cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A253914 Numbers that can be represented as both a^x + x and b^y + b, for some a, b, x, y > 1.

Original entry on oeis.org

6, 18, 20, 30, 66, 258, 260, 732, 1026, 3130, 4098, 4100, 16386, 19686, 46662, 65538, 65540, 65552, 262146, 531444, 823550, 1048578, 1048580, 4194306, 9765630, 14348910, 16777218, 16777220, 16777224, 67108866, 268435458, 268435460, 387420492, 387420498, 1073741826
Offset: 1

Views

Author

Alex Ratushnyak, Jan 18 2015

Keywords

Comments

Intersection of A099225 and A253913.
Includes a^(a*b)+a = (a^b)^a+a for a,b > 1. - Robert Israel, Apr 28 2019

Examples

			a(1) = 6 = 2^2 + 2, in this case a = b = x = y = 2.
a(2) = 18 = 2^4 + 2 = 4^2 + 2.
a(8) = 732 = 3^6 + 3 = 9^3 + 3.

Crossrefs

Cf. A099225, A253913, A099226.

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

Views

Author

Alex Ratushnyak, Jan 18 2015

Keywords

Comments

Intersection of A253913 and A253775.

Examples

			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

Links

Crossrefs

Cf. A253913, A253775, A253914, A253916.

Programs

  • Mathematica
    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 *)

Extensions

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

A309978 a(n) is the number of positive integers k such that there exists a nonnegative integer m with k + k^m = n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 2, 1, 3, 1, 3, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1
Offset: 1

Views

Author

Peter Kagey, Aug 28 2019

Keywords

Comments

Records occur at 1, 2, 4, 6, 30, ...
Does there exist n such that a(n) >= 5? Do there exist examples besides 30 and 130 such that a(n) = 4? If so in either case, n > A253913(10000) = 87469256.

Examples

			For n = 130 the a(130) = 4 positive integers with valid maps are
  129 via 129 + 129^0 = 130,
   65 via  65 +  65^1 = 130,
    5 via   5 +   5^3 = 130, and
    2 via   2 +   2^7 = 130.

Links

Crossrefs

Cf. A253913, A307074, A307092, A309997.

Programs

  • PARI
    a(n) = {if (n==1, return (0)); my(d = divisors(n)); 1 + sumdiv(n, d, if ((d>1) && (dMichel Marcus, Oct 16 2019

Formula

a(2n+1) = 1 for all n >= 1.
a(2n) >= 2 for all n >= 2.

A346156 Primes of the form x^k+x+1 where k >= 2 and x >= 1.

Original entry on oeis.org

3, 7, 11, 13, 19, 31, 43, 67, 73, 131, 157, 211, 223, 241, 307, 421, 463, 521, 601, 631, 733, 739, 757, 1123, 1303, 1483, 1723, 1741, 2551, 2971, 3307, 3391, 3541, 3907, 4099, 4423, 4831, 4931, 5113, 5701, 5851, 6007, 6163, 6481, 6571, 8011, 8191, 9283, 9901, 10303, 11131, 12211, 12433, 13807
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, Jul 07 2021

Keywords

Comments

Primes p such that p-1 is in A253913.
Primes with more than one representation of this form include 31 = 3^3+3+1 = 5^2+5+1 and 131 = 2^7+2+1 = 5^3+5+1. Are there any others?
There are no others with more than one representation (except 3, trivially) < 10^19 (first 170385840 terms). - Michael S. Branicky, Jul 08 2021

Examples

			a(3) = 11 is a term because 11 = 2^3+2+1 and is prime.

Links

Crossrefs

Cf. A253913, A346154.

Programs

  • Maple
    N:= 10^8: # for terms <= N
S:= {3}:
for k from 2 to ilog2(N-1) do
  S:= S union select(t -> t<= N and isprime(t),{seq(x^k+x+1,x=2..floor(N^(1/k)))}):
od:
sort(convert(S,list));
  • Python
    from sympy import isprime
def aupto(lim):
    xkx = set(x**k + x + 1 for k in range(2, lim.bit_length()) for x in range(int(lim**(1/k))+2))
    return sorted(filter(isprime, filter(lambda t: t<=lim, xkx)))
print(aupto(14000)) # Michael S. Branicky, Jul 07 2021

A346287 Numbers that are of both forms x^k+x+1 and x^k-(x+1) with k>=2 and x>=0.

Original entry on oeis.org

1, 11, 13, 19, 131, 5851, 416833471
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, Jul 12 2021

Keywords

Examples

			1 = 2^2-(2+1) = 0^2+(0+1)
11 = 4^2-(4+1) = 2^3+(2+1)
13 = 2^4-(2+1) = 3^2+(3+1)
19 = 5^2-(5+1) = 2^4+(2+1)
131 = 12^2-(12+1) = 5^3+(5+1)
5851 = 77^2-(77+1) = 18^3+(18+1)
416833471 = 20417^2-(20417+1) = 747^3+(747+1)

Crossrefs

Cf. A253913.

Programs

  • Maple
    N:= 10^11: # for terms <= N
R:= {3}:
for k from 2 to ilog2(N-1) do
  R:= R union {seq(x^k+x+1,x=2..floor(N^(1/k)))}
od:
A:= {1}:
for k from 2 to ilog2(N+3) do
  for x from 2 do
    r:= x^k-(x+1);
    if r > N then break fi;
    if member(r,R) then A:= A union {r} fi
od od:
sort(convert(A,list));
Showing 1-5 of 5 results.

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