A007340 -id:A007340 - cp's OEIS frontend

A328944 Arithmetic numbers (A003601) that are not harmonic (A001599).

3, 5, 7, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 27, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 77, 78, 79, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 96, 97, 99, 101

Offset: 1

Jaroslav Krizek, Oct 31 2019

nonn

Numbers m such that the arithmetic mean of the divisors of m is an integer but the harmonic mean of the divisors of m is not an integer.
Numbers m such that A(m) = A000203(m)/A000005(m) is an integer but H(m) = m * A000005(m)/A000203(m) is not an integer.
Corresponding values of A(m): 2, 3, 4, 6, 7, 6, 6, 9, 10, 7, 8, 9, 12, 10, 15, 9, 16, 12, 12, 19, 15, 14, 21, 12, 22, ...
Corresponding values of H(m): 3/2, 5/3, 7/4, 11/6, 13/7, 7/3, 5/2, 17/9, 19/10, 20/7, 21/8, 22/9, ...
Complement of A007340 with respect to A003601.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A003601, A007340, A046999, A328945.

[m: m in [1..10^5] | IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m)) and not IsIntegral(m * NumberOfDivisors(m) / SumOfDivisors(m))];

harm:= proc(S) local s; nops(S)/add(1/s,s=S) end proc:
filter:= proc(n) local S;
  S:= numtheory:-divisors(n);
  (convert(S,`+`)/nops(S))::integer and not harm(S)::integer
end proc:
select(filter, [$1..200]); # Robert Israel, May 04 2025

Select[Range[100], Divisible[DivisorSigma[1, #], DivisorSigma[0, #]] && !Divisible[# * DivisorSigma[0, #], DivisorSigma[1, #]] &] (* Amiram Eldar, Nov 01 2019 *)

A328945 Numbers m that are neither arithmetic (A003601) nor harmonic (A001599).

Original entry on oeis.org

2, 4, 8, 9, 10, 12, 16, 18, 24, 25, 26, 32, 34, 36, 40, 48, 50, 52, 58, 63, 64, 72, 74, 75, 76, 80, 81, 82, 84, 88, 90, 98, 100, 104, 106, 108, 112, 117, 120, 121, 122, 124, 128, 130, 136, 144, 146, 148, 152, 156, 160, 162, 170, 171, 172, 175, 176, 178, 180

Offset: 1

Jaroslav Krizek, Oct 31 2019

nonn

Numbers m such that neither the arithmetic mean of the divisors of m nor the harmonic mean of the divisors of m is an integer.
Numbers m such that neither A(m) = A000203(m)/A000005(m) nor H(m) = m * A000005(m)/A000203(m) is an integer.
Corresponding values of A(m): 3/2, 7/3, 15/4, 13/3, 9/2, 14/3, 31/5, 13/2, 15/2, 31/3, 21/2, 21/2, 27/2, ...
Corresponding values of H(m): 4/3, 12/7, 32/15, 27/13, 20/9, 18/7, 80/31, 36/13, 16/5, 75/31, 52/21, 64/21, ...

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A001599, A003601, A007340, A328944.

[m: m in [1..10^5] | not IsIntegral(m * NumberOfDivisors(m) / SumOfDivisors(m)) and not IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m))]

filter:= proc(n) local D,d,t;
  D:=numtheory:-divisors(n);
  d:= nops(D);
  convert(D,`+`) mod d <> 0 and not ((d/add(1/t,t=D))::integer)
end proc:
select(filter, [$1..200]); # Robert Israel, Dec 14 2023

Select[Range[180], !Divisible[DivisorSigma[1, #], DivisorSigma[0, #]] && !Divisible[# * DivisorSigma[0, #], DivisorSigma[1, #]] &] (* Amiram Eldar, Nov 01 2019 *)

isok(m) = my(f = factor(m), prd = sigma(f)/numdiv(f)); (denominator(prd) != 1) && (denominator(m/prd) != 1); \\ Michel Marcus, Nov 05 2019

A361387 Infinitary arithmetic numbers k whose mean infinitary divisor is an infinitary divisor of k.

Original entry on oeis.org

1, 6, 60, 270, 420, 630, 2970, 5460, 8190, 36720, 136500, 172900, 204750, 245700, 491400, 790398, 791700, 819000, 1037400, 1138320, 1187550, 1228500, 1801800, 2457000, 3767400, 4176900, 4504500, 5405400, 6397300, 6688500, 6741630, 7698600, 8353800, 10032750, 10228680

Offset: 1

Amiram Eldar, Mar 10 2023

nonn

Also, infinitary harmonic numbers k whose harmonic mean of the infinitary divisors of k is an infinitary divisor of k.

			6 is a term since the arithmetic mean of its infinitary divisors, {1, 2, 3, 6}, is 3, and 3 is also an infinitary divisor of 6.
60 is a term since the arithmetic mean of its infinitary divisors, {1, 3, 4, 5, 12, 15, 20, 60}, is 15, and 15 is also an infinitary divisor of 60.

