A146756 -id:A146756 - cp's OEIS frontend

A230477 Smallest number that is the sum of n positive n-th powers in >= n ways.

1, 50, 5104, 236674, 9006349824, 82188309244

Offset: 1

Jonathan Sondow, Oct 22 2013

Does a(6) exist? For which values of n does a(n) exist? Is there a proof that a(n) < a(n+1) when both exist?

			1 = 1^1.
50 = 1^2 + 7^2 = 5^2 + 5^2.
5104 = 1^3 + 12^3 + 15^3 = 2^3 + 10^3 + 16^3 = 9^3 + 10^3 + 15^3.
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.
9006349824 = 8^5 + 34^5 + 62^5 + 68^5 + 92^5 = 8^5 + 41^5 + 47^5 + 79^5 + 89^5 = 12^5 + 18^5 + 72^5 + 78^5 + 84^5 = 21^5 + 34^5 + 43^5 + 74^5 + 92^5 = 24^5 + 42^5 + 48^5 + 54^5 + 96^5.
82188309244 = 1^6 + 9^6 + 29^6 + 44^6 + 55^6 + 60^6 = 2^6 + 12^6 + 25^6 + 51^6 + 53^6 + 59^6 = 5^6 + 23^6 + 27^6 + 44^6 + 51^6 + 62^6 = 10^6 + 16^6 + 41^6 + 45^6 + 51^6 + 61^6 = 12^6 + 23^6 + 33^6 + 34^6 + 55^6 + 61^6 = 15^6 + 23^6 + 31^6 + 36^6 + 53^6 + 62^6.

A. H. Beiler, Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, Dover, NY, 1966, pp. 162-165, 290-291.
R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, D1.

Open Directory Project, Equal Sums of Like Powers

a(2) = A048610(2), a(3) = A025398(1), a(4) = A219921(1).

Cf. A146756 (smallest number that is the sum of n distinct positive n-th powers in exactly n ways), A230561 (smallest number that is the sum of two positive n-th powers in >= n ways), A091414 (smallest number that is the sum of n positive n-th powers in >= 2 ways).

a(n) <= A146756(n), with equality at least for n = 1, 3, 5 and inequality at least for n = 2, 4.

a(n) >= A091414(n) for n > 1, with equality at least for n = 2 and inequality at least for n = 3, 4, 5.

a(5) from Donovan Johnson, Oct 23 2013

a(6) from Michael S. Branicky, May 09 2021

A349764 a(n) is the smallest number which can be represented as the sum of n distinct n-th powers of primes in exactly n ways.

Original entry on oeis.org

2, 410, 8627527, 199898912404

Offset: 1

Ilya Gutkovskiy, Dec 20 2021

			For n = 2: 410 = 7^2 + 19^2 = 11^2 + 17^2.
For n = 3: 8627527 = 19^3 + 151^3 + 173^3 = 23^3 + 139^3 + 181^3 = 71^3 + 73^3 + 199^3.
From _Michael S. Branicky_, Dec 21 2021: (Start)
For n = 4: 199898912404 =  23^4 + 281^4 + 397^4 + 641^4,
                        = 137^4 + 383^4 + 467^4 + 601^4,
                        = 151^4 + 227^4 + 557^4 + 563^4,
                        = 257^4 + 317^4 + 347^4 + 643^4. (End)

Cf. A146756.

a(4) from Michael S. Branicky, Dec 21 2021

A146760 Last prime subtrahend at 10^n in A146759.

Original entry on oeis.org

5, 61, 997, 9929, 97283, 999983, 9999973, 99897341, 999999929, 9993948257, 99999999761, 999999999989, 9999516957181, 99999999999929, 999999999999989, 9999999999998857, 99999429057832259, 999999999999999989, 9999990391470218071

Offset: 1

Enoch Haga, Nov 02 2008

It is not necessary to compute A146759 to compute this sequence. a(n) is the largest prime p<=10^n such that c(p)-p is also a prime, where c(p) is the smallest cube exceeding p. - Sean A. Irvine, Mar 27 2013

			A(2)=61 because 61 is the 7th and last prime subtrahend under 10^3.

Sean A. Irvine, Table of n, a(n) for n = 1..150

Cf. A146756, A146757, A146759.

10 'cu less pr are prime
20 N=1:O=1:C=1
30 A=3:S=sqrt(N):if N>10^3 then print N,C-1:stop
40 B=N\A
50 if B*A=N then 100
60 A=A+2
70 if A<=S then 40
80 R=O^3:Q=R-N
90 if N1 print R;N;Q;C:N=N+2:C=C+1:goto 30
100 N=N+2:if N

More terms from Sean A. Irvine, Mar 27 2013

A374227 a(n) is the smallest number which can be represented as the sum of three distinct positive n-th powers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

6, 62, 5104, 5978882

Offset: 1

Ilya Gutkovskiy, Jul 01 2024

			a(3) = 5104 = 1^3 + 12^3 + 15^3 = 2^3 + 10^3 + 16^3 = 9^3 + 10^3 + 15^3.

Cf. A025415, A025419, A123075, A146756, A374226, A374228.

A374228 a(n) is the smallest number which can be represented as the sum of four distinct positive n-th powers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

10, 90, 1521, 300834

Offset: 1

Ilya Gutkovskiy, Jul 01 2024

			a(3) = 1521 = 1^3 + 2^3 + 8^3 + 10^3 = 1^3 + 4^3 + 5^3 + 11^3 = 4^3 + 6^3 + 8^3 + 9^3.

Cf. A025417, A025421, A146756, A374226, A374227.

