A006532 -id:A006532 - cp's OEIS frontend

A006534 Number of one-sided triangular polyominoes (n-iamonds) with n cells; turning over not allowed, holes are allowed.

1, 1, 1, 4, 6, 19, 43, 120, 307, 866, 2336, 6588, 18373, 52119, 147700, 422016, 1207477, 3471067, 9999135, 28893560, 83665729, 242826187, 706074369, 2056870697, 6001555275, 17538335077, 51323792789, 150390053432, 441210664337, 1295886453860, 3810208448847, 11214076720061, 33035788241735

Offset: 1

N. J. A. Sloane

The figures are formed by connecting n regular triangles by edges.

"Turning over not allowed" means that axial symmetric polyiamonds are counted separately, thus a(4) = 4 and a(5) = 6 while A000577(4) = 3 and A000577(5) = 4, cf. examples. - M. F. Hasler, Nov 12 2017

			From _M. F. Hasler_, Nov 12 2017: (Start)
Putting dots for the approximate center of the regular triangles (alternatively flipped up and down for neighboring dots), we have:
a(4) = #{ .... , .:. , ..: , :.. } = 4, while ..: and :.. are considered equivalent and not counted twice in A000577(4) = 3.
a(5) = #{ ..... , ...: , :... , ..:. , .:.. , :.: } = 6, and again the 2nd & 3rd and 4th & 5th are considered equivalent and not counted twice in A000577(5) = 4. (End)

F. Harary, Graphical enumeration problems; in Graph Theory and Theoretical Physics, ed. F. Harary, Academic Press, London, 1967, pp. 1-41.
W. F. Lunnon, Counting hexagonal and triangular polyominoes, pp. 87-100 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
P. J. Torbijn, Polyiamonds, J. Rec. Math. 2 (1969), 216-227.

John Mason, Table of n, a(n) for n = 1..52
R. K. Guy, O'Beirne's Hexiamond, in The Mathemagician and the Pied Puzzler - A Collection in Tribute to Martin Gardner, Ed. E. R. Berlekamp and T. Rogers, A. K. Peters, 1999, 85-96.
J. Meeus & N. J. A. Sloane, Correspondence, 1974-1975
Ed Pegg, Jr., Illustrations of polyforms
Eric Weisstein's World of Mathematics, Polyiamond.

Cf. A000577 (same with "turning over allowed"), A030223, A030224, A001420.

Corrected and extended by David W. Wilson
a(19) from Achim Flammenkamp, Feb 15 1999
a(20) to a(28) from Joseph Myers, Sep 24 2002
Edited by M. F. Hasler, Nov 12 2017
More terms from John Mason, Oct 28 2023

A019424 Numbers whose sum of divisors is a sixth power.

Original entry on oeis.org

1, 2667, 3937, 17490, 19410, 22578, 24610, 24910, 25466, 25910, 26554, 26818, 27285, 29342, 29733, 29762, 31102, 31535, 32043, 32997, 33985, 35585, 36635, 37985, 39697, 41393, 41837, 42347, 44047, 45637, 45739, 45937, 46117, 172011, 253921, 640737

Offset: 1

David W. Wilson

nonn

			sigma(2667) = 1+3+7+21+127+381+889+2667 = 4096 = 4^6.
sigma(3937) = 1+31+127+3937 = 4096 = 4^6.

Marius A. Burtea, Table of n, a(n) for n = 1..2010 (terms 1..1000 from Donovan Johnson)
Frits Beukers, Florian Luca and Frans Oort, Power Values of Divisor Sums, The American Mathematical Monthly, Vol. 119, No. 5 (May 2012), pp. 373-380.

Cf. A000203, A006532, A019422, A019423, A020477, A048257, A048258.

[n:n in [1..100000]| IsPower(SumOfDivisors(n),6)]; // Marius A. Burtea, Apr 17 2019

Select[Range[700000],IntegerQ[Surd[DivisorSigma[1,#],6]]&] (* Harvey P. Dale, Apr 19 2019 *)

c=0; for(n=1, 306455560, if(ispower(sigma(n), 6), c++; write("b019424.txt", c " " n))) /* Donovan Johnson, Jun 13 2013 */

A048256 Numbers whose sum of divisors is 6^6 = 46656.

