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 1-6 of 6 results.

A076871 Sum of two powerful numbers (definition (1), A001694).

Original entry on oeis.org

2, 5, 8, 9, 10, 12, 13, 16, 17, 18, 20, 24, 25, 26, 28, 29, 31, 32, 33, 34, 35, 36, 37, 40, 41, 43, 44, 45, 48, 50, 52, 53, 54, 57, 58, 59, 61, 63, 64, 65, 68, 72, 73, 74, 76, 80, 81, 82, 85, 88, 89, 90, 91, 96, 97, 98, 99, 100, 101, 104, 106, 108, 109, 112, 113, 116, 117
Offset: 1

Views

Author

N. J. A. Sloane, Nov 25 2002

Keywords

Comments

Complement of A085253. - Reinhard Zumkeller, Jun 23 2003

References

  • Aleksandar Ivić, The Riemann Zeta-Function, Wiley, NY, 1985, see p. 439.

Links

Crossrefs

Cf. A001694, A076872, A085252, A085253, A261883.
Different from A070049.

Programs

  • Mathematica
    With[{m = 120}, pow = Select[Range[m], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]; Select[Union[Plus @@@ Tuples[pow, {2}]], # <= m &]] (* Amiram Eldar, Jan 30 2023 *)

Formula

A085252(a(n)) > 0. - Reinhard Zumkeller, Jun 23 2003
Blomer shows that there are x/log^k x sums of two powerful numbers up to x, where k = 0.20629947... is A261883. - Charles R Greathouse IV, Sep 04 2015

Extensions

More terms from Vladeta Jovovic, Nov 25 2002

A351062 Numbers which are the sum of two perfect powers with different exponents: m = a^x + b^y with a > 0, b > 0, x > 1, y > 1 and x different from y, with m not a perfect power.

Original entry on oeis.org

2, 5, 10, 12, 17, 20, 24, 26, 28, 31, 33, 37, 40, 41, 43, 44, 48, 50, 52, 57, 59, 63, 65, 68, 72, 73, 76, 80, 82, 85, 89, 90, 91, 96, 97, 101, 106, 108, 113, 116, 117, 122, 126, 127, 129, 130, 132, 134, 136, 137, 141, 145, 148, 150, 152, 153, 155, 157, 160, 161, 162, 164, 170
Offset: 1

Views

Author

Alberto Zanoni, Jan 31 2022

Keywords

Examples

			2 is a term, as 2 = 1^2 + 1^3.
5 is a term, as 5 = 2^2 + 1^3.
17 is a term, as 17 = 1^2 + 2^4 or 3^2 + 2^3 or 4^2 + 1^3 (considering minimal possible exponents for bases equal to 1).

Links

Crossrefs

Cf. A001597, A070049, A327609.

Extensions

Definition clarified by Alberto Zanoni, Feb 28 2022

A074499 Sum of three perfect powers.

Original entry on oeis.org

3, 6, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79
Offset: 1

Views

Author

Rick L. Shepherd, Aug 24 2002

Keywords

Comments

It appears that every number >= 10 (and 4 and 7) is a sum of 4 perfect powers.

Crossrefs

Cf. A070049 (sum of two perfect powers), A001597 (perfect powers).

A075434 Numbers that are not the sum of two perfect powers.

Original entry on oeis.org

1, 3, 4, 6, 7, 11, 14, 15, 19, 21, 22, 23, 27, 30, 38, 39, 42, 46, 47, 49, 51, 55, 56, 60, 62, 66, 67, 69, 70, 71, 75, 77, 78, 79, 83, 84, 86, 87, 88, 92, 93, 94, 95, 99, 102, 103, 105, 107, 110, 111, 112, 114, 115, 118, 119, 120, 121, 123, 124, 131, 135, 138, 139, 140
Offset: 1

Views

Author

Zak Seidov, Oct 11 2002

Keywords

Crossrefs

Cf. A001597, complement of A070049.

Programs

  • PARI
    lista(nn) = {vec = vector(nn, i, i); pp = concat(1, select(i->ispower(i) , vec)); sumpp = Set(); for (i = 1, #pp, for (j = 1, #pp, sumpp = Set(concat(sumpp, pp[i]+pp[j])););); for (i = 1, nn, if (! setsearch(sumpp, i), print1(i, ", ")););} \\ Michel Marcus, Oct 04 2013

A290135 Numbers that are the sum of two proper prime powers (A246547).

Original entry on oeis.org

8, 12, 13, 16, 17, 18, 20, 24, 25, 29, 31, 32, 33, 34, 35, 36, 40, 41, 43, 48, 50, 52, 53, 54, 57, 58, 59, 64, 65, 68, 72, 73, 74, 76, 80, 81, 85, 89, 90, 91, 96, 97, 98, 106, 108, 113, 125, 128, 129, 130, 132, 133, 134, 136, 137, 141, 144, 145, 146, 148, 150, 152, 153, 155, 157, 160, 162, 170, 173, 174, 177, 178
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 20 2017

Keywords

Comments

Is 2213 the largest prime term that can be expressed as the sum of two proper prime powers in more than one way? - Altug Alkan, Jul 22 2017

Examples

			13 is in the sequence because 13 = 2^2 + 3^2.

Links

Crossrefs

Cf. A014091, A070049, A071330, A071331, A225102, A225103, A246547.

Programs

  • Maple
    N:= 1000: # to get all terms <= N
P:= select(isprime, [$2..floor(sqrt(N))]):
PP:= {seq(seq(p^j, j=2..floor(log[p](N))),p=P)}:
A:= select(`<=`,{seq(seq(PP[i]+PP[j],j=1..i),i=1..nops(PP))},N):
sort(convert(A,list)); # Robert Israel, Jul 21 2017
  • Mathematica
    nmax = 180; f[x_] := Sum[Boole[PrimePowerQ[k] && PrimeOmega[k] > 1] x^k, {k, 1, nmax}]^2; Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, nmax}]]

Formula

Exponents in expansion of (Sum_{k>=1} x^A246547(k))^2.

A327609 Numbers expressible as a^b + c^d with b&d > 1 and a&c >= 0.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 10, 12, 13, 16, 17, 18, 20, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 40, 41, 43, 44, 45, 48, 49, 50, 52, 53, 54, 57, 58, 59, 61, 63, 64, 65, 68, 72, 73, 74, 76, 80, 81, 82, 85, 89, 90, 91, 96, 97, 98, 100, 101, 104, 106, 108, 109, 113, 116, 117
Offset: 1

Views

Author

Elan Roth, Sep 18 2019

Keywords

Crossrefs

Cf. A070049.

Programs

  • Mathematica
    mx = 120; s = {0,1} ~Join~ Select[Range[2, mx], GCD @@ (Last /@ FactorInteger[#]) > 1 &]; Union[ Reap[Do[v = s[[i]] + s[[j]]; If[v <= mx, Sow@v], {i, Length@s}, {j, i}]][[2, 1]]] (* Giovanni Resta, Sep 19 2019 *)
  • PARI
    upto(n)={my(p=(1 + x + sum(k=2, sqrtint(n), sum(e=2, logint(n,k), x^(k^e))) + O(x*x^n))^2); select(i->polcoef(p,i), [0..n])} \\ Andrew Howroyd, Sep 22 2019
Showing 1-6 of 6 results.

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

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