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.

Previous Showing 31-40 of 52 results. Next

A162788 a(n) = A162528(n)/8.

Original entry on oeis.org

8, 9, 10, 11, 12, 13, 14, 16, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263
Offset: 1

Views

Author

Omar E. Pol, Jul 13 2009

Keywords

Links

Crossrefs

Cf. A161344, A162528, A162787, A162789.

A164005 Zero together with row 5 of the array in A163280.

Original entry on oeis.org

0, 7, 14, 21, 32, 45, 60, 77, 96, 117, 140, 165, 192, 221, 252, 285, 320, 357, 396, 437, 480, 525, 572, 621, 672, 725, 780, 837, 896, 957, 1020, 1085, 1152, 1221, 1292, 1365, 1440, 1517, 1596, 1677, 1760, 1845, 1932, 2021, 2112, 2205, 2300, 2397, 2496, 2597
Offset: 0

Views

Author

Omar E. Pol, Aug 08 2009

Keywords

Links

Crossrefs

Cf. A008578, A161344, A161345, A163280, A164000, A164004, A164006.
Cf. A028347.

Programs

  • Maple
    A033676 := proc(n) local a,d; a := 0 ; for d in numtheory[divisors](n) do if d^2 <= n then a := max(a,d) ; fi; od: a; end: A163280 := proc(n,k) local r,T ; r := 0 ; for T from k^2 by k do if A033676(T) = k then r := r+1 ; if r = n then RETURN(T) ; fi; fi; od: end: A164005 := proc(n) if n = 0 then 0; else A163280(5,n) ; fi; end: seq(A164005(n),n=0..80) ; # R. J. Mathar, Aug 09 2009
  • Mathematica
    Join[{0, 7, 14}, Table[n*(n + 4), {n, 3, 50}]] (* G. C. Greubel, Aug 28 2017 *)
  • PARI
    x='x+O('x^50); concat([0], Vec(x*(7 - 7*x + 4*x^3 - 2*x^4)/(1 - x)^3)) \\ G. C. Greubel, Aug 28 2017

Formula

Conjecture: a(n) = A100451(n+2). (See A163280.)
From G. C. Greubel, Aug 28 2017: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n >= 3.
a(n) = n*(n+4), n >= 3.
G.f.: x*(7 - 7*x + 4*x^3 - 2*x^4)/(1 - x)^3.
E.g.f.: x*(x+5)*exp(x) + 2*x + x^2. (End)

Extensions

Extended by R. J. Mathar, Aug 09 2009

A164012 Zero together with row 12 of the array in A163280.

Original entry on oeis.org

0, 31, 62, 87, 124, 115, 186, 147, 232, 207, 250, 242, 372, 312, 364, 405, 464, 476, 558, 570, 640, 693, 726, 782, 888, 925, 962, 1026, 1092, 1160, 1230, 1302, 1376, 1452, 1530, 1610, 1692, 1776, 1862, 1950, 2040, 2132, 2226, 2322, 2420, 2520, 2622, 2726
Offset: 0

Views

Author

Omar E. Pol, Aug 08 2009

Keywords

Links

Crossrefs

Cf. A008578, A161344, A161345, A163280, A164000, A164011.

Formula

Conjecture: a(n) = A119412(n), n > 36. [R. J. Mathar, Jul 31 2010]

Extensions

Terms beyond a(12) from R. J. Mathar, Jul 31 2010

A196226 m such that A054024(m) (sum of divisors of m reduced modulo m) is 3 + m/2.

Original entry on oeis.org

8, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514
Offset: 1

Views

Author

John W. Layman, Sep 29 2011

Keywords

Comments

This sequence appears to be identical to A073582 with its first term omitted and to A161344 with its first two terms omitted.
Conjectures. (1) If m>=14 is a term of this sequence, then sigma(2,m) is congruent to 5 + m/2 modulo m; (2) If m>=22 is a term of this sequence, then sigma(3,m) is congruent to 9 + m/2 modulo m; If m>=38 is a term of this sequence, then sigma(4,m) is congruent to 17 + m/2 modulo m. (sigma(k,m) denotes the sum of the k-th powers of the divisors of m.)
Similar conjectures can be made about sigma(k,m) congruent to 2^k+1 + m/2 modulo m, for m a sufficiently large term of this sequence..
The even semiprimes (A100484) m= 2*p with p>3, with sigma(2*p)= 3+p (mod 2p), are a subsequence. - R. J. Mathar, Oct 02 2011
The terms in this sequence which are not even semiprimes are 8, 690, 12978, 176946, ... - R. J. Mathar, Aug 24 2023

Crossrefs

Cf. A054024, A073582, A161344.