Original entry on oeis.org

17490, 19410, 22578, 24610, 24910, 25466, 25910, 26554, 26818, 27285, 29342, 29733, 29762, 31102, 31535, 32043, 32997, 33985, 35585, 36635, 37985, 39697, 41393, 41837, 42347, 44047, 45637, 45739, 45937, 46117

Offset: 1

Labos Elemer

Sequence has A048253(6)=30 terms from A048251(6)=17490 to A048252(6)=46117. - Ray Chandler, Sep 01 2010

			The divisors of 19410 are 1, 2, 3, 5, 6, 10, 15, 30, 647, 1294, 1941, 3235, 3882, 6470, 9705, and 19410; their sum is 46656, so 19410 is in the sequence.

Cf. A006532, A020477, A019422, A019423, A018427, A048251-A048255.

Select[Range[6^6], DivisorSigma[1, # ] == 6^6 &] (* Ray Chandler, Sep 01 2010 *)

A048257 Integers whose sum of divisors is a 7th power.

Original entry on oeis.org

1, 93, 127, 11811, 112890, 120054, 124338, 127330, 132770, 133998, 134090, 137058, 138754, 139962, 146710, 148665, 148810, 149534, 153986, 155510, 160215, 161194, 164985, 167134, 170986, 173098, 183687, 184682, 187143, 191913, 198485, 206823, 206965, 207687

Offset: 1

Labos Elemer

nonn

If m and n are coprime members of the sequence, then m*n is also a member. - Robert Israel, May 10 2018

			Divisors(11811) = {1,3,31,93,127,381,3937,11811} and sigma(11811) = 16384 = 4^7.

Donovan Johnson, Table of n, a(n) for n = 1..1000
Frits Beukers, Florian Luca and Frans Oort, Power Values of Divisor Sums, The American Mathematical Monthly, Vol. 119, No. 5 (May 2012), pp. 373-380.

Cf. A000203, A006532, A020477, A019422, A019423, A019424, A048251-A048256, A048258.

filter:= n -> type(map(t -> t[2]/7, ifactors(numtheory:-sigma(n))[2]),list(integer)):
select(filter, [$1..21*10^4]); # Robert Israel, May 09 2018

Select[Range[210000],IntegerQ[Surd[DivisorSigma[1,#],7]]&] (* Harvey P. Dale, Jun 09 2017 *)

isok(n) = ispower(sigma(n), 7); \\ Michel Marcus, Dec 20 2013

sigma(a(n)) = x^7, where the initial values of x are 1, 2, 4, 6 (48 times), ...

A048258 Integers whose sum of divisors is an 8th power.

Original entry on oeis.org

1, 217, 57337, 600270, 621690, 669990, 685290, 693294, 693770, 699810, 725934, 747670, 769930, 774894, 782598, 805970, 813378, 823938, 835670, 839802, 854930, 865490, 873334, 895594, 918435, 920414, 923410, 931634, 935715, 959565, 965174, 969034, 969206

Offset: 1

Labos Elemer

nonn

			Divisors(217) = {1,7,31,217}, sum = 256 = 2^8.
Divisors(57337) = {1,7,8191,57337}, sum = 65536 = 4^8.
Divisors(1676377) = {1,647,2591,1676377}, sum = 1679616 = 6^8.

Donovan Johnson, Table of n, a(n) for n = 1..1000
Frits Beukers, Florian Luca and Frans Oort, Power Values of Divisor Sums, The American Mathematical Monthly, Vol. 119, No. 5 (May 2012), pp. 373-380.

Cf. A006532, A020477, A019422, A019423, A019424, A048251-A048257.

Select[Range[10^6], IntegerQ@ Surd[DivisorSigma[1, #], 8] &] (* Michael De Vlieger, Aug 01 2017, after Harvey P. Dale at A048257 *)

isok(n) = ispower(sigma(n), 8); \\ Michel Marcus, Dec 30 2013

Sigma(1, a(n)) = x^8, where the initial values of x are 1, 2, 4, 6 (occurs 85 times), ...

