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.

User: Gerasimov Sergey

Gerasimov Sergey's wiki page.

Gerasimov Sergey has authored 69 sequences. Here are the ten most recent ones:

A241808 Numbers k such that (2*k)^3 - 3 is prime.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 13, 17, 19, 20, 37, 40, 53, 55, 58, 62, 68, 79, 89, 92, 95, 103, 112, 115, 119, 128, 137, 140, 158, 160, 169, 170, 193, 205, 214, 223, 229, 232, 235, 242, 248, 265, 272, 275, 278, 295, 313, 317, 322, 323, 337, 355, 359, 364
Offset: 1

Views

Author

Gerasimov Sergey, Apr 29 2014

Keywords

Links

Crossrefs

Cf. A006254, A153974.

Programs

Formula

a(n) = A153974(n)/2. - R. J. Mathar, May 14 2014

A241527 a(n) = n^3 + (3^n+1)/2.

Original entry on oeis.org

1, 3, 13, 41, 105, 247, 581, 1437, 3793, 10571, 30525, 89905, 267449, 799359, 2394229, 7177829, 21527457, 64574995, 193716077, 581137593, 1743400201, 5230185863, 15690540453, 47071601581, 141214782065, 423644320347, 1270932931741, 3812798762177, 11438396249433, 34315188706831
Offset: 0

Views

Author

Gerasimov Sergey, Apr 24 2014

Keywords

Links

Crossrefs

Cf. A000578, A007051.

Programs

  • Magma
    [n^3 + (3^n+1)/2: n in [0..29]]; // Juri-Stepan Gerasimov, Apr 25 2014
  • Mathematica
    Table[n^3+(3^n+1)/2,{n,0,40}] (* or *) LinearRecurrence[{7,-18,22,-13,3},{1,3,13,41,105},40] (* Harvey P. Dale, Mar 01 2024 *)

Formula

a(n) = A000578(n) + A007051(n).
G.f.: (x^4+18*x^3-10*x^2+4*x-1) / ((x-1)^4*(3*x-1)). - Colin Barker, Apr 25 2014

A240135 a(n) = composite(n)*2^(n - 3).

Original entry on oeis.org

1, 3, 8, 18, 40, 96, 224, 480, 1024, 2304, 5120, 10752, 22528, 49152, 102400, 212992, 442368, 917504, 1966080, 4194304, 8650752, 17825792, 36700160, 75497472, 159383552, 327155712, 671088640, 1409286144, 2952790016, 6039797760, 12348030976, 25769803776
Offset: 1

Views

Author

Gerasimov Sergey, Apr 02 2014

Keywords

Examples

			a(1)=1 because composite(1) * 2^(1-3) = 4 * 2^(-2) = 1;
a(2)=3 because composite(2) * 2^(2-3) = 6 * 2^(-1) = 3;
a(3)=8 because composite(3) * 2^(3-3) = 8 * 2^0 = 8.

Crossrefs

Cf. A002808.

Programs

  • Mathematica
    #[[1]]*2^(#[[2]]-3)&/@Module[{cs=Select[Range[50],CompositeQ]},Thread[ {cs,Range[Length[cs]]}]] (* Harvey P. Dale, Oct 08 2018 *)

A239885 a(n) = 2^(n-2) * prime(n).

Original entry on oeis.org

1, 3, 10, 28, 88, 208, 544, 1216, 2944, 7424, 15872, 37888, 83968, 176128, 385024, 868352, 1933312, 3997696, 8781824, 18612224, 38273024, 82837504, 174063616, 373293056, 813694976, 1694498816, 3456106496, 7180648448, 14629732352, 30333206528
Offset: 1

Views

Author

Gerasimov Sergey, Mar 29 2014

Keywords

Links

Crossrefs

Cf. A000040, A000079, A110295.

Programs

  • Magma
    [2^(n-2)*NthPrime(n): n in [1..50]]; // G. C. Greubel, Jan 04 2023
  • Mathematica
    Table[Prime[n]2^(n-2),{n,30}] (* Harvey P. Dale, Aug 19 2019 *)
  • PARI
    vector(50, n, prime(n)*2^(n-2)) \\ Colin Barker, Mar 29 2014
  • SageMath
    [2^(n-2)*nth_prime(n) for n in range(1,51)] # G. C. Greubel, Jan 04 2023

Formula

a(n) = A110295(n)/2.

A237251 Primes p such that p*2^(p-1)-1 is prime.

Original entry on oeis.org

2, 3, 5, 17, 257, 16487
Offset: 1

Views

Author

Gerasimov Sergey, Feb 05 2014

Keywords

Comments

The fifth Fermat prime, 65537, is not in the sequence: 65537*2^65536-1 is composite (per PFGW). - Michael B. Porter, Feb 11 2014
Also 65537*2^65536-1 is divisible by 16267 and 2058772459. - Jeppe Stig Nielsen, Jan 04 2020

Crossrefs

Cf. A019434, A092506, A230769, A236752.

Programs

  • PARI
    isok(p) = isprime(p) && isprime(p*2^(p-1) - 1); \\ Michel Marcus, Feb 06 2014

Extensions

a(5) from Ralf Stephan, Feb 03 2014
a(6) = A230769(26)+1 appended by Jeppe Stig Nielsen, Jan 04 2020

A236752 Primes of the form k*2^(k-1) - 1.

Original entry on oeis.org

3, 11, 31, 79, 191, 5119, 245759, 524287, 1114111, 3758096383, 1618481116086271, 653980173926178609468673073657929531391, 5359447279004780799548150067050349330431
Offset: 1

Views

Author

Gerasimov Sergey, Jan 30 2014

Keywords

Comments

Primes in A087323.
Corresponding values of k: 2, 3, 4, 5, 6, 10, 15, 16, 17, 28, 46, 123, ...
The values of k-1 are listed in A230769. - Jeppe Stig Nielsen, Oct 16 2019

Examples

			79 is in this sequence because it is prime and for k = 5, k*2^(k-1) - 1 = 5*2^(5-1) - 1 = 79.