Amiram Eldar, Table of n, a(n) for n = 1..135

Subsequence of A063947 and A361386.
Similar sequence: A007340, A353039.
Cf. A037445, A049417, A077609.

idivs[1] = {1}; idivs[n_] := Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, e_Integer} :> p^Select[Range[0, e], BitOr[e, #] == e &])]; Select[Range[10^5], IntegerQ[(r = Mean[(i = idivs[#])])] && MemberQ[i, r] &]

isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); } \\ Michel Marcus at A077609
is(n) = {my(f = factor(n), b, r); r = prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], (f[i, 1]^(2^(#b-k))+1)/2, 1))); denominator(r) == 1 && n%r==0 && isidiv(r, f); }

A157850 Numbers k such that are harmonic means of divisors of harmonic (Ore) numbers (harmonic numbers is A001599) and equal to one of the divisors of this harmonic numbers.

Original entry on oeis.org

1, 2, 5, 6, 8, 9, 11, 10, 15, 15, 14, 17, 24, 24, 21, 13, 19, 27, 25, 29, 26, 44, 44, 29, 46, 39, 46, 42, 47, 47, 35, 41, 60, 51, 37, 48, 45, 49, 50, 49, 53, 77, 86, 86, 51, 96, 75, 70, 80, 99, 110, 81, 84, 102, 82, 96, 114, 53, 108, 115, 105, 116, 91, 85, 105

Offset: 1

Jaroslav Krizek, Mar 07 2009, Apr 12 2009

nonn

			a(3) = 5, because 5 is harmonic mean of divisors of A007340(3)=140 and also is divisor of 140.

Amiram Eldar, Table of n, a(n) for n = 1..847 (Calculated from the b-file at A007340)

Cf. A007340, A000005, A000203, A001599.

a(n) = f(A007340(n)), where f(k) = k * tau(k)/sigma(k) = k * A000005(k)/A000203(k).

More terms from Amiram Eldar, Jul 09 2019

A331666 Refactorable numbers (A033950) that are simultaneously arithmetic (A003601) and harmonic (A001599).

Original entry on oeis.org

1, 672, 30240, 23569920, 45532800, 164989440, 447828480, 623397600, 1381161600, 1862023680, 2144862720, 3134799360, 3831421440, 13584130560, 14182439040, 16569653760, 21943595520, 22933532160, 34482792960, 35032757760, 40752391680, 53621568000, 56481384960

Offset: 1

Jaroslav Krizek, Jan 23 2020

nonn

Numbers m such that all values of sigma(m)/tau(m), m/tau(m) and m * tau(m)/sigma(m) are any integers (f, g, and h respectively).
Corresponding values of numbers f, g and h: (1, 84, 1260, 294624, 474300, 1178496, 2946240, 3298400, 5754840, 11784960, ...); (1, 28, 315, 73656, 118575, 257796, 699732, 721525, 1198925, 2909412, 1675674, ...); (1, 8, 24, 80, 96, 140, 152, 189, 240, 158, 260, 266, 220, 380, 384, 296, 392, ...).
Multiply-perfect numbers from this sequence are in A047728.

			For m = 672, f = sigma(m)/tau(m) = 2016/24 = 84; g = m/tau(m) = 672/24 = 28; h = m * tau(m)/sigma(m) = 672*24/2016 = 8.

Amiram Eldar, Table of n, a(n) for n = 1..118 (terms below 10^14)

Intersection of A033950 and A007340.

Cf. A000005, A000203, A001599, A003601, A047728.

[m: m in [1..10^6] | IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m)) and  IsIntegral(m / NumberOfDivisors(m)) and IsIntegral(m * NumberOfDivisors(m) / SumOfDivisors(m))]

Select[Range[3*10^7], Divisible[#, (d = DivisorSigma[0, #])] && Divisible[(s = DivisorSigma[1, #]), d] && Divisible[#*d, s] &] (* Amiram Eldar, Jan 24 2020 *)

is(k) = {my(f = factor(k), s = sigma(f), d = numdiv(f)); !(k % d) && !(s % d) && !((k * d) % s) ;} \\ Amiram Eldar, May 09 2024

A361787 Bi-unitary arithmetic numbers k whose mean bi-unitary divisor is a bi-unitary divisor of k.

Original entry on oeis.org

1, 6, 60, 270, 420, 630, 672, 2970, 5460, 8190, 10080, 22848, 30240, 99792, 136500, 172900, 204750, 208656, 245700, 249480, 312480, 332640, 342720, 385560, 491400, 695520, 708288, 791700, 819000, 861840, 1028160, 1037400, 1187550, 1228500, 1421280, 1528800, 1571328

Offset: 1

Amiram Eldar, Mar 24 2023

nonn

Also, bi-unitary harmonic numbers k whose harmonic mean of the bi-unitary divisors of k is a bi-unitary divisor of k.

