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 1-8 of 8 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

A114850 (n-th semiprime)^(n-th semiprime).

Original entry on oeis.org

256, 46656, 387420489, 10000000000, 11112006825558016, 437893890380859375, 5842587018385982521381124421, 341427877364219557396646723584, 88817841970012523233890533447265625
Offset: 1

Views

Author

Jonathan Vos Post, Feb 20 2006

Keywords

Comments

Semiprime analog of A051674. This is also a subset of A113877 "semiprimes to semiprime powers."

Links

Crossrefs

Cf. A001358, A051674, A113877.

Programs

  • Mathematica
    #^#&/@Select[Range[30],PrimeOmega[#]==2&] (* Harvey P. Dale, Nov 07 2016 *)

Formula

a(n) = A001358(n)^A001358(n).

Extensions

Corrected by Don Reble, Nov 22 2006

A114967 (n-th 3-almost prime)^(n-th 3-almost prime).

Original entry on oeis.org

16777216, 8916100448256, 39346408075296537575424, 104857600000000000000000000, 443426488243037769948249630619149892803, 33145523113253374862572728253364605812736
Offset: 1

Views

Author

Jonathan Vos Post, Feb 21 2006

Keywords

Comments

3-almost prime analog of A051674. A114850 is semiprime analog of A051674.

Examples

			a(1) = A014612(1)^A014612(1) = 8^8 = 16777216 = 2^24.
a(2) = A014612(2)^A014612(2) = 12^12 = 8916100448256 = 2^24 * 3^12.
a(3) = A014612(3)^A014612(3) = 18^18 = 39346408075296537575424 = 2^18 * 3^36.

Links

Crossrefs

Cf. A001358, A014612, A051674, A113877, A114850.

Programs

  • Mathematica
    #^#&/@Select[Range[40],PrimeOmega[#]==3&] (* Harvey P. Dale, Apr 25 2015 *)

Formula

a(n) = A014612(n)^A014612(n).

A114993 (n-th 4-almost prime)^(n-th 4-almost prime).

Original entry on oeis.org

18446744073709551616, 1333735776850284124449081472843776, 106387358923716524807713475752456393740167855629859291136, 12089258196146291747061760000000000000000000000000000000000000000
Offset: 1

Views

Author

Jonathan Vos Post, Feb 22 2006

Keywords

Comments

4-almost prime analog of A051674. A114850 is semiprime analog of A051674. a(5) = 56^56 has 94 digits.

Examples

			a(1) = A014613(1)^A014613(1) = 16^16 = 18446744073709551616 = 2^64.
a(2) = A014613(2)^A014613(2) = 24^24 = 1333735776850284124449081472843776 = 2^72 * 3^24.
a(3) = A014613(3)^A014613(3) = 36^36 = 2^72 * 3^72.

Crossrefs

Cf. A001358, A014613, A051674, A113877, A114850.

Formula

a(n) = A014613(n)^A014613(n).

A129539 Composite numbers to composite powers.

Original entry on oeis.org

256, 1296, 4096, 6561, 10000, 20736, 38416, 46656, 50625, 65536, 104976, 160000, 194481, 234256, 262144, 331776, 390625, 456976, 531441, 614656, 810000, 1000000, 1048576, 1185921, 1336336, 1500625, 1679616, 2085136, 2313441
Offset: 0

Views

Author

Tanya Khovanova, May 29 2007

Keywords

Comments

Only unique powers are shown here. E.g., 4096 = 4^6 = 8^4, but 4096 is only in the sequence once. - Franklin T. Adams-Watters, Sep 28 2011

Examples

			The sequence starts with 256, because 256 is the smallest composite number to the smallest composite power.

Crossrefs

Cf. A113877 - Semiprimes to semiprime powers - is a subsequence of this sequence.

Programs

  • Mathematica
    comp = Select[Range[2, 40], ! PrimeQ[ # ] &]; Select[ Union[Flatten[ Table[comp[[n]]^comp[[k]], {n, Length[comp]}, {k, Length[comp]}]]], # < comp[[Length[comp]]]^4 &]

A186637 Semiprime powers with special exponents: k^(j-1) where both j and k are arbitrary semiprime numbers.

Original entry on oeis.org

64, 216, 729, 1000, 1024, 2744, 3375, 7776, 9261, 10648, 15625, 17576, 35937, 39304, 42875, 54872, 59049, 59319, 65536, 97336, 100000, 117649, 132651, 166375, 185193, 195112, 238328, 262144, 274625, 328509, 405224, 456533, 537824, 551368, 614125, 636056, 658503, 753571, 759375, 804357, 830584, 857375
Offset: 1

Views

Author

Jonathan Vos Post, Feb 24 2011

Keywords

Comments

Semiprime analog of A036454: prime powers with special exponents: q^(p-1) where both p and q are arbitrary prime numbers.

