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 1-3 of 3 results.

A053810 Numbers of the form p^e where both p and e are prime numbers.

Original entry on oeis.org

4, 8, 9, 25, 27, 32, 49, 121, 125, 128, 169, 243, 289, 343, 361, 529, 841, 961, 1331, 1369, 1681, 1849, 2048, 2187, 2197, 2209, 2809, 3125, 3481, 3721, 4489, 4913, 5041, 5329, 6241, 6859, 6889, 7921, 8192, 9409, 10201, 10609, 11449, 11881, 12167
Offset: 1

Views

Author

Henry Bottomley, Mar 28 2000

Keywords

Comments

Possible orders of finite fields with exactly 2 subfields. In other words, possible orders of finite fields whose only proper subfield is the prime field. - Jianing Song, Jun 06 2025

Links

Crossrefs

Cf. A000040, A000961, A053811, A053812.
Cf. A203967; subsequence of A000961.
Cf. A113877 (similar for semiprimes).

Programs

  • Haskell
    a053810 n = a053810_list !! (n-1)
a053810_list = filter ((== 1) . a010051 . a100995) $ tail a000961_list
-- Reinhard Zumkeller, Jun 05 2013
  • Maple
    h := proc(n) local P; P := NumberTheory:-PrimeFactors(n); nops(P) = 1 and isprime(padic:-ordp(n, P[1])) end:
A053810List := upto -> seq(n, n = select(h, [seq(1..upto)])):  # Peter Luschny, Apr 14 2025
  • Mathematica
    pp={}; Do[if=FactorInteger[n]; If[Length[if]==1&&PrimeQ[if[[1, 1]]]&&PrimeQ[if[[1, 2]]], pp=Append[pp, n]], {n, 13000}]; pp
Sort[ Flatten[ Table[ Prime[n]^Prime[i], {n, 1, PrimePi[ Sqrt[12800]]}, {i, 1, PrimePi[ Log[ Prime[n], 12800]]}]]]
  • PARI
    is(n)=isprime(isprimepower(n)) \\ Charles R Greathouse IV, Mar 19 2013
  • Python
    from sympy import primepi, integer_nthroot, primerange
def A053810(n):
    def f(x): return int(n-1+x-sum(primepi(integer_nthroot(x, p)[0]) for p in primerange(x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 13 2024
  • SageMath
    def isA(n):
    p = prime_divisors(n)
    return len(p) == 1 and is_prime(valuation(n, p[0]))
print([n for n in srange(1, 12222) if isA(n)])  # Peter Luschny, Apr 14 2025

Formula

a(n) = A053811(n)^A053812(n). - David Wasserman, Feb 17 2006
A010055(a(n)) * A010051(A100995(a(n))) = 1. - Reinhard Zumkeller, Jun 05 2013
Sum_{n>=1} 1/a(n) = Sum_{p prime} P(p) = 0.6716752222..., where P is the prime zeta function. - Amiram Eldar, Nov 21 2020

Extensions

More terms from David Wasserman, Feb 17 2006
Name clarified by Peter Luschny, Apr 14 2025

A053812 Exponents occurring in A053810.

Original entry on oeis.org

2, 3, 2, 2, 3, 5, 2, 2, 3, 7, 2, 5, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 11, 7, 3, 2, 2, 5, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 13, 2, 2, 2, 2, 2, 3, 2, 2, 5, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 7, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 17, 2, 2, 2, 2, 3, 2, 2, 2
Offset: 1

Views

Author

Henry Bottomley, Mar 28 2000

Keywords

Links

Crossrefs

Cf. A000961, A000961, A001222, A025473, A053810, A053811.

Programs

  • PARI
    LIM = prime(80)^2; v = vector(400); count = 0; forprime (p = 2, prime(80), x = 2; while (p^x <= LIM, count++; v[count] = p^x; x = nextprime(x + 1))); v = vecsort(vector(count, i, v[i])); vector(count, i, bigomega(v[i])) \\ David Wasserman, Feb 17 2006
  • Python
    from sympy import primepi, integer_nthroot, primerange, factorint
def A053812(n):
    def f(x): return int(n-1+x-sum(primepi(integer_nthroot(x, p)[0]) for p in primerange(x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return list(factorint(kmax).values())[0] # Chai Wah Wu, Aug 13 2024

Formula

a(n) = A001222(A053810(n)). - David Wasserman, Feb 17 2006
a(n) = log(A053810(n))/log(A053811(n)). - Amiram Eldar, Nov 21 2020

Extensions

More terms from David Wasserman, Feb 17 2006
Offset corrected by Amiram Eldar, Nov 21 2020

A145521 Take the primes raised to prime exponents, arranged in numerical order (A053810). If A053810(n) = r(n)^q(n), where r(n) and q(n) are primes, then a(n) = q(n)^r(n).

Original entry on oeis.org

4, 9, 8, 32, 27, 25, 128, 2048, 243, 49, 8192, 125, 131072, 2187, 524288, 8388608, 536870912, 2147483648, 177147, 137438953472, 2199023255552, 8796093022208, 121, 343, 1594323, 140737488355328, 9007199254740992, 3125, 576460752303423488, 2305843009213693952, 147573952589676412928
Offset: 1

Views

Author

Leroy Quet, Oct 12 2008

Keywords

Comments

a(n) = A053812(n)^A053811(n).

Crossrefs

Cf. A053810, A053811, A053812, A145522.

Programs

  • PARI
    lista(nn) = for(k=1, nn, if(isprime(isprimepower(k, &p)), print1(bigomega(k)^p, ", "))); \\ Jinyuan Wang, Feb 25 2020
  • Python
    from math import prod
from sympy import primepi, integer_nthroot, primerange, factorint
def A145521(n):
    def f(x): return int(n-1+x-sum(primepi(integer_nthroot(x, p)[0]) for p in primerange(x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return prod(e**p for p,e in factorint(kmax).items()) # Chai Wah Wu, Aug 13 2024

Extensions

Extended by Ray Chandler, Nov 01 2008
More terms from Jinyuan Wang, Feb 25 2020
Showing 1-3 of 3 results.

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