A007532 -id:A007532 - cp's OEIS frontend

A055480 Energetic numbers.

24, 43, 63, 89, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 132, 135, 142, 153, 175, 209, 224, 226, 262, 264, 267, 283, 284, 332, 333, 334, 357, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 407, 445, 463, 518, 568, 598, 629, 739, 794, 809, 849, 935, 994, 1000

Offset: 1

Robert G. Wilson v, Jul 05 2000

Numbers that can be broken into two or more substrings and expressed as a sum of (possibly different) positive powers of those substrings.

			142 = 14^1 + 2^7, 8833 = 88^2 + 33^2.

Frank Rubin, Journal of Recreational Mathematics, Volume 12, Number 2, Page 139.

Jonathan Frech, Table of n, a(n) for n = 1..10000

This is a less stringent condition than that of a "powerful" number - compare A007532.

Cf. A007532, A072096.

More terms from Robert G. Wilson v, Mar 07 2002

Edited by David W. Wilson, Jan 29 2003

A192636 Powerful sums of two powerful numbers.

Original entry on oeis.org

8, 9, 16, 25, 32, 36, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000, 1024, 1089, 1125, 1152, 1156, 1225

Offset: 1

Charles R Greathouse IV, Jul 06 2011

nonn

Browning & Valckenborgh conjecture that a(n) ~ kn^2 with k approximately 0.139485255. See their Conjecture 1 and equation (14). Their Theorems 1 and 2 establish upper and lower asymptotic bounds.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Charles R Greathouse IV)
Tim D. Browning and K. Van Valckenborgh, Sums of three squareful numbers, Experimental Mathematics, Vol. 21, No. 2 (2012), pp. 204-211; arXiv preprint, arXiv:1106.4472 [math.NT], 2011.

Subsequence of A001694 and of A076871.

Cf. A001694, A007532, A005934, A005188, A003321, A014576, A023052, A046074, A013929, A076871, A143813. - Jonathan Vos Post, Jul 10 2011

With[{m = 1225}, pow = Select[Range[m], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]; Intersection[pow, Plus @@@ Tuples[pow, {2}]]] (* Amiram Eldar, Feb 12 2023 *)

isPowerful(n)=if(n>3,vecmin(factor(n)[,2])>1,n==1)
sumset(a,b)={
  my(c=vectorsmall(#a*#b));
  for(i=1,#a,
    for(j=1,#b,
      c[(i-1)*#b+j]=a[i]+b[j]
    )
  );
  vecsort(c,,8)
}; selfsum(a)={
  my(c=vectorsmall(binomial(#a+1,2)),k);
  for(i=1,#a,
    for(j=i,#a,
      c[k++]=a[i]+a[j]
    )
  );
  vecsort(c,,8)
};
list(lim)={
  my(v=select(isPowerful, vector(floor(lim),i,i)));
  select(n->n<=lim && isPowerful(n), Vec(selfsum(v)))
};

Numbers k such that there exists some a, b, c with A001694(a) + A001694(b) = k = A001694(c).

Corrected (on the advice of Donovan Johnson) by Charles R Greathouse IV, Sep 25 2012

A218539 Numbers that are equal to the sum of the uniform platonic polyhedral (figurate) numbers (tetrahedral, cubic, octahedral, dodecahedral, or icosahedral) on each of their digits.

Original entry on oeis.org

0, 1, 20, 21, 24, 153, 240, 241, 289, 304, 324, 370, 371, 407, 440, 441, 593, 739, 2167, 2284, 2348, 2484, 2583, 2860, 2861, 3009, 3029, 3093, 3249, 4288, 5859, 6888, 7996, 9898

Offset: 1

Thomas S. Pedigo, Nov 01 2012

153, 370, 371, and 407 are well known with regard to the cubic numbers.

			The octahedral numbers are represented by the formula, y(x)=(2x^3+x)/3; apply this formula to each of the digits in a(18)=739, i.e., y(7)=231, y(3)=19, y(9)=489; sum=739; the dodecahedral numbers are represented by the formula, y(x)=x(3x-1)(3x-2)/2; apply this formula to each of the digits in a(34)=9898, i.e., y(9)=2725, y(8)=2024; y(9)=2725, y(8)=2024; sum=9898.

Cf. A005188, A007532, A036057.

A279954 Prime d-powerful numbers: prime numbers which can be represented as the sum of some nonnegative powers of their digits.

Original entry on oeis.org

2, 3, 5, 7, 43, 89, 263, 283, 347, 373, 379, 463, 653, 733, 739, 2063, 2083, 2131, 2137, 2179, 2203, 2243, 2267, 2269, 2293, 2333, 2423, 2437, 2467, 2473, 2753, 2777, 2803, 2843, 2939, 2953, 3167, 3257, 3457, 3527, 3943, 3947, 4133, 4153, 4157, 4159, 4229, 4241, 4243, 4261, 4289, 4339, 4349, 4373, 4397

Offset: 1

Randy L. Ekl, Dec 23 2016

			43 is prime, and is 4^2 + 3^3, a sum of powers of its digits, and thus also a d-powerful number.

Cf. A007532, A061862, A134703.

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A055480 Energetic numbers.

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A192636 Powerful sums of two powerful numbers.

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A218539 Numbers that are equal to the sum of the uniform platonic polyhedral (figurate) numbers (tetrahedral, cubic, octahedral, dodecahedral, or icosahedral) on each of their digits.

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A279954 Prime d-powerful numbers: prime numbers which can be represented as the sum of some nonnegative powers of their digits.

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