Examples

			a(1) = smallest semiprime to power of (smallest semiprime - 1) = 4^(4-1) = 4^3 = 64.

Crossrefs

Cf. A001358, A036454, A113877, A186621.

Programs

  • Python
    from math import isqrt
from sympy import primepi, primerange, integer_nthroot, factorint
def A186637(n):
    def A072000(n): return int(-((t:=primepi(s:=isqrt(n)))*(t-1)>>1)+sum(primepi(n//p) for p in primerange(s+1)))
    def f(x): return int(n+x-sum(A072000(integer_nthroot(x, p-1)[0]) for p in range(4,x.bit_length()+1) if sum(factorint(p).values())==2))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f,n,n) # Chai Wah Wu, Sep 12 2024

Formula

{a(n)} = {A001358(i) ^ A186621(j)}.
{a(n)} = {a^b where a and b are elements of A001358} = {(p*q)^((r*s)-1) for primes p, q, r, s, not necessarily distinct}.

A217784 Triprimes to triprime powers.

Original entry on oeis.org

16777216, 429981696, 11019960576, 25600000000, 68719476736, 282429536481, 377801998336, 656100000000, 8916100448256, 9682651996416, 14048223625216, 16815125390625, 39062500000000, 53459728531456, 248155780267521, 360040606269696, 457163239653376, 576480100000000
Offset: 1

Views

Author

Kevin L. Schwartz and Christian N. K. Anderson, Mar 24 2013

Keywords

Comments

Triprimes are numbers with exactly three prime factors: A014612.
This is to triprimes as primes are to A053810 (Prime powers of prime numbers) and as semiprimes are to A113877 (Semiprimes to semiprime powers). - Jonathan Vos Post, Mar 26 2013
a(n) increases roughly as n^8, because 9669 of the first 10000 terms are powers of 8. - Kevin L. Schwartz and Christian N. K. Anderson, Jun 05 2013

Examples

			429981696 = 8^12.
a(10) = 9682651996416 = 42^8 = (2*3*7)^(2*2*2).

Links

Crossrefs

Cf. A014612, A053810, A113877, A129539.

Programs

  • Python
    from math import isqrt
from sympy import primepi, primerange, integer_nthroot, factorint
def A217784(n):
    def g(x): return int(sum(primepi(x//(k*m))-b for a, k in enumerate(primerange(integer_nthroot(x, 3)[0]+1)) for b, m in enumerate(primerange(k, isqrt(x//k)+1), a)))
    def f(x): return int(n+x-sum(g(integer_nthroot(x, k)[0]) for k in range(1,x.bit_length()) if sum(factorint(k).values())==3))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f,n,n) # Chai Wah Wu, Sep 12 2024
  • R
    library(gmp); istriprime=function(x) ifelse(as.bigz(x)<8, F, length(factorize(x))==3)as.bigz(which(sapply(1:200, istriprime)))->trp; maxy=tail(trp, 1)^trp[1]; len=0; y=as.bigz(rep(0, 100))
for(i in 1:length(trp)) { j=0; while((n=trp[i]^trp[(j=j+1)])<=maxy) y[(len=len+1)]=n }
y[1:len]->y; y[order(as.numeric(y))]
-- Kevin L. Schwartz and Christian N. K. Anderson, Jun 05 2013

A217908 Semiprime powers of distinct semiprimes.

Original entry on oeis.org

1296, 4096, 6561, 10000, 38416, 50625, 194481, 234256, 262144, 390625, 456976, 531441, 1000000, 1048576, 1185921, 1336336, 1500625, 2085136, 2313441, 4477456, 5764801, 6765201, 7529536, 9150625, 10077696, 10556001, 11316496, 11390625, 14776336, 17850625
Offset: 1

Views

Author

Kevin L. Schwartz and Christian N. K. Anderson, Mar 25 2013

Keywords

Comments

Subset of A113877.

Examples

			6561=9^4, and 9 and 4 are both semiprime. 46656 = 6^6 is excluded because the semiprimes are not distinct.

Links

Crossrefs

Cf. A113877.

Programs

  • Python
    from math import isqrt
from sympy import primepi, primerange, integer_nthroot, factorint
def A217908(n):
    def A072000(n): return int(-((t:=primepi(s:=isqrt(n)))*(t-1)>>1)+sum(primepi(n//p) for p in primerange(s+1)))
    def f(x): return int(n+x-sum(A072000(integer_nthroot(x, p)[0])-(p**p<=x) for p in range(4,x.bit_length()) if sum(factorint(p).values())==2))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f,n,n) # Chai Wah Wu, Sep 12 2024
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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