A116541 -id:A116541 - cp's OEIS frontend

A287472 Triangular numbers k such that phi(k) is also a triangular number, where phi(k) is the Euler totient function (A000010).

1, 231, 1035, 6786, 190036, 193131, 766941, 1237951, 1348903, 3069003, 3396921, 8034036, 9152781, 11875501, 15694003, 28001386, 29587278, 35149920, 61643856, 63196903, 130758706, 178161126, 198214005, 227751153, 268111746, 339210081, 402102261, 654224878

Offset: 1

Amiram Eldar, May 25 2017

nonn

The indices of these triangular numbers are: 1, 21, 45, 116, 616, 621, 1238, 1573, 1642, 2477, 2606, 4008, 4278, 4873, 5602, 7483, 7692, 8384, 11103, 11242, 16171, 18876, 19910, 21342, 23156, 26046, 28358, 36172, 46196, 46621, 67572, 72816, ...

The indices of the triangular phi values are: 1, 15, 32, 63, 384, 495, 927, 1440, 1599, 1856, 2015, 2240, 3200, 4640, 5375, 4895, 4095, 4095, 6400, 9855, 10880, 9855, 13824, 16128, 12095, 19520, 21504, 25344, 25983, 45584, 37184, 40959, ...

			231 = 21*22/2 is triangular, phi(231)=120=15*16/2 is also triangular, thus 231 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A000010, A000217, A083674, A083675, A115910, A116541.

triQ[n_] := IntegerQ@Sqrt[8n+1]; Select[Accumulate[Range[1000]], triQ[EulerPhi[#]]&]

isok(n) = ispolygonal(n, 3) && ispolygonal(eulerphi(n), 3); \\ Michel Marcus, May 25 2017

A292063 Triangular numbers n such that psi(n) is also a triangular number, where psi is the Dedekind psi function (A001615).

Original entry on oeis.org

1, 780, 2775, 5050, 474825, 681528, 1727011, 5286126, 5911641, 6604795, 17325441, 21612025, 27799696, 45025305, 386767578, 1538599128, 2086160121, 3679490220, 5718242211, 7092226351, 8019794628, 16505718895, 36604197735, 55541611986, 56693041356, 89369984476

Offset: 1

Amiram Eldar, Sep 08 2017

nonn

The indices of these triangular numbers are 1, 39, 74, 100, 974, 1167, 1858, 3251, 3438, 3634, 5886, 6574, 7456, 9489, ...

The indices of the triangular psi values are 1, 63, 95, 135, 1280, 1664, 2015, 4607, 4095, 4095, 7424, 7424, 9152, 12543, ...

			780 is in the sequence since 780 = 39*40/2 is triangular and psi(780) = 2016 = 63*64/2 is also triangular.

Giovanni Resta, Table of n, a(n) for n = 1..100

Cf. A000217, A001615, A083674, A083675, A115910, A116541, A287472.

psi[n_] := If[n<1, 0, n*Sum[MoebiusMu[d]^2/d, {d, Divisors @ n}]]; triQ[n_] := IntegerQ@ Sqrt[8n+1]; Select[Accumulate[Range[1000]], triQ[psi[#]]&]

a(18)-a(26) from Giovanni Resta, Sep 11 2017

A323747 Smallest triangular number whose number of divisors is the n-th triangular number, or 0 if no such number exists.

Original entry on oeis.org

1, 0, 28, 496, 1631432881, 0, 8256, 2016, 41616, 0, 169878528, 2717872128, 0

Offset: 1

Jon E. Schoenfield, May 25 2019

Additional known terms include a(15)=270480, a(16)=77309214720, a(19)=117261433825538425475625, a(20)=7874496, a(22)=0, a(23)=316659361382400, a(24)=100472400, a(25)=0, a(27)=18951806016, a(28)=35184372088827805696000000, a(31)=20752587086144471040, a(32)=3877678080.
It is known (see the comments and links at A081978) that a(n)=0 for every n such that n*(n+1)/2 is an odd composite not divisible by 3; this includes n = 10, 13, 22, 25, ..., i.e., all n such that n mod 12 is 1 or 10.
Conjectures:
1. a(n) > 0 for every n such that n*(n+1)/2 is even.
2. a(n) = 0 for every n such that n*(n+1)/2 is odd except n = 1, 5, and 9 (whose corresponding values of n*(n+1)/2 are 1, 15, and 45, respectively). Can this be proved for any of the values of n in {14, 17, 18, 21, 26, 29, 30}?

			a(1) = 1 because 1 is the only triangular number having A000217(1)=1 divisors.
a(2) = 0 because no triangular number has A000217(2)=3 divisors. (Each number with 3 divisors is the square of a prime, and no such number can be of the form k*(k+1)/2.)
a(3) = 28 because 28 = 7*(7+1)/2 = 2^2 * 7 is the smallest triangular number with A000217(3)=6 divisors.
a(5) = 1631432881 = 13^4 * 239^2 is the only triangular with A000217(5)=15 divisors.

Cf. A000217, A081978, A116541, A242585, A292989.

a(6)-a(13) and updated comments from Jon E. Schoenfield, Jan 29 2021

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A287472 Triangular numbers k such that phi(k) is also a triangular number, where phi(k) is the Euler totient function (A000010).

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A292063 Triangular numbers n such that psi(n) is also a triangular number, where psi is the Dedekind psi function (A001615).

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A323747 Smallest triangular number whose number of divisors is the n-th triangular number, or 0 if no such number exists.

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