A024050 -id:A024050 - cp's OEIS frontend

A385773 Primes having only {1, 2, 5} as digits.

2, 5, 11, 151, 211, 251, 521, 1151, 1511, 2111, 2221, 2251, 2521, 2551, 5521, 11251, 11551, 12211, 12251, 12511, 15121, 15511, 15551, 21121, 21211, 21221, 21521, 22111, 22511, 25111, 25121, 51151, 51511, 51521, 51551, 52121, 52511, 55511, 111121, 111211

Offset: 1

Jason Bard, Jul 09 2025

Supersequence of A024050, A020453.

Cf. A000040, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [1, 2, 5]];

Flatten[Table[ Select[FromDigits /@ Tuples[{1, 2, 5}, n], PrimeQ], {n, 7}]]

primes_with(, 1, [1, 2, 5]) \\ uses function in A385776

print(list(islice(primes_with("125"), 41))) # uses function/imports in A385776

A068916 Smallest positive integer that is equal to the sum of the n-th powers of its prime factors (counted with multiplicity).

Original entry on oeis.org

2, 16, 1096744, 3125, 256, 823543, 19683

Offset: 1

Dean Hickerson, Mar 07 2002

Does a(n) exist for all n?
a(12)=65536, a(27)=4294967296. a(n) exists for all n of the form n=p^i-i, where p is prime and i > 0, since p^p^i is an example (see A067688 and A081177). - Jud McCranie, Mar 16 2003
a(23) <= 298023223876953125. a(24) <= 7625597484987. - Jud McCranie, Jan 18 2016
a(10) = 285311670611. - Jud McCranie, Jan 25 2016
a(24) = 7625597484987. - Jud McCranie, Jan 30 2016

			a(3) = 1096744 = 2^3*11^3*103; the sum of the cubes of the prime factors is 3*2^3 + 3*11^3 + 103^3 = 1096744.

S. P. Hurd and J. S. McCranie, Integers that are Sums of Uniform Powers of all their Prime Factors: the sequence A068916, J. of Int. Seq., vol 22, article 19.3.4.

Cf. A067688, A268036.

Cf. A081177, A000325, A024024, A024050.

a[n_] := For[x=1, True, x++, If[x==Plus@@(#[[2]]#[[1]]^n&/@FactorInteger[x]), Return[x]]]

isok(k, n) = {my(f=factor(k)); sum(j=1, #f~, f[j,2]*f[j,1]^n) == k;}
a(n) = {my(k = 1); while(! isok(k,n), k++); k;} \\ Michel Marcus, Jan 25 2016

from sympy import factorint
def a(n):
  k = 1
  while True:
    f = factorint(k)
    if k == sum(f[d]*d**n for d in f): return k
    k += 1
for n in range(1, 8):
  print(a(n), end=", ") # Michael S. Branicky, Feb 16 2021

A273940 Primes of the form 5^m - m.

Original entry on oeis.org

23, 15619, 244140613

Offset: 1

Vincenzo Librandi, Jun 05 2016

Corresponding values of m are given in A058046.

The next term has 254 digits.

Amiram Eldar, Table of n, a(n) for n = 1..5

Primes of the form k^m - m: A081296 (k=2), A224420 (k=3), A224451 (k=4), this sequence (k=5), A273941 (k=6), A224468 (k=7), A224469 (k=8).

Cf. A024050, A058046.

[a: n in [0..400] | IsPrime(a) where a is 5^n-n];

Select[Table[5^n - n, {n, 400}], PrimeQ]

forstep(n=2,1e4,2, if(ispseudoprime(t=5^n-n), print1(t", "))) \\ Charles R Greathouse IV, Jun 08 2016

a(n) = A024050(A058046(n)). - Amiram Eldar, Jul 27 2025

A268414 a(n) = 5a(n-1) - 2n for n > 0, a(0) = 1.

Original entry on oeis.org

1, 3, 11, 49, 237, 1175, 5863, 29301, 146489, 732427, 3662115, 18310553, 91552741, 457763679, 2288818367, 11444091805, 57220458993, 286102294931, 1430511474619, 7152557373057, 35762786865245, 178813934326183, 894069671630871, 4470348358154309, 22351741790771497, 111758708953857435

Offset: 0

Ilya Gutkovskiy, Feb 04 2016

In general, the ordinary generating function for the recurrence relation b(n) = k*b(n - 1) - m*n, with n > 0 and b(0)=1, is (1 - (m + 2)*x + x^2)/((1 - x)^2*(1 - k*x)). This recurrence gives the closed form b(n) = ((k^2 - k*(m + 2) + 1)*k^n + m*((k - 1)*n + k))/(k - 1)^2.

Index entries for linear recurrences with constant coefficients, signature (7,-11,5).

Cf. A014827, A024050, A094195, A104745, A107585, A164045, A176916, A221907.

[(4*n + 3*5^n + 5)/8: n in [0..30]]; // Vincenzo Librandi, Feb 06 2016

Table[(4 n + 3 5^n + 5)/8, {n, 0, 23}]
LinearRecurrence[{7, -11, 5}, {1, 3, 11}, 24]

Vec((1-4*x+x^2)/((1-x)^2*(1-5*x)) + O(x^100)) \\ Altug Alkan, Feb 04 2016

G.f.: (1 - 4*x + x^2)/((1 - x)^2*(1 - 5*x)).
a(n) = (4*n + 3*5^n + 5)/8.
Sum_{n>=0} 1/a(n) = 1.449934283402232875...
Lim_{n -> oo} a(n + 1)/a(n) = 5.
From Elmo R. Oliveira, Sep 10 2024: (Start)
E.g.f.: exp(x)*(3*exp(4*x) + 4*x + 5)/8.
a(n) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3) for n > 2. (End)

a(24)-a(25) from Elmo R. Oliveira, Sep 10 2024

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A385773 Primes having only {1, 2, 5} as digits.

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A068916 Smallest positive integer that is equal to the sum of the n-th powers of its prime factors (counted with multiplicity).

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A273940 Primes of the form 5^m - m.

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A268414 a(n) = 5*a(n-1) - 2*n for n > 0, a(0) = 1.

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A268414 a(n) = 5a(n-1) - 2n for n > 0, a(0) = 1.