A083674 -id:A083674 - cp's OEIS frontend

A256149 Square numbers n such that sigma(n) is a triangular number.

1, 36, 441, 5625, 6084, 407044, 8444836, 17388900, 35070084, 40729924, 57790404, 80138304, 537822481, 588159504, 659821969, 918999225, 1820387556, 2179862721, 2599062361, 5110963081, 28816420516, 36144473689, 46082779561, 55145598561, 147225690000, 163405126756

Offset: 1

Antonio Roldán, Mar 16 2015

nonn

This sequence is the intersection of A000290 and A045746.

			441 is in the sequence because 441 = 21^2 is square number, and sigma(441) = 441 + 147 + 63 + 49 + 21 + 9 + 7 + 3 + 1 = 741 = 38*39/2 is triangular number.

Amiram Eldar, Table of n, a(n) for n = 1..72

Cf. A000290, A000217, A045746, A083674, A256150, A256151, A256152.

t = Accumulate[Range@ 10000]; Select[Range[10000]^2, MemberQ[t, DivisorSigma[1, #]] &] (* Michael De Vlieger, Mar 17 2015 *)
Select[Range[500000]^2,OddQ[Sqrt[8DivisorSigma[1,#]+1]]&] (* Harvey P. Dale, Feb 25 2017 *)

{for(i=1,10^6,n=i*i;if(ispolygonal(sigma(n), 3),print1(n,", ")))}

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

Original entry on oeis.org

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

A226363 Oblong numbers (A002378) whose sum of divisors is also an oblong number.

Original entry on oeis.org

6, 20, 30, 306, 1722, 2862, 11772, 13572, 28730, 29756, 40602, 54056, 219492, 351056, 463080, 947702, 1391220, 1546292, 2043470, 7174362, 7703400, 11400752, 15104882, 19127502, 20155610, 113667582, 172173762, 314299712, 475654290, 555238532, 558447792, 562519806

Offset: 1

Alex Ratushnyak, Jun 05 2013

nonn

Donovan Johnson, Table of n, a(n) for n = 1..200

Cf. A000203, A002378, A083674.

oblongQ[n_] := IntegerQ[(-1 + Sqrt[1 + 4*n])/2]; s = Select[Range[24000], oblongQ[DivisorSigma[1, # (# + 1)]] &]; s * (s + 1) (* T. D. Noe, Jun 12 2013 *)

A259240 Least n-gonal number greater than 1 such that sigma(n) is also n-gonal.

Original entry on oeis.org

36, 81, 590, 5286126, 15880, 1932821, 37990539325, 6280, 234222782808, 3350529, 931738, 455621651099, 3312, 2680, 373569353, 1128231876, 47531850550, 601657, 4609261, 115668, 164642040082433296, 336577944, 40161257476, 5031720, 31424211, 25785, 12670237746

Offset: 3

Michel Marcus, Jun 22 2015

nonn

			For n=4, 81 is a square and sigma(81)=121 is also a square.

Hiroaki Yamanouchi, Table of n, a(n) for n = 3..85
Eric Weisstein's World of Mathematics, Polygonal Number

Cf. A000203, A008848, A083674.

a(n) = {k = 2; while(! ((p = k*((n-2)*k-(n-4))/2) && ispolygonal(sigma(p), n)), k++); k;}

a(23)-a(29) from Hiroaki Yamanouchi, Sep 26 2015

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

A317478 Triangular numbers whose sum of divisors is an oblong number.

Original entry on oeis.org

6, 28, 55, 496, 666, 780, 1540, 2145, 6441, 6903, 8128, 15051, 21736, 36585, 44551, 232903, 234955, 644680, 2258875, 3186550, 3462396, 6211050, 22174470, 33550336, 48437403, 62591266, 107538445, 134898525, 153554050, 624157446, 1309312378, 1339937028

Offset: 1

Amiram Eldar, Jul 29 2018

Includes all the even perfect numbers.
The indices of these triangular numbers are 3, 7, 10, 31, 36, 39, 55, 65, 113, 117, 127, 173, 208, 270, 298, 682, 685, 1135, 2125, 2524, 2631, 3524, 6659, 8191, 9842, 11188, 14665, 16425, 17524, 35331, 51172, 51767, 52019, 52486, 58993, 65585, 97532.
The indices of the corresponding oblong numbers are 3, 7, 8, 31, 38, 48, 63, 63, 95, 104, 127, 144, 224, 255, 224, 512, 575, 1215, 1728, 2448, 3072, 3968, 7695, 8191, 9215, 9792, 12159, 15872, 17576, 37296, 46656, 58239, 63855, 40959, 46080, 62720, 102960.
Number of terms < 10^k, k=1,2,3...: 1, 3, 6, 11, 15, 18, 22, 26, 30, 40, 52, 64, 80, 90, 110, 128, ..., . - Robert G. Wilson v, Jul 31 2018

			55 is in the sequence since sigma(55) = 72 = 8 * 9 is an oblong number.

Robert G. Wilson v, Table of n, a(n) for n = 1..129

Cf. A000203, A000217, A000396, A002378, A083674, A083675, A175849.

Intersection of A000217 and A236387. - Michel Marcus, Jul 30 2018

tri[n_] := n(n+1)/2; aQ[n_] := IntegerQ[Sqrt[4 * DivisorSigma[1, tri[n]] + 1]]; tri[Select[Range[52000], aQ]]
Module[{nn=60000,obno},obno=Table[n(n+1),{n,nn}];Select[Accumulate[Range[nn]],MemberQ[ obno,DivisorSigma[1,#]]&]] (* Harvey P. Dale, Aug 26 2024 *)

cp's OEIS Frontend

A256149 Square numbers n such that sigma(n) is a triangular number.

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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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A226363 Oblong numbers (A002378) whose sum of divisors is also an oblong number.

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A259240 Least n-gonal number greater than 1 such that sigma(n) is also n-gonal.

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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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A317478 Triangular numbers whose sum of divisors is an oblong number.

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