A172461 -id:A172461 - cp's OEIS frontend

A319745 Nonunitary harmonic numbers: numbers such that the harmonic mean of their nonunitary divisors is an integer.

4, 9, 12, 18, 24, 25, 45, 49, 54, 60, 112, 121, 126, 150, 168, 169, 270, 289, 294, 336, 361, 529, 560, 594, 637, 726, 841, 961, 1014, 1232, 1369, 1638, 1680, 1681, 1734, 1849, 1984, 2166, 2184, 2209, 2430, 2520, 2688, 2700, 2809, 2850, 3174, 3481, 3721, 3780

Offset: 1

Amiram Eldar, Sep 27 2018

nonn

Includes all the numbers with a single nonunitary divisor. Those with more than one: 12, 18, 24, 45, 54, 60, 112, ...
Supersequence of A064591 (nonunitary perfect numbers).
Ligh & Wall showed that if p, 2p-1 and 2^p-1 are distinct primes (A172461, except for 2), then the following numbers are in the sequence: 6*p^2, p^2*(2p-1), 6*p^2*(2p-1), 2^(p+1)*3*(2^p-1), 2^(p+1)*15*(2^p-1) and 2^(p+1)*(2p-1)*(2^p-1).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Steve Ligh and Charles R. Wall, Functions of Nonunitary Divisors, Fibonacci Quarterly, Vol. 25 (1987), pp. 333-338.

Cf. A001599, A006086, A063947, A064591, A172461, A286325.

nudiv[n_] := Block[{d = Divisors[n]}, Select[d, GCD[#, n/#] > 1 &]]; nhQ[n_]:= Module[ {divs=nudiv[n]}, Length[divs] > 0 && IntegerQ[HarmonicMean[divs]]]; Select[Range[30000], nhQ]

hm(v) = #v/sum(k=1, #v, 1/v[k]);
vnud(n) = select(x->(gcd(x, n/x)!=1), divisors(n));
isok(n) = iferr(denominator(hm(vnud(n))) == 1, E, 0); \\ Michel Marcus, Oct 28 2018

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

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]

A154895 Perfect numbers whose number of proper divisors is prime.

Original entry on oeis.org

6, 28, 8128, 137438691328, 2305843008139952128

Offset: 1

Omar E. Pol, Jan 25 2009

nonn

The next term is too large to include in the data section. If there are no odd perfect numbers then the next term is a(6) = 2^606 * (2^607 - 1) = 1.410... * 10^365. - Amiram Eldar, Jul 29 2020

			28 is member because the number of proper divisors of 28 is 5, a prime number.

Cf. A000396, A006516, A032741, A133033, A154896, A172461.

Table[2^(p-1)*(2^p-1), {p, Select[MersennePrimeExponent[Range[8]], PrimeQ[2# - 1] &]}] (* Amiram Eldar, Jul 29 2020 *)

a(n) = A006516(A172461(n)), assuming that odd perfect numbers do not exist. - Amiram Eldar, Jul 29 2020

A383065 Integers k such that (k/rad(k))*2^rad(k) - 1 is prime where rad = A007947.

Original entry on oeis.org

2, 3, 4, 5, 7, 9, 12, 13, 16, 17, 18, 19, 27, 31, 36, 50, 60, 61, 64, 80, 89, 107, 108, 112, 127, 135, 147, 189, 200, 212, 243, 252, 343, 448, 464, 500, 521, 576, 600, 607, 612, 648, 675, 688, 756, 768, 784, 800, 832, 875, 900, 1058, 1212, 1279, 1280

Offset: 1

Juri-Stepan Gerasimov, Apr 19 2025

nonn

			12 is a term because (12/6)*2^6 - 1 = 127 is prime, where d = 6 is largest squarefree divisor d of k = 12.

Supersequence of A000043 and A172461.

Cf. A002234, A003557, A007947.

[k: k in [1..1300] | IsPrime((k div &*PrimeDivisors(k))*2^&*PrimeDivisors(k)-1)];

s={};Do[r=Last[Select[Divisors[n], SquareFreeQ]];If[PrimeQ[2^r*n/r-1],AppendTo[s,n]],{n,1280}];s (* James C. McMahon, May 01 2025 *)

isok(k) = my(r=factorback(factorint(k)[, 1])); ispseudoprime((k/r)*2^r - 1); \\ Michel Marcus, Apr 20 2025

cp's OEIS Frontend

A319745 Nonunitary harmonic numbers: numbers such that the harmonic mean of their nonunitary divisors is an integer.

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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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A154895 Perfect numbers whose number of proper divisors is prime.

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A383065 Integers k such that (k/rad(k))*2^rad(k) - 1 is prime where rad = A007947.

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