A045746 -id:A045746 - 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,", ")))}

A256150 Oblong numbers n such that sigma(n) is a triangular number.

Original entry on oeis.org

2, 12, 56, 342, 992, 16256, 17822, 169332, 628056, 1189190, 2720850, 11085570, 35599122, 67100672, 1147210770, 1317435912, 1707135806, 7800334080, 11208986256, 13366943840, 17109032402, 17179738112, 46343540900, 58413331032, 83717924940, 204574837700, 274877382656, 445968192672, 589130699852

Offset: 1

Antonio Roldán, Mar 16 2015

nonn

The numbers 12, 56, 992, 16256, 67100672, ... (A139256(n), twice even perfect numbers) are in the sequence because they are oblong (A139256(n) = 2^k*(2^k-1)) and sigma(A139256(n)) = sigma(2^k*(2^k-1)) = sigma(2^k)*sigma(2^k-1) = (2^(k+1)-1)*2^(k+1)/2 is a triangular number.

This sequence is the intersection of A002378 and A045746.

			2 is in the sequence because 2=1*2 is oblong, and sigma(2) = 1+2 = 3 = 2*3/2 is triangular.

Cf. A000217, A002378, A139256, A045746, A256149, A256151, A256152.

Select[2 Accumulate[Range@10000], MemberQ[Accumulate[Range@10000], DivisorSigma[1, #]] &] (* Michael De Vlieger, Mar 17 2015 *)

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

A180929 Numbers whose sum of divisors is a pentagonal number.

Original entry on oeis.org

1, 6, 11, 104, 116, 129, 218, 363, 408, 440, 481, 534, 566, 568, 590, 638, 646, 684, 718, 807, 895, 979, 999, 1003, 1007, 1137, 1251, 1282, 1557, 1935, 2197, 2367, 2571, 2582, 2808, 2855, 3132, 3283, 3336, 3578, 3737, 3891, 3946, 3980, 4172, 4484, 4886, 5158

Offset: 1

Jonathan Vos Post, Sep 25 2010

			a(1) = 1 because the sum of divisors of 1 is the pentagonal number 1.
a(2) = 6 because the sum of divisors of 6 is the pentagonal number 12.
a(3) = 11 because the sum of divisors of 11 is the pentagonal number 12.
a(4) = 104 because the sum of divisors of 104 is the pentagonal number 210.
a(5) = 116 because the sum of divisors of 116 is the pentagonal number 210.
a(6) = 129 because the sum of divisors of 129 is the pentagonal number 176.
a(7) = 218 because the sum of divisors of 218 is the pentagonal number 330.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000203, A000326.

Numbers whose sum of divisors is a ...: A045746 (triangular number), A006532 (square), A180930 (hexagonal number).

isA000326 := proc(n) if not issqr(24*n+1) then false; else sqrt(24*n+1)+1 ; (% mod 6) = 0 ; end if; end proc:
for n from 1 to 5000 do if isA000326(numtheory[sigma](n)) then printf("%d,",n) ; end if; end do: # R. J. Mathar, Sep 26 2010

pnos=Table[(n (3n-1))/2,{n,500}]; okQ[n_]:=Module[{ds=DivisorSigma[1,n]},MemberQ[pnos,ds]] Select[Range[5000],okQ] (* Harvey P. Dale, Sep 26 2010 *)
Select[Range[5200],IntegerQ[(Sqrt[1+24DivisorSigma[1,#]]+1)/6]&] (* Harvey P. Dale, Jun 14 2013 *)

is(n)=ispolygonal(sigma(n),5) \\ Jason Yuen, Oct 14 2024

A000203(a(n)) is in A000326.

Corrected and extended by R. J. Mathar and Harvey P. Dale, Sep 26 2010

A180930 Numbers whose sum of divisors is a hexagonal number.

Original entry on oeis.org

1, 5, 8, 12, 36, 54, 56, 87, 95, 160, 212, 328, 342, 356, 427, 531, 660, 672, 843, 852, 858, 909, 910, 940, 992, 1002, 1012, 1162, 1222, 1245, 1353, 1417, 1435, 1495, 1509, 1547, 1757, 1837, 1909, 1927, 1998, 2072, 2274, 2793, 2983, 3051, 3212, 3219, 3515, 3548, 3870

Offset: 1

Jonathan Vos Post, Sep 26 2010

nonn

54, 56, 87, and 95 are the smallest four numbers whose sum of divisors is the same hexagonal number (120).

			a(1) = 1 because the sum of divisors of 1 is the hexagonal number 1.
a(2) = 5 because the sum of divisors of 5 is the hexagonal number 6.
a(3) = 8 because the sum of divisors of 8 is the hexagonal number 15.
a(4) = 12 because the sum of divisors of 12 is the hexagonal number 28.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000203, A000384.

Numbers whose sum of divisors is a ...: A045746 (triangular number), A006532 (square), A180929 (pentagonal number).

isA000384 := proc(n) if not issqr(8*n+1) then false; else sqrt(8*n+1)+1 ; (% mod 4) = 0 ; end if; end proc:
for n from 1 to 4000 do if isA000384(numtheory[sigma](n)) then printf("%d,",n) ; end if; end do: # R. J. Mathar, Sep 26 2010

hnos=Table[n (2n-1),{n,500}]; okQ[n_]:=Module[{ds=DivisorSigma[1,n]},MemberQ[hnos,ds]] Select[Range[5000],okQ] (* Harvey P. Dale, Sep 26 2010 *)

is(n)=ispolygonal(sigma(n),6) \\ Jason Yuen, Oct 14 2024

A000203(a(n)) is in A000384.

