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.

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A075079 Numbers k in A001597 such that 2*k + 1 is prime.

Original entry on oeis.org

1, 8, 9, 36, 81, 125, 128, 216, 243, 441, 576, 729, 900, 1089, 1296, 1331, 1728, 1764, 2025, 4356, 5184, 5625, 7569, 8000, 8649, 9216, 9261, 9801, 10404, 11025, 15129, 17424, 17576, 18225, 19683, 23409, 24336, 24389, 26244, 27000, 31329, 32768, 34596, 35721
Offset: 1

Views

Author

Zak Seidov, Oct 11 2002

Keywords

Examples

			2*8 + 1 = 17 is prime, so 8 is a term.

Crossrefs

Cf. A001597, A075081.

Programs

  • PARI
    lista(nn) = {vec = vector(nn, i, i); pp = select(i->((ispower(i) || (i==1)) && isprime(2*i+1)), vec); for (i = 1, #pp, print1(pp[i], ", "));} \\ Michel Marcus, Oct 02 2013

Extensions

More terms from Michel Marcus, Oct 02 2013

A075453 Numerator of 1/pp(n) + 1/pp(n+1), where pp(n) = A001597(n-1) is the n-th perfect power.

Original entry on oeis.org

5, 3, 17, 25, 41, 52, 59, 17, 85, 113, 145, 181, 221, 246, 253, 17, 313, 365, 103, 49, 52, 499, 545, 613, 667, 704, 761, 841, 925, 249, 1041, 1105, 1201, 1301, 1405, 1513, 1625, 1741, 1861, 1961, 253, 2113, 2245, 2381, 2521, 2627, 2700, 2813, 2965, 3121
Offset: 1

Views

Author

Zak Seidov, Oct 11 2002

Keywords

Examples

			a(2) = 3 because 1/pp(2) + 1/pp(3) = 1/4 + 1/8 = 3/8.

Crossrefs

Cf. A001597.

Extensions

Edited by Dean Hickerson, Oct 15 2002

A075455 Prime averages of two successive perfect powers (A001597(k) + A001597(k+1))/2.

Original entry on oeis.org

2203, 77983, 90863, 185477, 371087, 388573, 613607, 912349, 1293899, 1600919, 2146457, 30661333, 35608189, 48823147, 81190429, 105823093, 122753857, 204341747, 338602837, 368601707, 374788121, 426958673, 498675409, 586371239, 656232799, 665360321, 674509487, 693132527
Offset: 1

Views

Author

Zak Seidov, Oct 11 2002

Keywords

Examples

			2203 = (A001597(61)+A001597(62))/2 = (2197+2209)/2.

Crossrefs

Cf. A001597.

Programs

  • Mathematica
    a = Select[Range[2, 50000000], GCD @@ (Transpose[FactorInteger[ # ]][[2]]) > 1 &]; sizea = Length[a]; (a[[Select[Range[sizea - 1], PrimeQ[(a[[ # ]] + a[[ # + 1]])/2] &]]] + a[[Select[Range[sizea - 1], PrimeQ[(a[[ # ]] + a[[ # + 1]])/2] &] + 1]])/2 (* Tanya Khovanova, Sep 15 2007 *)

Extensions

More terms from Tanya Khovanova, Sep 15 2007
More terms from Amiram Eldar, Feb 18 2022

A075773 Let {b(n)} be the sequence of perfect powers (A001597); then a(n) = max { b(n)-b(n-1), b(n+1)-b(n) }.

Original entry on oeis.org

3, 4, 4, 7, 9, 9, 5, 5, 13, 15, 17, 19, 21, 21, 4, 16, 25, 27, 27, 20, 18, 18, 33, 35, 35, 19, 39, 41, 43, 43, 28, 47, 49, 51, 53, 55, 57, 59, 61, 61, 39, 65, 67, 69, 71, 71, 38, 75, 77, 79, 81, 81, 47, 85, 87, 89, 89, 68, 71, 71, 12, 95, 97, 99, 101, 103, 103
Offset: 1

Views

Author

Neil Fernandez, Oct 09 2002

Keywords

Examples

			The perfect powers are 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, etc. The 7th is 27. This is 2 larger than the 6th (25) and 5 smaller than the 8th (32). So a(7)=5.

