A253775 -id:A253775 - cp's OEIS frontend

A253776 Primes representable as x^y + x + y, where x>1, y>1 are integers.

13, 71, 137, 251, 353, 523, 1013, 1033, 2213, 4933, 24421, 32803, 39341, 59063, 78137, 117701, 125053, 140663, 148933, 205441, 274693, 279949, 343073, 371311, 456613, 493121, 524309, 571873, 681563, 912773, 1225153, 1594339, 1953253, 2406241, 2924353, 3241943, 3652421

Offset: 1

Alex Ratushnyak, Jan 12 2015

nonn

Robert Israel and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 800 terms from Israel)

Cf. A253775, A253777.

N:= 10^6; # to get all terms <= N
A:= select(t -> t <= N and isprime(t), {seq(seq(x^y+x+y, y = 2..floor(log[x](N-x))), x=2..floor(sqrt(N)))}, N):
sort(convert(A,list)); # Robert Israel, Sep 18 2024

list(lim)=my(v=List()); forstep(y=3,oo,2, if(2^y+y+2>lim, break); for(x=2,oo, my(p=x^y+x+y); if(p>lim, break); if(isprime(p), listput(v,p)))); Set(v) \\ Charles R Greathouse IV, Sep 18 2024

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

A253777 Numbers representable as x^y + x + y in two or more ways, where x>1, y>1 are integers.

Original entry on oeis.org

22, 523, 531456, 16777232, 281474976710684, 150094635296999160

Offset: 1

Alex Ratushnyak, Jan 12 2015

The sequence is infinite since it contains all the numbers (k^2)^(k^2-k)+k^2+k^2-k = k^(2k^2-2k)+k+2k^2-2k for k>1. - Giovanni Resta, May 19 2015

Let a, b, and k be integers such that m = ab(k^a-k^b)/(a-b) is an integer. Then, the number given by (x,y) = (k^a,m/a) is the same as that given by (k^b,m/b). The given terms correspond to (a,b,k) = (2,1,2), (3,1,2), (2,1,3), (3,2,2), (4,2,2)/(2,1,4), and (3,1,3). - Charlie Neder, Apr 19 2019

			a(1) = 22 = 2^4 + 2 + 4 = 4^2 + 4 + 2.
a(2) = 523 = 8^3 + 8 + 3 = 2^9 + 2 + 9.
a(3) = 531456 = 3^12 + 3 + 12 = 9^6 + 9 + 6.
a(4) = 16777232 = 4^12 + 4 + 12 = 8^8 + 8 + 8.
a(5) = 281474976710684 = 4^24 + 4 + 24 = 16^12 + 16 + 12.
a(6) = 150094635296999160 = 3^36 + 3 + 36 = 27^12 + 27 + 12.

Cf. A253775, A253776.

a(6) from Lars Blomberg, May 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.

A250004 Numbers representable as x^y + x*y, where x>1, y>1 are integers (without multiplicity).

Original entry on oeis.org

8, 14, 15, 24, 35, 36, 42, 48, 63, 76, 80, 93, 99, 120, 140, 142, 143, 168, 195, 224, 234, 255, 258, 272, 288, 323, 360, 364, 399, 440, 483, 528, 530, 536, 575, 624, 645, 675, 728, 747, 756, 783, 840, 899, 960, 1023, 1030, 1044, 1088, 1155, 1224, 1295, 1320, 1364, 1368

Offset: 1

Alex Ratushnyak, Jan 14 2015

nonn

The sequence of numbers representable as x^y + x*y in two or more ways begins: 24, 76, 272, 1044, 2208, 4120, 16412. Example: 2208 = 46^2 + 46*2 = 3^7 + 3*7.

The subsequence of squares begins: 36, 1764.

			a(2) = 14 = 2^3 + 2*3.
a(3) = 15 = 3^2 + 3*2.
a(22) = 255 = 15^2 + 15*2.

Cf. A253775.

Cf. A024036 is a subsequence, except first 2 terms.

isok(n) = {for (p=2, floor(log(n)/log(2)), for (k=2, sqrtnint(n, p), if (n == k^p + p*k, return (1)););); return (0);} \\ Michel Marcus, Jan 16 2015

A255804 Numbers representable as xy(x+y), b*c+b+c, and d^e+d+e, where d>1, e>1, b>=c>1 and x>=y>1.

Original entry on oeis.org

264, 308, 8192, 16400, 88508, 236684, 504812, 12127808, 22491308, 82310258, 227240552, 385278014, 1069061114, 2363758544, 2591166314, 2985365684, 3310448834, 4042988642, 4791339182, 5712714308, 7553782658, 8626601522, 12494656622, 14498688512, 15165306758, 15445891244

Offset: 1

Alex Ratushnyak, Mar 07 2015

nonn

Intersection of A253775, A254671, A255265.

			a(2) = 308 = 17^2 + 17 + 2 = 7 * 4 * (7 + 4) = 102 * 2 + 102 + 2.

David A. Corneth, Table of n, a(n) for n = 1..160
David A. Corneth, PARI program

Cf. A255267, A254034, A255265, A254671, A253775.

PARI
```
\\ See Corneth link
```

TOP = 100000000
a = [0]*TOP
c = []
for y in range(2, TOP//2):
  if 2**y + 2 + y >= TOP: break
  for x in range(2, TOP//2):
    k = x**y+(x+y)
    if k>=TOP: break
    c.append(k)
for y in range(2, TOP//2):
  if 2*y*y*y >= TOP: break
  for x in range(y, TOP//2):
    k = x*y*(x+y)
    if k>=TOP: break
    a[k]=1
for y in range(2, TOP//2):
  if y*(y+2) >= TOP: break
  for x in range(y, TOP//2):
    k = x*y+(x+y)
    if k>=TOP: break
    a[k]|=2
    # if a[k]==3 and (k in c): print(k, end=', ')
print([n for n in range(TOP) if a[n]==3 and (n in c)])

More terms from David A. Corneth, Oct 18 2024

A253287 Numbers representable as x^y + x*y and as b^c + b + c, where x, y, b, c are integers > 1.

Original entry on oeis.org

8, 14, 3135, 10714, 70342709283

Offset: 1

Alex Ratushnyak, Jan 18 2015

Intersection of A250004 and A253775.

No more terms < 10^20. - Lars Blomberg, Dec 13 2015

			a(2) = 14 = 2^3 + 2*3 = 3^2 + 2 + 3.
a(3) = 3135 = 55^2 + 55*2 = 5^5 + 5 + 5.
a(4) = 10714 = 22^3 + 22*3 = 103^2 + 103 + 2.
a(5) = 70342709283 = 265221^2 + 265221*2 = 4128^3 + 4128 + 3.

Cf. A250004, A253775.

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A253776 Primes representable as x^y + x + y, where x>1, y>1 are integers.

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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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A253777 Numbers representable as x^y + x + y in two or more ways, where x>1, y>1 are integers.

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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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A250004 Numbers representable as x^y + x*y, where x>1, y>1 are integers (without multiplicity).

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A255804 Numbers representable as x*y*(x+y), b*c+b+c, and d^e+d+e, where d>1, e>1, b>=c>1 and x>=y>1.

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A253287 Numbers representable as x^y + x*y and as b^c + b + c, where x, y, b, c are integers > 1.

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A255804 Numbers representable as xy(x+y), b*c+b+c, and d^e+d+e, where d>1, e>1, b>=c>1 and x>=y>1.