Corrected and extended by several authors, Sep 27 2010

A113930 Numbers k such that sigma(k) and phi(k) are both triangular numbers.

Original entry on oeis.org

1, 2, 22, 3051, 3219, 3393, 5057, 8653, 75618, 95675, 100503, 102949, 104714, 287826, 438547, 522339, 537159, 688050, 2191200, 2317118, 2418548, 2507683, 2599128, 3212964, 4534573, 5367797, 6047913, 6302639, 7689149, 13758296, 14380145, 15342050, 16148979

Offset: 1

Giovanni Resta, Jan 30 2006

nonn

phi(k) = A000010(k) is the Euler totient function, while sigma(k) = A000203(k) is the sum of divisors of k.

			sigma(100503) = 156520 = T(559) and phi(100503) = 61776 = T(351).

Donovan Johnson, Table of n, a(n) for n = 1..207 (terms < 10^12)

Cf. A000010, A000203, A000217.

Subsequence of A045746.

isok(n) = ispolygonal(sigma(n), 3) && ispolygonal(eulerphi(n), 3); \\ Michel Marcus, Jan 26 2014

A291485 Numbers m such that sigma(x) = m*(m+1)/2 has at least one solution.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 12, 13, 15, 18, 20, 24, 27, 30, 31, 32, 35, 38, 39, 47, 48, 51, 55, 56, 62, 63, 64, 79, 80, 84, 90, 92, 95, 96, 104, 111, 116, 119, 120, 128, 135, 140, 142, 143, 144, 147, 152, 155, 156, 159, 160, 167, 168, 170, 171, 175, 176, 182, 184, 188, 191, 192, 195, 203, 207, 208

Offset: 1

Altug Alkan, Aug 24 2017

Let b(n) be the smallest k such that sigma(k) is the n-th triangular number, or 0 if no such k exists. For n >= 1, b(n) sequence is 1, 2, 5, 0, 8, 0, 12, 22, 0, 0, 0, 45, 36, 0, 54, 0, 0, 98, 0, 104, 0, 0, 0, 152, 0, 0, 160, 0, 0, 200, ...

			15 is a term because sigma(54) = sigma(56) = sigma(87) = sigma(95) = A000217(15).

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

Cf. A000203, A000217, A045746.

N:= 1000: # to get all terms <= N
Sigmas:= {seq(numtheory:-sigma(x),x=1..N*(N+1)/2)}:
select(t -> member(t*(t+1)/2, Sigmas), [$1..N]); # Robert Israel, Aug 25 2017

invT[n_] := (Sqrt[8*n+1]-1)/2; Union@ Select[invT /@ DivisorSigma[1, Range[ 208*209/2]], IntegerQ[#] && # <= 208 &] (* Giovanni Resta, Aug 25 2017 *)

A329704 Numbers k such that the sum of divisors of k (A000203) and the sum of proper divisors of k (A001065) are both triangular numbers (A000217).

Original entry on oeis.org

1, 2, 5, 36, 54, 441, 473, 6525, 52577, 124025, 683820, 1513754, 1920552, 6079931, 6762923, 14751657, 17052782, 17310942, 36543714, 49919939, 60260967, 251849052, 364535720, 372476909, 562047389, 670395564, 670440852, 783856979, 824626800, 1084201689, 1122603809

Offset: 1

Amiram Eldar, Feb 28 2020

nonn

Are 1 and 36 the only terms that are also triangular numbers?

No other triangular terms up to A000217(10^8). - Michel Marcus, Mar 01 2020

			5 is a term since sigma(5) = 6 and sigma(5) - 5 = 1 are both triangular numbers.

Amiram Eldar, Table of n, a(n) for n = 1..48

Intersection of A045745 and A045746.

Cf. A000203, A000217, A001065.

triQ[n_] := IntegerQ @ Sqrt[8*n+1]; Select[Range[10^5], triQ[(s = DivisorSigma[1, #])] && triQ[s - #] &]

isok(k) = my(s=sigma(k)); ispolygonal(s, 3) && ispolygonal(s-k, 3); \\ Michel Marcus, Feb 29 2020

A232619 Numbers k such that sigma(k) and sigma(2*k) are triangular numbers.

Original entry on oeis.org

1, 3199808, 36210072, 48138616, 106449517, 109040509, 109903789, 110003837, 204102760401, 3050610977780, 3655577208308, 3660654715316, 3660725004788, 5607890017600, 6123048180800, 7412048520512, 7471435008064, 53218856249194, 53221469311786, 71505497845764, 103285924642372, 176553289677888, 176596975245888, 176600346573888

Offset: 1

Alex Ratushnyak, Nov 27 2013

			sigma(3199808) = 6784086 = triangular(3683),  sigma(3199808*2) = 13621590 = triangular(5219), so 3199808 is in the sequence.

A subsequence of A045746.

Cf. A000203, A000217, A232445.

is(n)=issquare(8*sigma(n)+1)&&issquare(8*sigma(2*n)+1) \\ Ralf Stephan, Nov 30 2013

a(9) from Hiroaki Yamanouchi, Oct 02 2014

Terms a(10) onward from Max Alekseyev, Feb 21 2024

cp's OEIS Frontend

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

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A256150 Oblong numbers n such that sigma(n) is a triangular number.

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A180929 Numbers whose sum of divisors is a pentagonal number.

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A180930 Numbers whose sum of divisors is a hexagonal number.

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A113930 Numbers k such that sigma(k) and phi(k) are both triangular numbers.

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A291485 Numbers m such that sigma(x) = m*(m+1)/2 has at least one solution.

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A329704 Numbers k such that the sum of divisors of k (A000203) and the sum of proper divisors of k (A001065) are both triangular numbers (A000217).

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