Crossrefs

Cf. A002054, A003261, A196273, A230769.

Extensions

More terms and corrections of terms and comments by Ralf Stephan, Feb 03 2014

A234037 The union of odious numbers with evil squares and evil numbers with odious squares.

Original entry on oeis.org

5, 9, 10, 13, 17, 18, 20, 21, 23, 26, 29, 33, 34, 36, 37, 39, 40, 42, 43, 46, 47, 51, 52, 58, 61, 65, 66, 68, 69, 71, 72, 73, 74, 75, 77, 78, 80, 81, 84, 85, 86, 89, 92, 93, 94, 95, 101, 102, 104, 107, 109, 113, 115, 116, 122, 125, 129, 130, 132, 133, 135, 136, 137
Offset: 1

Views

Author

Gerasimov Sergey, Jan 13 2014

Keywords

Comments

Numbers n with odd sum of binary weight of n and binary weight of n^2.
Primes are in this sequence: 5, 13, 17, 23, 29, 37, 43, 47, 61, 71, 73,....
Evil numbers with odious squares: 5, 9, 10, 17, 18, 20, 23, 29, 33, 34,...
Odious numbers with evil squares: 13, 21, 26, 37, 42, 47, 52, 61, 69,...

Examples

			a(3) = 10 because 2 (binary weight of 10) + 3 (binary weight of 100) = 5 (odd).
a(4) = 13 because 3 (binary weight of 13) + 4 (binary weight of 169) = 7 (odd).

Links

Crossrefs

Cf. A000069, A000120, A001969, A159918, A231431, A235001.

Programs

  • Mathematica
    Select[Range[100], Mod[DigitCount[#, 2, 1], 2] != Mod[DigitCount[#^2, 2, 1], 2] &] (* Amiram Eldar, Aug 31 2020 *)

Extensions

Wrong term removed by Amiram Eldar, Aug 31 2020

A235331 Numbers with odious squares.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 9, 10, 11, 14, 16, 17, 18, 19, 20, 22, 23, 25, 28, 29, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 43, 44, 46, 49, 50, 51, 55, 56, 58, 59, 62, 64, 65, 66, 67, 68, 70, 71, 72, 75, 76, 77, 78, 79, 80, 82, 85, 86, 87, 88, 89, 91, 92, 95, 97, 98, 100
Offset: 1

Views

Author

Gerasimov Sergey, Jan 06 2014

Keywords

Comments

Numbers that are sqrt(odious squares).

Examples

			1 = sqrt(A235001(1)) = sqrt(1).
2 = sqrt(A235001(2)) = sqrt(4).
4 = sqrt(A235001(3)) = sqrt(16).
5 = sqrt(A235001(4)) = sqrt(25).

Links

Crossrefs

Cf. A235001, A194991.

Programs

  • Magma
    [n : n in [0..130] |IsOdd(&+Intseq(n^2, 2))]; // Vincenzo Librandi, Jan 31 2018
  • Mathematica
    Select[Range[200], OddQ[DigitCount[#^2, 2][[1]]] &] (* Vincenzo Librandi, Jan 31 2018 *)
  • PARI
    isok(n) = (hammingweight(n^2) % 2) == 1; \\ Michel Marcus, Jan 31 2018

Formula

a(n) = sqrt(A235001(n)).

A235001 Odious squares.

Original entry on oeis.org

1, 4, 16, 25, 49, 64, 81, 100, 121, 196, 256, 289, 324, 361, 400, 484, 529, 625, 784, 841, 961, 1024, 1089, 1156, 1225, 1296, 1444, 1521, 1600, 1681, 1849, 1936, 2116, 2401, 2500, 2601, 3025, 3136, 3364, 3481, 3844, 4096, 4225, 4356, 4489, 4624, 4900, 5041, 5184, 5625
Offset: 1

Views

Author

Gerasimov Sergey, Jan 02 2014

Keywords

Comments

This sequence is the intersection of A000069 and A000290. Numbers n^2 such that A159918 is odd.

Examples

			16 is a square 4^2 and 16 in base 2 is a 10000, having an odd number of 1's, thus 16 is in this sequence.

Links

Crossrefs

Cf. A231431 (evil squares).

Programs

  • Mathematica
    Select[Range[200]^2,OddQ[DigitCount[#,2,1]]&] (* Harvey P. Dale, Jan 14 2014 *)

A234648 Even sums of 2 consecutive odious numbers (A000069).

Original entry on oeis.org

6, 24, 30, 40, 54, 72, 86, 96, 102, 120, 126, 136, 150, 160, 166, 184, 198, 216, 222, 232, 246, 264, 278, 288, 294, 312, 326, 344, 350, 360, 374, 384, 390, 408, 414, 424, 438, 456, 470, 480, 486, 504, 510, 520, 534, 544, 550, 568, 582, 600, 606, 616, 630
Offset: 1

Views

Author

Gerasimov Sergey, Dec 29 2013

Keywords

Comments

All the terms in this sequence are evil numbers (A001969).

Links

Crossrefs

Intersection of A005843 and A234011.
Cf. A000069, A001969, A036554, A131323, A225822.

Programs

  • Mathematica
    odQ[n_] := OddQ @ DigitCount[n, 2, 1]; Select[Plus @@@ Partition[Select[ Range[320], odQ], 2, 1], EvenQ] (* Amiram Eldar, Aug 31 2020 *)

Formula

a(n) = A234011(A036554(n)) = A225822(n) + (-1)^n.

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

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