}]; seq

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

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

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A335370 Harmonic numbers m with a record number k of distinct prime numbers p_i (i = 1..k) that do not divide m such that mp_1, mp_1p_2, ... , mp_1...p_k are all harmonic numbers.

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cp's OEIS Frontend

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

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Mathematica

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

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Mathematica

A335370 Harmonic numbers m with a record number k of distinct prime numbers p_i (i = 1..k) that do not divide m such that m*p_1, m*p_1*p_2, ... , m*p_1*...*p_k are all harmonic numbers.

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A335370 Harmonic numbers m with a record number k of distinct prime numbers p_i (i = 1..k) that do not divide m such that mp_1, mp_1p_2, ... , mp_1...p_k are all harmonic numbers.