Crossrefs

Cf. A001597, A053289, A075772.

Extensions

Missing 4 inserted and more terms from Sean A. Irvine, Mar 06 2025

A077286 Primes which are not the difference between two successive perfect powers (A001597).

Original entry on oeis.org

29, 31, 179, 181, 281, 313, 379, 397, 487, 563, 839, 883, 907, 929, 953, 977, 997, 1049, 1171, 1567, 1823, 2213, 2339, 2371, 2383, 2729, 2749, 2897, 2999, 3067, 3137, 3313, 3529, 3637, 3823, 4591, 4789, 4871, 4951, 5197, 5237, 5279, 5531, 5573, 5741
Offset: 1

Views

Author

Robert G. Wilson v, Oct 31 2002

Keywords

Examples

			29 is not the difference between two successive perfect powers.

Crossrefs

Cf. A001597, A023057.

Programs

  • Mathematica
    pp = Union[ Join[{1}, Flatten[ Table[n^i, {n, 2, Sqrt[10^12]}, {i, 2, Log[n, 10^12]}]]]]; l = Length[pp]; d = Sort[Take[pp, -l + 1] - Take[pp, l - 1]]; a = {}; Do[ If[ PrimeQ[ d[[n]]], a = Append[a, d[[n]]]], {n, 1, l - 1}]; Complement[ Table[ Prime[i], {i, 1, 760}], Take[ Union[a], 760]]

A099997 Bisection of A001597.

Original entry on oeis.org

1, 8, 16, 27, 36, 64, 100, 125, 144, 196, 225, 256, 324, 361, 441, 512, 576, 676, 784, 900, 1000, 1089, 1225, 1331, 1444, 1600, 1728, 1849, 2025, 2116, 2197, 2304, 2500, 2704, 2809, 3025, 3136, 3364, 3481, 3721, 3969, 4225, 4489, 4761, 4913, 5184, 5476
Offset: 1

Views

Author

N. J. A. Sloane, Nov 20 2004

Keywords

Programs

  • Mathematica
    t = Union@ Flatten@ Join[{1}, Table[ n^i, {n, 2, Sqrt[5775]}, {i, 2, Log[n, 5775]}]]; t[[2# - 1]] & /@ Range@(Length@t/2) (* Robert G. Wilson v *)
  • Python
    from sympy import mobius, integer_nthroot
def A099997(n):
    def f(x): return int((n<<1)-3+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,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 14 2024

Extensions

More terms from Robert G. Wilson v, Dec 14 2005

A099998 Bisection of A001597.

Original entry on oeis.org

4, 9, 25, 32, 49, 81, 121, 128, 169, 216, 243, 289, 343, 400, 484, 529, 625, 729, 841, 961, 1024, 1156, 1296, 1369, 1521, 1681, 1764, 1936, 2048, 2187, 2209, 2401, 2601, 2744, 2916, 3125, 3249, 3375, 3600, 3844, 4096, 4356, 4624, 4900, 5041, 5329, 5625, 5832
Offset: 1

Views

Author

N. J. A. Sloane, Nov 20 2004

Keywords

Programs

  • Mathematica
    t = Union@ Flatten@ Table[ n^i, {n, 2, Sqrt[6083]}, {i, 2, Log[n, 6083]}]; t[[2# - 1]] & /@ Range@(Length@t/2)
  • Python
    from sympy import mobius, integer_nthroot
def A099998(n):
    def f(x): return int((n<<1)-2+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,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 14 2024

Extensions

More terms from Robert G. Wilson v, Dec 14 2005

A111026 Perfect powers (A001597) of the form 3p + q + 3, p & q are primes.

