A011541 -id:A011541 - cp's OEIS frontend

A305130 Numbers k with the property that there exists a positive integer M, called multiplier, such that the sum of the digits of k times the multiplier added to the reversal of this product gives k.

10, 11, 12, 18, 22, 33, 44, 55, 66, 77, 88, 99, 101, 110, 121, 132, 141, 165, 181, 201, 202, 221, 222, 261, 262, 282, 302, 303, 322, 323, 342, 343, 363, 403, 404, 423, 424, 444, 463, 483, 504, 505, 525, 545, 564, 584, 585, 605, 606, 645, 646, 666, 686, 706

Offset: 1

Viorel Nitica, May 26 2018

These numbers are related to the taxicab number 1729. This is why they might be called "additive Hardy-Ramanujan numbers".

			For k = 11 the sum of the digits is 2 and the multiplier is 5: 2 * 5 = 10 and 10 + 01 = 11.
For k = 747 the sum of the digits is 18 and the multiplier is 7: 18 * 7 = 126 and 126 + 621 = 747.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Viorel Nitica, About some relatives of the taxicab numbers, submitted to Journal of Integer Sequences, (2018). [Where these numbers are introduced.]
Viorel Niţică, Jeroz Makhania, About the Orbit Structure of Sequences of Maps of Integers, Symmetry (2019), Vol. 11, No. 11, 1374.

Subsequence of A067030.

Cf. A004086, A011541, A305131.

Block[{k, d, j}, Reap[Do[k = 1; d = Total@ IntegerDigits[i]; While[Nor[k > i, Set[j, # + IntegerReverse@ #] == i &[d k]], k++]; If[j == i, Sow[{i, k}]], {i, 720}]][[-1, 1, All, 1]] ] (* Michael De Vlieger, Jan 28 2020 *)

A374226 a(n) is the smallest number which can be represented as the sum of two distinct positive n-th powers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

3, 65, 87539319

Offset: 1

Ilya Gutkovskiy, Jul 01 2024

			a(3) = 87539319 = 167^3 + 436^3 = 228^3 + 423^3 = 255^3 + 414^3.

Cf. A011541, A093195, A146756, A374227, A374228.

A061798 Number of sums i^3 + j^3 that occur more than once for 1<=i<=j<=n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 7, 7, 8, 8, 8, 9, 10, 10, 10, 10, 10, 10, 10, 10, 12, 12, 12, 13, 13, 14, 15, 16, 16, 16, 17, 17, 19, 19, 19, 19, 20, 20, 20, 21, 23, 24, 24, 24, 25, 25, 25, 25

Offset: 1

Labos Elemer, Jun 22 2001

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A000217, A011541, A061791.

a(n) = my(v=vector(2*n^3, i, 0)); for(i=1, n, for(j=i, n, v[i^3+j^3]+=1)); sum(i=1, #v, v[i]>1); \\ Seiichi Manyama, May 14 2024

def A(n)
  h = {}
  (1..n).each{|i|
    (i..n).each{|j|
      k = i * i * i + j * j * j
      if h.has_key?(k)
        h[k] += 1
      else
        h[k] = 1
      end
    }
  }
  h.to_a.select{|i| i[1] > 1}.size
end
def A061798(n)
  (1..n).map{|i| A(i)}
end
p A061798(80) # Seiichi Manyama, May 14 2024

A098110 Smallest number that is the difference between two positive cubes in n ways.

Original entry on oeis.org

7, 721, 3367, 4118877, 1412774811, 424910390480793

Offset: 1

Jeff Burch, Sep 23 2004

a(7) <= 15490327057569000, a(8) <= 123922616460552000. - Giovanni Resta, Mar 19 2020

			Pairs (x, y) such that x^3 - y^3 = a(1), ..., a(6):
7 = (2, 1);
721 = (16, 15), (9, 2);
3367 = (34, 33), (16, 9), (15, 2)l
4118877 = (162, 51), (165, 72), (178, 115), (678, 675);
1412774811 = (1134, 357), (1155, 504), (1246, 805), (2115, 2004), (4746, 4725);
424910390480793 = (596001, 595602), (317982, 316575), (141705, 134268), (83482, 53935), (77385, 33768), (75978, 23919).

