A253914 -id:A253914 - cp's OEIS frontend

A253913 Numbers of the form m^k + m, with m >= 0 and k > 1.

0, 2, 6, 10, 12, 18, 20, 30, 34, 42, 56, 66, 68, 72, 84, 90, 110, 130, 132, 156, 182, 210, 222, 240, 246, 258, 260, 272, 306, 342, 350, 380, 420, 462, 506, 514, 520, 552, 600, 630, 650, 702, 732, 738, 756, 812, 870, 930, 992, 1010, 1026, 1028, 1056, 1122, 1190, 1260, 1302

Offset: 1

Alex Ratushnyak, Jan 18 2015

nonn

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

Cf. A099225, A253914.

N:= 10000: # for terms <= N
S:= 0, 2:
for k from 2 to floor(log[2](N)) do
  for m from 2 do
    v := m^k+m; if v > N then break fi;
    S:= S, v;
od od:
sort(convert({S}, list)): # Robert Israel, Apr 28 2019, changed Jul 8 2021

max = 1000; Sort[Flatten[Table[m^k + m, {m, 2, Floor[Sqrt[max]]}, {k, 2, Floor[Log[m, max]]}]]] (* Alonso del Arte, Jan 18 2015 *)

def aupto(lim):
    xkx = set(x**k + x for k in range(2, lim.bit_length()) for x in range(int(lim**(1/k))+2))
    return sorted(filter(lambda t: t<=lim, xkx))
print(aupto(1500)) # Michael S. Branicky, Jul 08 2021

Changed to include 0 and 2 by Robert Israel, Jul 08 2021

A253917 Numbers that can be represented as both x^y + x and b^c + b + c, for some b, c, x, y > 1.

Original entry on oeis.org

72, 738, 2758, 16777232, 1073741856, 282429536508, 95367431640650, 150094635296999148, 221073919720733357899812, 311973482284542371301330321821976098, 1329227995784915872903807060280344640, 85070591730234615865843651857942052992

Offset: 1

Alex Ratushnyak, Jan 18 2015

nonn

Intersection of A253913 and A253775.

			72 = 2^6+2+6 = 8^2+8,
738 = 3^6+3+6 = 9^3+9,
2758 = 52^2+52+2 = 14^3+14,
16777232 = 4^12+4+12 = 8^8+8,
1073741856 = 2^30+2+30 = 32^6+32,
282429536508 = 3^24+3+24 = 27^8+27,
95367431640650 = 5^20+5+20 = 25^10+25,
150094635296999148 = 9^18+9+18 = 27^12+27,
221073919720733357899812 = 6^30+6+30 = 30^15+36,
311973482284542371301330321821976098 = 7^42+7+42 = 49^21+49,
1329227995784915872903807060280344640 = 4^60+4+60 = 64^20+64,
85070591730234615865843651857942052992 = 2^126+2+126 = 128^18+128,
etc. - _Robert G. Wilson v_, Jan 19 2015

Robert G. Wilson v, Table of n, a(n) for n = 1..17
Robert G. Wilson v, 51 identified terms with no assurance that these are the only terms less than 10^1000.

Cf. A253913, A253775, A253914, A253916.

f[n_] := Block[{t = Transpose@ Flatten[ Table[{m^k + m, m^k + m + k}, {m, 2, Floor@ Sqrt[2^n]}, {k, Floor@ Log[m, 2^(n - 1)] + 1, Floor@ Log[m, 2^n]}], 1]}, Intersection[ t[[1]], t[[2]]]]; f[1] = {}; Array[f, 50] // Flatten (* Robert G. Wilson v, Jan 19 2015 *)

a(7)-a(12) from Robert G. Wilson v, Jan 19 2015

A253916 Numbers that can be represented as both x^y + y and b^c + b + c, for some b, c, x, y > 1.

Original entry on oeis.org

264, 1334, 4108, 373323, 6436371, 387420507, 1099511627816

Offset: 1

Alex Ratushnyak, Jan 18 2015

Intersection of A099225 and A253775.

			264 is in the list since 264 = 2^8 + 8 and 264 = 4^4 + 4 + 4.
a(2) = 1334 = 11^3 + 3 = 36^2 + 36 + 2.

Cf. A099225, A253775, A253914, A253917.

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A253913 Numbers of the form m^k + m, with m >= 0 and k > 1.

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A253917 Numbers that can be represented as both x^y + x and b^c + b + c, for some b, c, x, y > 1.

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A253916 Numbers that can be represented as both x^y + y and b^c + b + c, for some b, c, x, y > 1.

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