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 1-7 of 7 results.

A138537 Primes p_n for which A140141(n) = 2p_n, where p_n = n-th prime (A000040).

Original entry on oeis.org

11, 23, 29, 31, 47, 53, 59, 67, 71, 79, 83, 103, 107, 127, 131, 137, 139, 149, 151, 167, 173, 179, 191, 197, 199, 211, 223, 227, 229, 239, 251, 263, 269, 271, 283, 293, 307, 311, 317, 331, 347, 359, 367, 373, 379, 383, 389, 419, 431, 439, 443, 463, 467, 479
Offset: 1

Views

Author

Vladimir Shevelev, May 10 2008

Keywords

Comments

Perhaps the same as A058340, but need proof. - Ray Chandler, May 20 2008
The first member of this sequence not in A058340 is 295937. - Robert Israel, Aug 12 2016

Links

Crossrefs

Cf. A000010, A000040, A039649, A058340, A140141.

Programs

  • Maple
    filter:= n -> isprime(n) and numtheory:-invphi(numtheory:-phi(n))[2] = 2*n:
select(filter, [seq(i,i=2..1000)]); # Robert Israel, Aug 12 2016

Extensions

Corrected and extended by Ray Chandler, May 20 2008

A138539 Primes p_n for which A140141(n) < 2p_n, where p_n = n-th prime (A000040).

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 37, 41, 43, 61, 73, 89, 97, 101, 109, 113, 157, 163, 181, 193, 233, 241, 257, 277, 281, 313, 337, 349, 353, 397, 401, 409, 421, 433, 449, 457, 461, 487, 521, 541, 577, 593, 601, 613, 617, 641, 661, 673, 701, 733, 757, 761, 769, 821, 829
Offset: 1

Views

Author

Vladimir Shevelev, May 10 2008

Keywords

Comments

Apparently this is the union of {2} and A058341. - R. J. Mathar, Jul 03 2009
This sequence is the complement of A058340. - Torlach Rush, Jun 29 2018

Crossrefs

Cf. A140141, A039649, A000010, A000040, A058340.

Extensions

Corrected and extended by Ray Chandler, May 20 2008

A138786 "Left" odd composite numbers n for which n < A140607((n-1)/2).

Original entry on oeis.org

9, 21, 25, 27, 35, 45, 49, 55, 69, 75, 77, 81, 93, 95, 99, 105, 115, 119, 121, 125, 133, 135, 141, 143, 147, 153, 155, 161, 165, 169, 175, 187, 189, 203, 207, 209, 213, 215, 217, 219, 221, 225, 231, 235, 237, 243, 245, 247, 253, 259, 261, 267, 279, 285, 287
Offset: 1

Views

Author

Vladimir Shevelev, May 18 2008

Keywords

Comments

There are odd composite numbers which are neither in this sequence nor in A140608. The first such number is 91, see A140667.

Links

Crossrefs

Cf. A140607, A039649, A140140, A138193, A138217, A138227, A138535, A138536, A138537, A138539, A140141, A000010, A000040, A140608, A140667.

Extensions

Extended by Ray Chandler, May 20 2008

A140607 (A039649(2n+1)+A137576(n))/2.

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 13, 17, 19, 22, 23, 31, 37, 29, 31, 31, 43, 37, 37, 41, 43, 55, 47, 64, 45, 53, 61, 55, 59, 61, 55, 61, 67, 78, 71, 73, 91, 106, 79, 136, 83, 77, 85, 89, 91, 96, 109, 97, 136, 101, 103, 109, 107, 109, 109, 113, 155, 103, 145, 166, 111, 201, 127, 113
Offset: 1

Views

Author

Vladimir Shevelev, May 18 2008

Keywords

Comments

If 2n+1 is a prime then a(n) = 2n+1.

Links

Crossrefs

Cf. A039649, A137576, A000010, A000040, A140140, A138193, A138217, A138227, A138535, A138536, A138537, A138539, A140141, A140608, A138786, A140667.