Original entry on oeis.org

16, 25, 27, 32, 49, 121, 125, 128, 169, 225, 243, 289, 343, 361, 512, 529, 625, 729, 841, 961, 1000, 1225, 1331, 1369, 1681, 1849, 2025, 2048, 2187, 2197, 2209, 2401, 2809, 3025, 3125, 3375, 3481, 3721, 3969, 4225, 4489, 4913, 5041, 5329, 5625, 5929, 6241
Offset: 1

Views

Author

Walter Kehowski, Oct 05 2005

Keywords

Comments

The sequence has repetitions since different p's and q's will give the same perfect power. Remove the andmap in the program if you want the repetitions.
Includes all perfect powers, pp, (A001597) congruent +/- 1 (modulo 6). Also if pp-9 or pp-12 is a prime or if (pp -2)/3 or (pp-3)/3 is a prime.
The number of perfect powers of the form 3p + q + 3 <= 10^n: 0,5,21,56,157,433,...,. - Robert G. Wilson v, Jun 21 2006
In the first one million integers there are 1111 perfect powers (A070428) of which only 433 of them are of the form 3p + q + 3.

Examples

			a(5)=49 since 3*3+37+3=49 = 5*3+31+3 = 3*11+13+3 = 3*13+7+7 = 7^2.
6859 = 19^3 is in the sequence because there are 116 different ways to combine primes of the form 3p + q + 3, beginning with p=5 & q=6841 and ending with p=2281 & q=13.

Crossrefs

Cf. A000040, A001597.

Programs

  • Maple
    with(numtheory); egcd := proc(n) local L; L:=map(proc(z) z[2] end, ifactors(n)[2]); igcd(op(L)) end: PW:=[]: for z to 1 do for j from 1 to 100 do for k from 1 to 100 do p:=ithprime(j); q:=ithprime(k); x:=3*p+q+3; if egcd(x)>1 and andmap(proc(w) not(w[3]=x) end, PW) then PW:=[op(PW), [p,q,x]] fi od od od; PW; map(proc(z) z[3] end, PW);
  • Mathematica
    fQ[n_] := GCD @@ Last /@ FactorInteger@n > 1; lst = {}; Do[p = Prime@j; q = Prime@k; x = 3p + q + 3; If[fQ@x, AppendTo[lst, x]], {j, 340}, {k, PrimePi[6856 - 3Prime@j]}]; Union@lst (* Robert G. Wilson v *)

Formula

a(n)=3p+q+3 where p and q are primes and a(n) is a perfect power.

Extensions

Edited, corrected and extended by Robert G. Wilson v, Jun 21 2006

A143863 Primes such that the sum of digits is a perfect power (A001597).

Original entry on oeis.org

13, 17, 31, 53, 71, 79, 97, 103, 107, 211, 233, 251, 277, 349, 367, 431, 439, 457, 503, 521, 547, 619, 673, 691, 701, 709, 727, 853, 907, 997, 1021, 1061, 1069, 1087, 1151, 1201, 1223, 1249, 1429, 1447, 1483, 1511, 1601, 1609, 1627, 1663, 1699, 1753, 1789
Offset: 1

Views

Author

Giovanni Teofilatto, Sep 04 2008

Keywords

Links

Crossrefs

Cf. A062338.

Programs

  • Mathematica
    Select[Prime[Range[300]],GCD@@FactorInteger[Total[IntegerDigits[#]]][[;;,2]]>1&] (* Harvey P. Dale, Sep 18 2023 *)

Formula

Union of A062339, A062343, A106757, A106768, A107618 etc. [From R. J. Mathar, Sep 13 2008]

Extensions

389, 569, 581, 659, 677 etc. removed by R. J. Mathar, Sep 13 2008

A171755 Least positive integer k where k is a perfect power and the number of divisors of k = A001597(n).

Original entry on oeis.org

1, 8, 128, 36, 216, 1296, 900, 2147483648, 7776, 46656, 27000, 44100, 10077696, 60466176, 810000, 170141183460469231731687303715884105728, 362797056, 2176782336, 5832000, 24300000, 1587600, 2822400, 9261000, 2821109907456
Offset: 1

Views

Author

Ray Chandler, Dec 17 2009

Keywords

Comments

a(n) is the least member of A001597 such that A000005(a(n)) = A001597(n).

Links

Crossrefs

Cf. A175050, A175051.
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