A374256 a(n) is the smallest number which can be represented as the sum of n distinct positive n-th powers in exactly 2 ways, or -1 if no such number exists.

Original entry on oeis.org

-1, 65, 1009, 6834, 1158224, 19198660, 1518471174, 301963223843, 14599274102522, 1601155487573222

Offset: 1

Ilya Gutkovskiy, Jul 01 2024

			a(2) = 65 = 1^2 + 8^2 = 4^2 + 7^2.
a(3) = 1009 = 1^3 + 2^3 + 10^3 = 4^3 + 6^3 + 9^3.

Cf. A025303, A025400, A091414, A146756, A374226, A374257.

f:= proc(n) uses priqueue;
  local pq,w,t,g,i,count,newt;
  g:= proc(t) local i; [-add((t[i]+i)^n,i=1..n),op(t)] end proc;
  w:= [0$(n+1)];
  initialize(pq);
  insert(g([0$n]),pq);
  do
    t:= extract(pq);
    if t[1] = w[1] then return -t[1] fi;
    w:= t;
    for i from 2 to n+1 do
        if t[i]=t[-1] then
          newt:= g(t[2..-1] + [0$(i-2),1$(n+2-i)]);
        insert(newt,pq);
  fi od od;
end proc:
-1, seq(f(n),n=2..10); # Robert Israel, Jul 01 2024

a(9)-a(10) from Robert Israel, Jul 01 2024

A374257 a(n) is the smallest number which can be represented as the sum of n distinct positive n-th powers in exactly 3 ways, or -1 if no such number exists.

Original entry on oeis.org

-1, 325, 5104, 16578, 70211956, 201968338, 1690592199245

Offset: 1

Ilya Gutkovskiy, Jul 01 2024

			a(2) = 325 = 1^2 + 18^2 = 6^2 + 17^2 = 10^2 + 15^2.
a(3) = 5104 = 1^3 + 12^3 + 15^3 = 2^3 + 10^3 + 16^3 = 9^3 + 10^3 + 15^3.

Cf. A025304, A025401, A146756, A374227, A374256.

from itertools import count
from sympy.solvers.diophantine.diophantine import power_representation
def A374257(n): return next(m for m in count(1) if len(list(power_representation(m,n,n)))==3) if n>1 else -1 # Chai Wah Wu, Jul 01 2024

a(7) from Michael S. Branicky, Jul 09 2024

A146759 Number of primes p < 10^n such that c - p is prime, where c is the next cube greater than p.

Original entry on oeis.org

2, 7, 43, 224, 1355, 9306, 66200, 500249, 3883527, 31081813, 254358928, 2120975833

Offset: 1

Enoch Haga, Nov 02 2008

			a(2) = 7 because at 10^2 there are 7 primes that, subtracted from the next higher value cube, produce prime differences: {3, 5, 41, 47, 53, 59, 61}.

Cf. A006880, A146756, A146757, A146760.

Table[Length[Select[Prime[Range[PrimePi[10^n]]], PrimeQ[Ceiling[#^(1/3)]^3 - #] &]], {n, 6}] (* T. D. Noe, Mar 31 2013 *)
cpQ[n_]:=PrimeQ[Ceiling[Surd[n,3]]^3-n]; nn=9; Module[{c=Table[If[ cpQ[n],1,0], {n, Prime[ Range[ PrimePi[ 10^nn]]]}]}, Table[ Total[ Take[c,PrimePi[10^p]]],{p,nn}]] (* Harvey P. Dale, Aug 13 2014 *)

a(n) = {my(nb = 0); forprime(p=2, 10^n, if (isprime((sqrtnint(p,3)+1)^3 - p), nb++);); nb;} \\ Michel Marcus, Jun 22 2019

list(nmax) = {my(m = 0, c = 2, cc = c^3, n = 0, pow = 10); forprime(p = 1, , if(p > pow, print1(m, ", "); n++; if(n == nmax, break); pow *= 10); if(p > cc, c++; cc = c^3); if(isprime(cc - p), m++));} \\ Amiram Eldar, Jan 20 2025

10 'cu less pr are prime
20 N=1:O=1:C=1
30 A=3:S=sqrt(N):if N>10^3 then print N,C-1:stop
40 B=N\A
50 if B*A=N then 100
60 A=A+2
70 if A<=S then 40
80 R=O^3:Q=R-N
90 if N1 print R;N;Q;C:N=N+2:C=C+1:goto 30
100 N=N+2:if N

Better name and more terms from Sean A. Irvine, Mar 27 2013
a(10)-a(11) from Chai Wah Wu, Jun 21 2019
a(12) from Amiram Eldar, Jan 20 2025

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.

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.

cp's OEIS Frontend

A230477 Smallest number that is the sum of n positive n-th powers in >= n ways.

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A349764 a(n) is the smallest number which can be represented as the sum of n distinct n-th powers of primes in exactly n ways.

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A146760 Last prime subtrahend at 10^n in A146759.

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A374227 a(n) is the smallest number which can be represented as the sum of three distinct positive n-th powers in exactly n ways, or -1 if no such number exists.

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A374228 a(n) is the smallest number which can be represented as the sum of four distinct positive n-th powers in exactly n ways, or -1 if no such number exists.

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A374256 a(n) is the smallest number which can be represented as the sum of n distinct positive n-th powers in exactly 2 ways, or -1 if no such number exists.

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A374257 a(n) is the smallest number which can be represented as the sum of n distinct positive n-th powers in exactly 3 ways, or -1 if no such number exists.

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A146759 Number of primes p < 10^n such that c - p is prime, where c is the next cube greater than p.

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

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