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.

A247094 Integers of the form (2^k + 1)/(2k + 1).

Original entry on oeis.org

1, 2, 3, 5, 27, 565, 7085, 48771, 1266205, 9099507, 17602325, 128207979, 26494256091, 11147523830125, 84179432287299, 165269711096165, 281629680514649643, 4246732448623781667, 126774939137440139965, 1925041114036033717685, 14833445639443302757131
Offset: 1

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Author

Juri-Stepan Gerasimov, Nov 18 2014

Keywords

Comments

a(A103579(n)) is a subsequence.
Numbers n such that 2n + 1 divides 2^n + 1: 0, 1, 2, 5, 6, 9, 14, 18, 21, 26, 29, 30, 33, 41, 50, 53, ...

Examples

			1 is in this sequence because (2^1 + 1)/(2*1 + 1) = 1,
2 is in this sequence because (2^0 + 1)/(2*0 + 1) = 2,
3 is in this sequence because (2^5 + 1)/(2*5 + 1) = 3.

Links

Crossrefs

Cf. A081856, A081858, A103579, A224486, A247132.

Programs

  • PARI
    s=[]; for(k=0, 100, t=(2^k + 1)/(2*k + 1); if(type(t)=="t_INT", s=concat(s, t))); s=vecsort(s,,8) \\ Colin Barker, Nov 18 2014

Extensions

a(19) corrected by Colin Barker, Nov 18 2014

A307176 Number of Sophie Germain primes of the form 4k + 1 less than 10^n.

Original entry on oeis.org

1, 5, 17, 89, 589, 3833, 27940, 211439, 1653257, 13283194, 109058142, 911411528, 7731354496
Offset: 1

Views

Author

Rodolfo Ruiz-Huidobro, Mar 27 2019

Keywords

Comments

Sophie Germain primes can alternatively be Lucasian primes, primes of the form 4k + 1, or, the individual prime 2.

Examples

			There are five Sophie Germain Primes of the form 4k + 1 below 10^2: {5, 29, 41, 53, 89}, therefore a(2) = 5.

Crossrefs

Cf. A091098, A092816, A002515, A307121, A002144, A005384, A103579.

Programs

  • Mathematica
    nonLucSophies = Select[4Range[2500000] + 1, PrimeQ[#] && PrimeQ[2# + 1] &]; Table[Length[Select[nonLucSophies, # < 10^n &]], {n, 0, 7}]

Formula

a(n) < A092816(n).
a(n) <= A091098(n) (with equality for n = 1).
a(n) = A092816(n) - A307121(n) - 1.

A185343 Least positive number k such that k*p+1 divides 2^p+1 where p is prime(n), or 0 if no such number exists.

Original entry on oeis.org

2, 0, 2, 6, 62, 210, 2570, 9198, 121574, 2, 23091222, 48, 2, 68186767614, 6, 2, 48, 12600235023025650, 109368, 794502, 24, 2550476412689091085878, 6, 2, 10, 8367330694575771627040945250, 4030501264, 6, 955272, 2, 446564785985483547852197647548252246, 8, 8, 32424, 8
Offset: 1

Views

Author

Bill McEachen, Feb 26 2011

Keywords

Comments

Akin to A186283 except for 2^p+1 and restricted to primes.
The larger terms of this sequence occur for the primes p > 3 in sequence A000978. These large terms are (2^p-2)/(3p).
a(n) = 2 iff prime(n) is in A103579. - Robert Israel, Jul 17 2023

Examples

			2^3+1 = 9 has no factor of the form k*3+1 except 1, so a(primepi(3)) = a(2) = 0.
2^29+1 = 536870913 has factor 2*29+1=59, so a(primepi(29)) = a(10) = 2.

Crossrefs

Cf. A098268, A103579, A186283.

Programs

  • Maple
    f:= proc(n) local p,F;
  p:= ithprime(n);
  F:= select(t -> t mod p = 1, numtheory:-divisors(2^p+1) minus {1});
  if F = {} then 0 else (min(F)-1)/p; fi
end proc:
map(f, [$1..50]); # Robert Israel, Jul 17 2023
  • Mathematica
    Table[q = First /@ FactorInteger[2^p + 1]; s = Select[q, Mod[#1, p] == 1 &, 1]; If[s == {}, 0, (s[[1]] - 1)/p], {p, Prime[Range[30]]}]

A362140 Numbers k in A224486 for which the arithmetic derivative k' (A003415) is also in A224486.

Original entry on oeis.org

2, 5, 6, 9, 14, 18, 29, 33, 41, 53, 54, 65, 69, 89, 113, 134, 141, 158, 173, 198, 209, 221, 233, 249, 278, 281, 293, 326, 329, 338, 393, 473, 506, 509, 545, 581, 593, 614, 629, 641, 653, 713, 729, 749, 761, 809, 846, 905, 950, 953, 965, 986, 1013, 1014, 1026, 1041, 1049
Offset: 1

Views

Author

Marius A. Burtea, May 03 2023

Keywords

Comments

Sophie Germain primes p that are not Lucasian primes (A103579) are terms because p' = 1 = A224486(1).

Examples

			6 = A224486(4) and 6' = 5 = A224486(3), so 6 is a term.
9 = A224486(5) and 9' = 6 = A224486(4), so 9 is a term.
14 = A224486(6) and 14' = 9 = A224486(5), so 14 is a term.

Crossrefs

Cf. A003415, A103579, A224486.

Programs

  • Magma
    czn:=func; f:=func; [n:n in [2..5000]|czn(n) and czn(Floor(f(n)))];
  • Mathematica
    d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); curzonQ[n_] := PowerMod[2, n, 2*n + 1] == 2*n; Select[Range[2, 1050], curzonQ[#] && curzonQ[d[#]] &] (* Amiram Eldar, May 05 2023 *)
Showing 1-4 of 4 results.

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