Cf. A014439, A014440, A014441, A181123, A034179, A265625, A333376, A333377.

Cf. A011541,A047696.

a(6) from Giovanni Resta, Mar 19 2020

A180099 Primes which are the sum of three distinct positive cubes of prime numbers in two or more distinct ways.

Original entry on oeis.org

185527, 451837, 591751, 1265471, 1266929, 1618973, 1626227, 1664713, 2586277, 2754683, 2765519, 2805193, 3422303, 3740309, 3748499, 4154779, 5336479, 5483953, 5557987, 6130151, 6586091, 7231013, 7361801, 7726571, 8205553

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 09 2010

nonn

			185527 = 47^3+43^3+13^3=53^3+31^3+19^3.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A011541, A024974, A025400, A122723, A180088, A180089.

lst={};Do[Do[Do[If[PrimeQ[p=Prime[a]^3+Prime[b]^3+Prime[c]^3],AppendTo[lst,p]],{c,b-1,1,-1}],{b,a-1,1,-1}],{a,88}];lst1=Sort@lst; lst={};Do[If[lst1[[n]]==lst1[[n+1]],AppendTo[lst,lst1[[n]]]],{n,Length[lst1]-1}];lst
Select[Tally[Select[Total/@Subsets[Prime[Range[50]]^3,{3}],PrimeQ]],#[[2]]> 1&] [[All,1]]//Sort (* Harvey P. Dale, Sep 26 2020 *)

A219921 Numbers expressible as the sum of four nonnegative fourth-powers in four different ways.

Original entry on oeis.org

236674, 260658, 282018, 300834, 334818, 478338, 637794, 650034, 650658, 671778, 708483, 708834, 729938, 789378, 811538, 816578, 832274, 849954, 941859, 989043, 1042083, 1045539, 1099203, 1099458, 1102258, 1179378, 1243074, 1257954, 1283874, 1323234, 1334979

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, Apr 12 2013

nonn

A natural extension of the two-sets-of-two-cubes taxi-cab numbers (A001235).

a(4) is the first number which contains distinct fourth-powers in all four sets of four, and is therefore also A146756(4).

			a(1) = 236674 = 1^4+2^4+7^4+22^4 = 3^4+6^4+18^4+19^4 = 7^4+14^4+16^4+19^4 = 8^4+16^4+17^4+17^4.

Christian N. K. Anderson, Table of n, a(n) for n = 1..1000
Christian N. K. Anderson, Decomposition of the first 1000 terms into four sets of four fourth powers.

Other sums of four fourth powers: A176197, A133526.
Cf. A146756, A230477.
Cf. A001235, A011541.

A272892 Taxi-cab numbers n such that n-1 and n+1 are both prime.

Original entry on oeis.org

32832, 513000, 2101248, 8647128, 43570872, 46661832, 152275032, 166383000, 175959000, 351981000, 543449088, 610991208, 809557632, 970168752, 1710972648, 2250265752, 2262814272, 2560837032, 3222013032, 3308144112, 3582836712, 4505949000, 4543936488, 4674301632, 4868489178

Offset: 1

Altug Alkan, May 09 2016

nonn

Taxi-cab numbers that are in A014574.
There are two versions of "taxicab numbers" that are A001235 and A011541. This sequence focuses on the version A001235.
First six terms are 2^6*3^3*19, 2^3*3^3*5^3*19, 2^12*3^3*19, 2^3*3^3*7^2*19*43, 2^3*3^6*31*241, 2^3*3^8*7*127.
This sequence contains many terms that are divisible by 6^3 = 216. But there are also terms that are not divisible by 6^3. For example, 166383*10^3 and 351981*10^3 are terms that are not divisible by 216.

			Taxi-cab number 32832 is a term because 32831 and 32833 are twin primes.