A186438 Positive numbers whose squares end in two identical digits.

Original entry on oeis.org

10, 12, 20, 30, 38, 40, 50, 60, 62, 70, 80, 88, 90, 100, 110, 112, 120, 130, 138, 140, 150, 160, 162, 170, 180, 188, 190, 200, 210, 212, 220, 230, 238, 240, 250, 260, 262, 270, 280, 288, 290, 300, 310, 312, 320, 330, 338, 340, 350, 360, 362, 370, 380, 388, 390, 400, 410, 412

Offset: 1

Michel Lagneau, Feb 21 2011

The numbers are of the form : 10k, or 50k - 12, or 50k + 12, or 50k + 38.

			62 is in the sequence because 62^2 = 3844.

Jean Meeus, Letter to N. J. A. Sloane, Dec 26 1974.

J. Meeus & N. J. A. Sloane, Correspondence, 1974-1975
Ana Rechtman, Juillet 2021, 4e défi, Images des Mathématiques, CNRS, 2021 (in French).
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Cf. A016742 (even squares), A123912.

with(numtheory):T:=array(1..10):for p from 1 to 1000 do:n:=p^2:l:=length(n):n0:=n:for
  m from 1 to l do:q:=n0:u:=irem(q,10):v:=iquo(q,10):n0:=v :T[m]:=u:od:if T[1]=T[2]
  then printf(`%d, `,p):else fi:od:

tidQ[n_]:=Module[{idn=IntegerDigits[n^2]},idn[[-1]]==idn[[-2]]]; Select[ Range[ 4,500],tidQ] (* or *) LinearRecurrence[{1,0,0,0,0,0,1,-1},{10,12,20,30,38,40,50,60},60] (* Harvey P. Dale, Jan 25 2014 *)

G.f.: 2*x*(5*x^6+x^5+4*x^4+5*x^3+4*x^2+x+5)/((x-1)^2 * (x^6+x^5+x^4+x^3+x^2+x+1)). [Colin Barker, Jul 02 2012]

A256151 Triangular numbers n such that sigma(n) is a square number.

Original entry on oeis.org

1, 3, 66, 210, 820, 2346, 4278, 22578, 27966, 32131, 35511, 51681, 53956, 102378, 169653, 173755, 177906, 223446, 241860, 256686, 306153, 310866, 349866, 431056, 434778, 470935, 491536, 512578, 567645, 579426, 688551, 799480, 845650, 893116, 963966, 1031766, 1110795, 1200475, 1613706, 1719585

Offset: 1

Antonio Roldán, Mar 16 2015

nonn

This sequence is the intersection of A000217 and A006532.

The corresponding triangular indices are in A116990. - Michel Marcus, Mar 17 2015

			3 is in the sequence because 3=2*3/2 is triangular, and sigma(3)=1+3=4=2^2 is square.

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

Cf. A000290, A000217, A006532, A074285, A116990, A256149, A256150, A256152.

[n*(n+1) div 2: n in [1..2000] | IsSquare(SumOfDivisors(n*(n+1) div 2))]; // Vincenzo Librandi, Mar 17 2015

Select[Accumulate[Range[0, 2000]], IntegerQ@Sqrt@DivisorSigma[1, #] &] (* Michael De Vlieger, Mar 17 2015 *)

{for(i=1,2*10^3,n=i*(i+1)/2;if(issquare(sigma(n)),print1(n,", ")))}

A048252 Largest number whose sum of divisors is 6^n.

