A342103 -id:A342103 - cp's OEIS frontend

A342104 Balanced numbers (A020492) that are not arithmetic numbers (A003601).

2, 12, 18630, 27000, 443394, 6242022, 14412720, 22315419, 26744100, 44630838, 50496960, 106034880, 128710944, 148536990, 162907584, 212072880, 218470770, 296259930, 349444530, 397253968, 535267776, 641250900, 641418960, 666274653, 684165552, 688208724, 709639408

Offset: 1

Bernard Schott, Feb 28 2021

nonn

Equivalently, numbers m such that phi(m) divides sigma(m) but tau(m) does not divide sigma(m), the corresponding quotients sigma(m)/phi(m) = A023897(m).

The only prime in the sequence is 2, because sigma(2)/phi(2) = 3 and sigma(2)/tau(2) = 3/2; then, if p odd prime, sigma(p)/phi(p) = (p+1)/(p-1) is an integer iff p = 3, but for p = 3, tau(3) divides sigma(3) with sigma(3)/tau(3) = 4/2 = 2.

			Sigma(12) = 28, phi(12) = 4 and tau(12) = 6, hence phi(12) divides sigma(12), but tau(12) does not divide sigma(12), so 12 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..5620

Equals A020492 \ A003601.
Cf. A000005 (tau), A000010 (phi), A000203 (sigma), A023897 (sigma/phi).
Cf. A342103, A342105, A342106.

with(numtheory): filter:= q -> (sigma(q) mod phi(q) = 0) and (sigma(q) mod tau(q) <> 0) : select(filter, [$1..500000]);

Select[Range[500000], Divisible[DivisorSigma[1, #], {DivisorSigma[0, #], EulerPhi[#]}] == {False, True} &] (* Amiram Eldar, Feb 28 2021 *)

isok(m) = my(s=sigma(m)); !(s % eulerphi(m)) && (s % numdiv(m)); \\ Michel Marcus, Mar 01 2021

a(5)-a(27) from Amiram Eldar, Feb 28 2021

A342105 Arithmetic numbers (A003601) that are not balanced numbers (A020492).

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 20, 21, 22, 23, 27, 29, 31, 33, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 51, 53, 54, 55, 57, 59, 60, 61, 62, 65, 66, 67, 68, 69, 71, 73, 77, 79, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 96, 97, 99, 101, 102, 103, 107, 109, 110, 111, 113, 114, 115, 116

Offset: 1

Bernard Schott, Mar 05 2021

nonn

Equivalently, numbers m such that tau(m) divides sigma(m) but phi(m) does not divide sigma(m), the corresponding quotients sigma(m)/tau(m) = A102187(m).

Primes in the sequence are primes >= 5; proof: 2 is in A342104 and 3 is in A342103, then for p prime >= 5, phi(p) = p-1 >= 4, tau(p) = 2, sigma(p) = p+1 >= 6; hence 2 divides p+1 but p-1 does not divide p+1.

			Sigma(21) = 32, tau(21) = 4 and phi(21) = 12, hence tau(21) divides sigma(21), but phi(21) does not divide sigma(21), so 21 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Equals A003601 \ A020492.
Cf. A000005 (tau), A000010 (phi), A000203 (sigma), A102187 (sigma/tau).
Cf. A342103, A342104, A342106.

with(numtheory): filter:= q -> (sigma(q) mod tau(q) = 0) and (sigma(q) mod phi(q) <> 0) : select(filter, [$1..120]);

Select[Range[120], Divisible[DivisorSigma[1, #], {DivisorSigma[0, #], EulerPhi[#]}] == {True, False} &] (* Amiram Eldar, Mar 05 2021 *)

isok(m) = my(s=sigma(m)); !(s % numdiv(m)) && (s % eulerphi(m)); \\ Michel Marcus, Mar 05 2021

A342106 Numbers that are not arithmetic and not balanced.

Original entry on oeis.org

4, 8, 9, 10, 16, 18, 24, 25, 26, 28, 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, 192, 194, 196, 200

Offset: 1

Bernard Schott, Mar 08 2021

nonn

Equivalently, numbers m such that neither tau(m) (A000005) nor phi(m) (A000010) divide sigma(m) (A000203).
A342103, A342104, A342105 and this sequence form a partition of the set of positive integers N* (A000027).
There are no primes in the sequence since 2 is in A342104, 3 is in A342103, and odd primes >= 5 are in A342105.

			Sigma(9) = 13, tau(9) = 3 and phi(9) = 6, hence sigma(9)/tau(9) = 13/3 and sigma(9)/phi(9) = 13/6, 9 is a term.

Cf. A000005 (tau), A000010 (phi), A000203 (sigma).
Cf. A020492, A003601.
Cf. A342103, A342104, A342105.

with(numtheory): filter:= q -> (sigma(q) mod phi(q) <> 0) and (sigma(q) mod tau(q) <> 0) : select(filter, [$1..200]);

Select[Range[200], ! Or @@ Divisible[DivisorSigma[1, #], {DivisorSigma[0, #], EulerPhi[#]}] &] (* Amiram Eldar, Mar 08 2021 *)

isok(m) = my(s=sigma(m)); (s % numdiv(m)) && (s % eulerphi(m)); \\ Michel Marcus, Mar 08 2021

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

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A342104 Balanced numbers (A020492) that are not arithmetic numbers (A003601).

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A342105 Arithmetic numbers (A003601) that are not balanced numbers (A020492).

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A342106 Numbers that are not arithmetic and not balanced.

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

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