Chai Wah Wu, Table of n, a(n) for n = 1..6385 a(n) for n = 1..88 from Charles R Greathouse IV

Cf. A001235, A014574.

T=thueinit(x^3+1,1);
isA001235(n)=my(v=thue(T,n)); sum(i=1,#v,v[i][1]>=0 && v[i][2]>=v[i][1])>1
p=2; forprime(q=3,1e9, if(q-p==2 && isA001235(p+1), print1(p+1", ")); p=q) \\ Charles R Greathouse IV, May 09 2016

a(7)-a(25) from Charles R Greathouse IV, May 09 2016

A371602 Taxicab numbers that are sandwiched between squarefree numbers.

Original entry on oeis.org

4104, 32832, 39312, 110808, 171288, 262656, 314496, 373464, 513000, 886464, 1016496, 1075032, 1195112, 1331064, 1370304, 1407672, 1609272, 1728216, 1734264, 1774656, 2101248, 2515968, 2864288, 2987712, 2991816, 3511872, 3512808, 3551112, 4104000, 4342914, 4467528, 4511808, 4607064

Offset: 1

Massimo Kofler, Mar 29 2024

nonn

All terms are even numbers.

			4104 = 2^3 * 3^3 * 19 (between 4103 = 11 * 373 and 4105 = 5 * 821).
32832 = 2^6 * 3^3 *19 (between 32831 and 32833, which are twin primes).
39312 = 2^4 * 3^3 * 7 * 13 (between 39311 = 19 * 2069 and 39313, which is prime).

Christian Boyer, Les nombres Taxicabs, in Dossier Pour La Science, pp. 26-28, Volume 59 (Jeux math') April/June 2008 Paris.

Intersection of A001235 and A067874.
A272892 is a subsequence.
Cf. A011541, A005117.

Select[Range[300000], And @@ SquareFreeQ /@ (# + {-1, 1}) && Length[PowersRepresentations[#, 2, 3]] > 1 &] (* Amiram Eldar, Mar 29 2024 *)

A374168 a(n) is the smallest number which can be represented as the sum of two nonzero square pyramidal numbers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

2, 60, 9692375

Offset: 1

Ilya Gutkovskiy, Jun 30 2024

There are no further positive terms <= 10^15. - Michael S. Branicky, Jul 01 2024

			a(2) = 60 = 5 + 55 = 30 + 30.

Eric Weisstein's World of Mathematics, Square Pyramidal Number.

Cf. A000330, A011541, A016032, A334013, A374193, A374194.

A374193 a(n) is the smallest number which can be represented as the sum of two nonzero pentagonal pyramidal numbers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

2, 1471, 269406

Offset: 1

Ilya Gutkovskiy, Jun 30 2024

There are no further positive terms <= 10^15. - Michael S. Branicky, Jul 01 2024

			a(2) = 1471 = 1 + 1470 = 288 + 1183.

Eric Weisstein's World of Mathematics, Pentagonal Pyramidal Number

Cf. A002411, A011541, A332989, A334013, A374168, A374194.

cp's OEIS Frontend

A305130 Numbers k with the property that there exists a positive integer M, called multiplier, such that the sum of the digits of k times the multiplier added to the reversal of this product gives k.

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A374226 a(n) is the smallest number which can be represented as the sum of two distinct positive n-th powers in exactly n ways, or -1 if no such number exists.

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A061798 Number of sums i^3 + j^3 that occur more than once for 1<=i<=j<=n.

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A098110 Smallest number that is the difference between two positive cubes in n ways.

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A180099 Primes which are the sum of three distinct positive cubes of prime numbers in two or more distinct ways.

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A219921 Numbers expressible as the sum of four nonnegative fourth-powers in four different ways.

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A272892 Taxi-cab numbers n such that n-1 and n+1 are both prime.

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PARI

Extensions

A371602 Taxicab numbers that are sandwiched between squarefree numbers.

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Mathematica

A374168 a(n) is the smallest number which can be represented as the sum of two nonzero square pyramidal numbers in exactly n ways, or -1 if no such number exists.

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