Original entry on oeis.org

1, 5, 22, 187, 1219, 7597, 46117, 278857, 1676377, 10067797, 60450517, 362758177, 2176626817, 13060193977, 78363525817, 470183516857, 2820894903487, 16926601754197, 101559860054047, 609359671998037, 3656158318966357

Offset: 0

Labos Elemer

nonn

Terms of this sequence are products of distinct terms in A005105. - Ray Chandler, Sep 01 2010

Ray Chandler, Table of n, a(n) for n = 0..1000

Cf. A006532, A020477, A019422, A019423, A018427, A048251-A048256.

a(n) = {sn = 6^n; forstep(x=sn, 1, -1, if (sigma(x) == sn, return (x)););} \\ Michel Marcus, Dec 15 2013

a(9)-a(14) from Donovan Johnson, Sep 02 2008
a(15)-a(20) from Donovan Johnson, Nov 22 2008
Edited and extended by Ray Chandler, Sep 01 2010

A256152 Numbers k such that k is the product of two distinct primes and sigma(k) is a square number.

Original entry on oeis.org

22, 94, 115, 119, 214, 217, 265, 382, 497, 517, 527, 679, 745, 862, 889, 1174, 1177, 1207, 1219, 1393, 1465, 1501, 1649, 1687, 1915, 1942, 2101, 2159, 2201, 2359, 2899, 2902, 2995, 3007, 3143, 3383, 3401, 3427, 3937, 4039, 4054, 4097, 4315, 4529, 4537, 4702, 4741, 5029, 5065, 5398, 5587

Offset: 1

Antonio Roldán, Mar 16 2015

nonn

This sequence is the intersection of A006881 and A006532.

			199 is in the sequence because 119=7*17 (the product of two distinct primes) and sigma(119)=8*18=144=12^2 (a square number).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A006881, A006532, A256149, A256150, A256151.

Cf. A020639, A010052.

a256152 n = a256152_list !! (n-1)
256152_list = filter f a006881_list where
   f x = a010052' ((spf + 1) * (x `div` spf + 1)) == 1
         where spf = a020639 x
-- Reinhard Zumkeller, Apr 06 2015

f[n_] := Block[{pf = FactorInteger@ n}, Max @@ Last /@ pf == 1 && Length@ pf == 2]; Select[Range@ 6000, IntegerQ@ Sqrt@ DivisorSigma[1, #] && f@ # &] (* Michael De Vlieger, Mar 17 2015 *)

{for(i=1,10^4,if(omega(i)==2&&issquarefree(i)&&issquare(sigma(i)),print1(i,", ")))}

A291167 Numbers k such that psi(k) is a perfect square where psi(k) = A001615(k).

Original entry on oeis.org

1, 3, 18, 20, 22, 27, 60, 66, 70, 72, 80, 88, 92, 94, 99, 115, 119, 162, 170, 210, 212, 214, 217, 240, 243, 252, 264, 265, 276, 280, 282, 288, 308, 310, 315, 320, 322, 345, 352, 357, 368, 376, 382, 385, 423, 497, 500, 510, 517, 527, 540, 594, 596, 612, 636, 637, 642, 648, 651, 679, 680, 710, 725, 742

Offset: 1

Altug Alkan, Aug 19 2017

The product of an even number of distinct members of A066436 is in the sequence. - Robert Israel, Aug 22 2017

			60 is a term because psi(60) = 144 is a perfect square.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001615, A006532, A066436.

filter:= proc(n) issqr(n*mul(1+1/p,p=numtheory:-factorset(n))) end proc:
select(filter, [$1..1000]); # Robert Israel, Aug 22 2017

Select[Range@ 750, IntegerQ@ Sqrt[# Sum[MoebiusMu[d]^2/d, {d, Divisors@ #}]] &] (* Michael De Vlieger, Aug 19 2017 *)

a001615(n) = n*sumdivmult(n, d, issquarefree(d)/d);
is(n) = issquare(a001615(n));

cp's OEIS Frontend

A006534 Number of one-sided triangular polyominoes (n-iamonds) with n cells; turning over not allowed, holes are allowed.

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A019424 Numbers whose sum of divisors is a sixth power.

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Magma

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A048256 Numbers whose sum of divisors is 6^6 = 46656.

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A048257 Integers whose sum of divisors is a 7th power.

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A048258 Integers whose sum of divisors is an 8th power.

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A186438 Positive numbers whose squares end in two identical digits.

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Maple

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A256151 Triangular numbers n such that sigma(n) is a square number.

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Magma

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