			6 is a term since the arithmetic mean of its bi-unitary divisors, {1, 2, 3, 6}, is 3, and 3 is also a bi-unitary divisor of 6.
60 is a term since the arithmetic mean of its bi-unitary divisors, {1, 3, 4, 5, 12, 15, 20, 60}, is 15, and 15 is also a bi-unitary divisor of 60.

Amiram Eldar, Table of n, a(n) for n = 1..236

Subsequence of A286325 and A361786.
Similar sequence: A007340, A353039, A361387.
Cf. A188999, A222266, A286324.

biudivQ[f_, d_] := AllTrue[f, OddQ[Last[#]] || IntegerExponent[d, First[#]] != Last[#]/2 &]; biuDivs[n_] := Module[{d = Divisors[n], f = FactorInteger[n]}, Select[d, biudivQ[f, #] &]]; Select[Range[10^5], IntegerQ[(r = Mean[(i = biuDivs[#])])] && MemberQ[i, r] &]

isbdiv(f, d) = {for (i=1, #f~, if(f[i, 2]%2 == 0 && valuation(d, f[i, 1]) == f[i, 2]/2, return(0))); 1;}
is(n) = {my(f = factor(n), r, p, e); r = prod(i=1, #f~, p = f[i, 1]; e = f[i, 2]; if(e%2, (p^(e+1)-1)/((e + 1)*(p-1)), ((p^(e+1)-1)/(p-1)-p^(e/2))/e)); denominator(r) == 1 && n%r==0 && isbdiv(f, r); }

A342358 Balanced numbers (A020492) that are also arithmetic numbers (A003601) and harmonic numbers (A001599).

Original entry on oeis.org

1, 6, 140, 270, 2970, 332640, 14303520, 5297292000

Offset: 1

Bernard Schott, Mar 09 2021

Equivalently, numbers m such that sigma(m)/phi(m), sigma(m)/tau(m) and m*tau(m)/sigma(m) are all integers where phi = A000010, tau = A000005 and sigma = A000203.
Conjecture: 1 would be the only odd term of this sequence, because Oystein Ore conjectured that 1 is the only odd harmonic number (see link), and 1 is an arithmetic and balanced number (A342103).
Proposition: there are no primes in the sequence. Proof: the only prime that is both arithmetic and balanced is 3 (A342103), but 3 is not an harmonic number.
As Hans-Joachim Kanold (1957) proved that the asymptotic density of the harmonic numbers is 0 (see link), the asymptotic density of this sequence is also 0.
a(9) > 6.5*10^14 (verified using list of balanced numbers from Jud McCranie). All the numbers in this range that are both balanced and harmonic numbers are also arithmetic numbers. - Amiram Eldar, Mar 09 2021

			For 6: tau(6) = 4, phi(6) = 2, sigma(6) = 12, 6*tau(6)/sigma(6) = 6*4/12 = 2, sigma(6)/tau(6) = 3 and sigma(6)/phi(6) = 2, hence 6 is a term.

Hans-Joachim Kanold, Über das harmonische Mittel der Teiler einer natürlichen Zahl, Math. Ann., Vol. 133 (1957), pp. 371-374.
Oystein Ore, On the averages of the divisors of a number, Amer. Math. Monthly, Vol. 55, No. 10 (1948), pp. 615-619.
Oystein Ore, On the averages of the divisors of a number (annotated scanned copy).

Intersection of A001599, A003601 and A020492.
Intersection of A001599 and A342103.
Intersection of A007340 and A020492.
Cf. A000005, A000010, A000203.

with(numtheory): filter:= q -> (sigma(q) mod phi(q) = 0) and (sigma(q) mod tau(q) = 0 and (q*tau(q) mod sigma(q) = 0) : select(filter, [$1..300000]);

Select[Range[350000], And @@ Divisible[(s = DivisorSigma[1, #]), {(d = DivisorSigma[0, #]), EulerPhi[#]}] && Divisible[#*d, s] &] (* Amiram Eldar, Mar 09 2021 *)

isok(m) = my(s=sigma(m), t=numdiv(m)); !(s % eulerphi(m)) && !(s % t) && !((m*t) % s); \\ Michel Marcus, Mar 09 2021

a(6)-a(8) from Amiram Eldar, Mar 09 2021

cp's OEIS Frontend

A328944 Arithmetic numbers (A003601) that are not harmonic (A001599).

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A328945 Numbers m that are neither arithmetic (A003601) nor harmonic (A001599).

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A361387 Infinitary arithmetic numbers k whose mean infinitary divisor is an infinitary divisor of k.

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A157850 Numbers k such that are harmonic means of divisors of harmonic (Ore) numbers (harmonic numbers is A001599) and equal to one of the divisors of this harmonic numbers.

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A331666 Refactorable numbers (A033950) that are simultaneously arithmetic (A003601) and harmonic (A001599).

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A361787 Bi-unitary arithmetic numbers k whose mean bi-unitary divisor is a bi-unitary divisor of k.

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A342358 Balanced numbers (A020492) that are also arithmetic numbers (A003601) and harmonic numbers (A001599).

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