Extensions

Extended by Ray Chandler, May 20 2008, May 24 2008

A140608 "Right" odd composite numbers n for which n > A140607((n-1)/2).

Original entry on oeis.org

15, 33, 39, 51, 57, 63, 65, 85, 87, 111, 117, 123, 129, 145, 159, 171, 177, 183, 185, 195, 201, 205, 249, 255, 265, 273, 275, 291, 303, 305, 315, 321, 327, 333, 339, 341, 393, 399, 411, 417, 435, 447, 451, 455, 465, 471, 481, 485, 489, 505, 511, 513, 519, 537
Offset: 1

Views

Author

Vladimir Shevelev, May 18 2008

Keywords

Comments

Conjecture. The sequence is infinite.

Links

Crossrefs

Cf. A140607, A039649, A140140, A138193, A138217, A138227, A138535, A138536, A138537, A138539, A140141, A000010, A000040, A138786, A140667.

Extensions

Extended by Ray Chandler, May 20 2008

A140667 Odd composite numbers k for which k = A140607((k-1)/2).

Original entry on oeis.org

91, 1581, 2465, 8481, 25761, 31609, 33355, 34945, 118405, 146611, 319507, 736291, 994507, 3270403, 3375487, 5176153, 6186403, 6228685, 8650951, 10679131, 22028203, 26017291, 31470211, 33796531, 41710411, 42149971, 42474547, 46672291, 48316969, 49019851, 58986091, 68182003, 69885649
Offset: 1

Views

Author

Ray Chandler, May 20 2008

Keywords

Crossrefs

Cf. A140607, A039649, A140140, A138193, A138217, A138227, A138535, A138536, A138537, A138539, A140141, A000010, A000040, A140608, A138786.

Programs

  • PARI
    f(n) = (eulerphi(2*n+1) + 1 + g(n))/2; \\ A140607
g(n) = sumdiv(2*n+1, d, eulerphi(d)/(t=znorder(Mod(2, d))))*t-t+1; \\ A137576
isok(c) = if (!isprime(c) && (c%2), f((c-1)/2) == c); \\ Michel Marcus, Jan 31 2023

Extensions

More terms from Michel Marcus, Jan 31 2023

A280450 Smallest k > 2n+1 such that phi(k) = phi(2n+1).

Original entry on oeis.org

4, 8, 9, 14, 22, 21, 16, 32, 27, 26, 46, 33, 38, 58, 62, 44, 39, 57, 45, 55, 49, 52, 94, 86, 64, 106, 75, 63, 118, 77, 74, 104, 134, 92, 142, 91, 82, 93, 158, 162, 166, 128, 116, 115, 95, 99, 111, 119, 122, 125, 206, 112, 214, 133, 117, 145, 178, 135, 153, 242
Offset: 1

Views

Author

Thomas Ordowski, Jan 03 2017

Keywords

Comments

All terms are composite.

Links

Crossrefs

Cf. A000010, A001274, A005384, A140141.

Programs

  • Maple
    f:= n -> min(select(`>`,numtheory:-invphi(numtheory:-phi(2*n+1)),2*n+1)):
map(f, [$1..100]); # Robert Israel, Jan 03 2017
  • Mathematica
    Table[k = 2 n + 2; While[EulerPhi@ k != #, k++] &@ EulerPhi[2 n + 1]; k, {n, 120}] (* Michael De Vlieger, Jan 03 2017 *)
  • PARI
    a(n) = my(k=2*n+2); while(eulerphi(k)!=eulerphi(2*n+1), k++); k \\ Felix Fröhlich, Jan 05 2017

Formula

2n+1 < a(n) < 4n+3.
From Robert Israel, Jan 03 2017: (Start)
a(n)=2n+2 if and only if 2n+1 is in A001274.
If n > 3 is in A005384, then a(n)=4n+2. (End)
Showing 1-7 of 7 results.

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