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 1-4 of 4 results.

A094473 Smallest prime factor of 2^n+3^n.

Original entry on oeis.org

5, 13, 5, 97, 5, 13, 5, 17, 5, 13, 5, 97, 5, 13, 5, 3041, 5, 13, 5, 41, 5, 13, 5, 17, 5, 13, 5, 97, 5, 13, 5, 1153, 5, 13, 5, 97, 5, 13, 5, 17, 5, 13, 5, 89, 5, 13, 5, 193, 5, 13, 5, 97, 5, 13, 5, 17, 5, 13, 5, 41, 5, 13, 5, 769, 5, 13, 5, 97, 5, 13, 5, 17, 5, 13, 5
Offset: 1

Views

Author

Labos Elemer, Jun 02 2004

Keywords

Comments

If n = 4*k+1 or 4*k+3 then 2^n+3^n is divisible by 5.
If n = 4*k+2 then 2^n+3^n is divisible by 13.
Case n = 4*k including especially n = 2^j cannot be discussed with elementary tools and primality of 2^n+3^n remains open.
a(n) = 17 for n == 8 (mod 16). - Bruno Berselli, Dec 23 2019

Links

Crossrefs

Cf. A007689, A020639, A050244, A082101, A094474-A094494.

Programs

  • GAP
    List([1..80],n->Factors(2^n+3^n)[1]); # Muniru A Asiru, Nov 01 2018
  • Magma
    [Min(PrimeFactors(2^n+3^n)): n in[1..100]]; // Vincenzo Librandi, Dec 23 2019
  • Magma
    [PrimeFactors(2^n+3^n)[1]: n in[1..600]]; // Bruno Berselli, Dec 23 2019
  • Mathematica
    mif[x_]:=Part[Flatten[FactorInteger[x]], 1] Table[mif[2^w+3^w], {w, 1, 75}]
FactorInteger[#][[1,1]]&/@Table[2^n+3^n,{n,80}] (* Harvey P. Dale, Mar 26 2019 *)
  • PARI
    a(n)=factor(2^n+3^n)[1,1] \\ Charles R Greathouse IV, Apr 29 2015
  • PARI
    A094473(n) = { my(k=(2^n+3^n)); forprime(p=2,k,if(!(k%p),return(p))); }; \\ Antti Karttunen, Nov 01 2018

Formula

a(n) = A020639(A007689(n)). - Antti Karttunen, Nov 01 2018

A294132 Sorted list of prime factors of numbers of the form 3^(2^m) + 2^(2^m) with m >= 0.

Original entry on oeis.org

5, 13, 17, 97, 257, 401, 769, 1153, 3041, 14177, 65537, 286721, 1810433, 2752513, 4043777, 7340033, 13631489, 23068673, 72222721, 319291393, 348061697, 625483777, 3937533953, 54498164737, 106216554497, 121899667073, 151597350913, 342456532993
Offset: 1

Views

Author

Arkadiusz Wesolowski, Oct 23 2017

Keywords

Comments

Primes p such that the multiplicative order of 3/2 (mod p) is a power of 2.

Examples

			The first 5 such numbers are 5, 13, 97, 6817, 43112257, 1853024483819137. Their prime factorizations are (5), (13), (97), (17) (401), (14177) (3041), (1153) (1607133116929). - _N. J. A. Sloane_, Oct 29 2017

Links

Crossrefs

Cf. A050244, A094499, A082101, A294133, A294134, A294135, A294136.

Programs

  • PARI
    print1(5, ", "); forprime(p=13, 342456532993, z=znorder(Mod(3/2, p)); if(2^ispower(z)==z, print1(p, ", ")));

A082869 3^n - 2^n is a semiprime.

Original entry on oeis.org

4, 7, 9, 13, 19, 23, 37, 71, 89, 97, 131, 167, 193, 227, 229, 257, 263, 269, 271
Offset: 1

Views

Author

Hugo Pfoertner, May 24 2003

Keywords

Comments

a(20) >= 653. - Max Alekseyev, Aug 26 2021

Examples

			a(1) = 4 because 3^4 - 2^4 = 5 * 13
a(2) = 7 because 3^7 - 2^7 = 29 * 71
a(3) = 9 because 3^9 - 2^9 = 1009 * 19
a(4) = 13 because 3^13 - 2^13 = 53 * 29927
a(5) = 19 because 3^19 - 2^19 = 1559 * 745181
a(6) = 23 because 3^23 - 2^23 = 47 * 2002867877
a(7) = 37 because 3^37 - 2^37 = 8891471 * 50642213021
a(8) = 71 because 3^71 - 2^71 = 67049419 * 111998979662707645844109121
a(9) = 89 because 3^89 - 2^89 = 4120081168939 * 706132008101135602203621405289
a(10) = 97 because 3^97 - 2^97 = 319128643 * 59813046375181769306016700165290169537
a(11) = 131 because 3^131 - 2^131 = 263 * 1210399177182288006201752262354382648158190136861552303421773
a(12) = 167 because 3^167 - 2^167 = 167884386911 * 284602839755962600307038183361142274453177384697761703968640951718869
a(13) = 193 because 3^193 - 2^193 = 773 * 157116815095122696291789672145814943987605497895096234870661710074857006307174092298131047
a(14) = 227 because 3^227 - 2^227 = 167360891302418779411 * 12102381564694515014432350438002672779054341887509579790377508212702751544613632122970969
a(15) = 229 because 3^229 - 2^229 = 271117470516046849 * 67237232094433305864393166477037402086197319313004074022941345112953840883539481643687544179
a(16) = 257 because 3^257 - 2^257 = 3650201327 * 114247220844165289049224917003868019618046824570124111266639206512722372880755761151052076786187552795911804402733
a(17) = 263 because 3^263 - 2^263 = 1789696394587605010251024191 * 169867630212703250249981022070263878299079238108093021871181171428200213741587995035055139427113909
a(18) = 269 because 3^269 - 2^269 = 3767 * 58833122596041019850277965408508940208380870952125838087379156948993498251689923575161076689330121444393974916753840891087813
a(19) = 271 because 3^271 - 2^271 = 2711 * 735750407736473144959046057264728365874119021724332398617327934122565857164514694088659506296666818455309890905079155116415309

Links

Crossrefs

Cf. A001047, A001358, A057468, A058765, A050244, A095027.

Extensions

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 30 2007

A094499 Smallest prime factor of 2^(2^n)+3^(2^n), i.e., exponents are powers of 2.

Original entry on oeis.org

13, 97, 17, 3041, 1153, 769, 257, 72222721, 4043777, 2330249132033, 625483777, 286721, 14496395542529, 2752513, 65537, 319291393, 54498164737
Offset: 1

Views

Author

Labos Elemer, Jun 02 2004

Keywords

Comments

Factors are of the form k*2^(n+1)+1.

Links

Crossrefs

Cf. A050244, A294132.

Programs

  • Mathematica
    f[n_] := Block[{k = 1, m = 2^(n + 1)}, While[ Mod[ PowerMod[2, 2^n, k*m + 1] + PowerMod[3, 2^n, k*m + 1], k*m + 1] != 0, k++ ]; k*m + 1]; Table[ f[n], {n, 9}] (* Robert G. Wilson v, Jun 03 2004 *)

Extensions

Edited and extended by Robert G. Wilson v, Jun 03 2004
Showing 1-4 of 4 results.

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

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