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.

A342103 Balanced numbers (A020492) that are also arithmetic numbers (A003601).

Original entry on oeis.org

1, 3, 6, 14, 15, 30, 35, 42, 56, 70, 78, 105, 140, 168, 190, 210, 248, 264, 270, 357, 418, 420, 570, 594, 616, 630, 714, 744, 812, 840, 910, 1045, 1240, 1254, 1485, 1672, 1848, 2090, 2214, 2376, 2436, 2580, 2730, 2970, 3080, 3135, 3339, 3596, 3720, 3828, 3956, 4064, 4180
Offset: 1

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Author

Bernard Schott, Feb 28 2021

Keywords

Comments

Equivalently, numbers m such that phi(m) (A000010) and tau(m) (A000005) both divide sigma(m) (A000203). In this case, the quotients sigma(m)/phi(m) = A023897(m) and sigma(m)/tau(m) = A102187(m).
Phi, tau and sigma are multiplicative functions and for this reason if k and q are coprime and included in this sequence then k*q is another term.
The only prime in the sequence is 3, because sigma(2)/tau(2) = 3/2 and when p is an odd prime, sigma(p)/phi(p) = (p+1)/(p-1) is an integer iff p=3 with sigma(3)/phi(3) = 4/2 = 2, and also sigma(3)/tau(3) = 4/2 = 2.

Examples

			phi(30) = tau(30) = 8, sigma(30) = 72 and 72/8 = 9, hence 30 is a term.
phi(12) = 4, tau(12) = 6, sigma(12) = 28, phi(12) divides sigma(12), but tau(12) does not divide sigma(12), hence 12 is a balanced number but is not an arithmetic number, and 12 is not a term.
phi(20) = 8, tau(20) = 6, sigma(20) = 42, tau(20) divides sigma(20), but phi(20) does not divide sigma(20), hence 20 is an arithmetic number but is not a balanced number, and 20 is not a term.

Links

Crossrefs

Intersection of A003601 and A020492.
Cf. A000005 (tau), A000010 (phi), A000203 (sigma), A023897 (sigma/phi), A102187 (sigma/tau).
Cf. A342104, A342105, A342106.

Programs

  • Maple
    with(numtheory): filter:= q -> (sigma(q) mod phi(q) = 0) and (sigma(q) mod tau(q) = 0) : select(filter, [$1..5000]);
  • Mathematica
    Select[Range[5000], And @@ Divisible[DivisorSigma[1, #], {DivisorSigma[0, #], EulerPhi[#]}] &] (* Amiram Eldar, Feb 28 2021 *)
  • PARI
    isok(m) = my(s=sigma(m)); !(s % eulerphi(m)) && !(s % numdiv(m)); \\ Michel Marcus, Mar 01 2021

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

Original entry on oeis.org

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

Views

Author

Bernard Schott, Feb 28 2021

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

Programs

  • Maple
    with(numtheory): filter:= q -> (sigma(q) mod phi(q) = 0) and (sigma(q) mod tau(q) <> 0) : select(filter, [$1..500000]);
  • Mathematica
    Select[Range[500000], Divisible[DivisorSigma[1, #], {DivisorSigma[0, #], EulerPhi[#]}] == {False, True} &] (* Amiram Eldar, Feb 28 2021 *)
  • PARI
    isok(m) = my(s=sigma(m)); !(s % eulerphi(m)) && (s % numdiv(m)); \\ Michel Marcus, Mar 01 2021

Extensions

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

Views

Author

Bernard Schott, Mar 05 2021

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

Programs

  • Maple
    with(numtheory): filter:= q -> (sigma(q) mod tau(q) = 0) and (sigma(q) mod phi(q) <> 0) : select(filter, [$1..120]);
  • Mathematica
    Select[Range[120], Divisible[DivisorSigma[1, #], {DivisorSigma[0, #], EulerPhi[#]}] == {True, False} &] (* Amiram Eldar, Mar 05 2021 *)
  • PARI
    isok(m) = my(s=sigma(m)); !(s % numdiv(m)) && (s % eulerphi(m)); \\ Michel Marcus, Mar 05 2021
Showing 1-3 of 3 results.

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