Programs

  • Maple
    isA196226 := proc(n)
    sigmar := modp(numtheory[sigma](n),n) ;
    if sigmar = 3+n/2 then
        true;
    else
        false;
    end if;
end proc:
A196226 := proc(n)
     option remember;
     if n =1 then
        8;
    else
        for a from procname(n-1)+1 do
            if isA196226(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A196226(n),n=1..100) ; # R. J. Mathar, Aug 24 2023
  • PARI
    lista(nn) = {for(n=1, nn, if ((sigma(n) % n) == (3 + n/2), print1(n, ", ")););} \\ Michel Marcus, Jul 12 2014

A232803 Odd primes, twice odd primes, 4, and 8.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62, 67, 71, 73, 74, 79, 82, 83, 86, 89, 94, 97, 101, 103, 106, 107, 109, 113, 118, 122, 127, 131, 134, 137, 139, 142, 146, 149, 151, 157, 158, 163, 166
Offset: 1

Views

Author

J. Lowell, Nov 30 2013

Keywords

Comments

Also numbers n for which all possible n X n magic squares are prime.
Note that there are no 2 X 2 magic squares.
All primes, except 2, belong to this sequence. This is because p X p magic squares, with p primes, cannot be derived from smaller magic squares. Otherwise p would be equal to a product of smaller integers. Also, since there are no 2 X 2 magic squares, we cannot have a (2p) X (2p) that could be derived from smaller magic squares. And also we cannot have an 8 X 8 derived magic square (see first example). So this sequence is A065091 (odd primes) U A100484 (even semiprimes) U {8}. And A100484 U {8} is also A161344 (see second comment by Zak Seidov there). So this sequence is: A065091 U A161344. - Michel Marcus, Dec 07 2013

Examples

			8 qualifies because a composite 8 X 8 magic square is impossible, such a square would require a 2 X 2 magic square, and there are none (see 2nd link).
9 is not part of sequence because a 9 X 9 magic square can be created by multiplying a 3 X 3 magic square by itself.
Similarly 12 is not part of sequence because a 12 X 12 magic square can be created by multiplying a 3 X 3 magic square and a 4 X 4 magic square (see 3rd and 4th links).

Links

Programs

  • PARI
    isok(n) = (isprime(n) && (n%2)) || (n==8) || (!(n%2) && isprime(n/2)); \\ Michel Marcus, Dec 07 2013

Extensions

More terms from Michel Marcus, Dec 07 2013
Replaced definition with a more explicit one, following the comments of Michel Marcus. - N. J. A. Sloane, Dec 19 2019

A161428 a(n) = A161424(n)/4.

Original entry on oeis.org

4, 5, 6, 7, 8, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277
Offset: 1

Views

Author

Omar E. Pol, Jun 20 2009

Keywords

Crossrefs

Cf. A018253, A160811, A160812, A161205, A161344, A161345, A161346, A161424, A161425.

Extensions

Terms beyond a(8) from R. J. Mathar, Jun 24 2009

A161827 Complement of A006446.

Original entry on oeis.org

5, 7, 10, 11, 13, 14, 17, 18, 19, 21, 22, 23, 26, 27, 28, 29, 31, 32, 33, 34, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 98
Offset: 1

Views

Author

Omar E. Pol, Jun 21 2009, Jun 28 2009, Feb 08 2010

Keywords

Comments

The asymptotic density of this sequence is 1 (Cooper and Kennedy, 1989). - Amiram Eldar, Jul 10 2020

Links

Crossrefs

Cf. A000005, A018253, A161205, A161344, A161345, A161424, A006446, A161828, A161835.

Extensions

More terms from N. J. A. Sloane, Feb 08 2010

A162190 Triangle read by rows in which row n lists the divisors of n, the n-th prime and the consecutive composites that are greater than the n-th prime, with a(0)=1.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 4, 1, 3, 5, 6, 1, 2, 4, 7, 8, 9, 10, 1, 5, 11, 12, 1, 2, 3, 6, 13, 14, 15, 16, 1, 7, 17, 18, 1, 2, 4, 8, 19, 20, 21, 22, 1, 3, 9, 23, 24, 25, 26, 27, 28, 1, 2, 5, 10, 29, 30, 1, 11, 31, 32, 33, 34, 35, 36, 1, 2, 3, 4, 6, 12, 37, 38, 39, 40
Offset: 0

Views

Author

Omar E. Pol, Jun 30 2009

Keywords

Examples

			Triangle begins:
1;
1,(2);
1,.2,(3),4;
1,....3,...(5),6;
1,.2,....4,......(7),8,.9,10;
1,..........5,..............(11),12;
1,.2,.3,.......6,..................(13),14,15,16;
1,................7,............................(17),18;
1,.2,....4,..........8,................................(19),20,21,22;

Links

Crossrefs

Cf. A000005, A000040, A000720, A027750, A018253, A160811, A160812, A161205, A161344, A161345, A161424, A006446, A161827, A161828, A161835.

A162789 a(n) = A162529(n)/9.

Original entry on oeis.org

9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 21, 23, 27, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233
Offset: 1

Views

Author

Omar E. Pol, Jul 13 2009

Keywords

Crossrefs

Cf. A161344, A162529, A162788, A162790.

A162790 a(n) = A162530(n)/10.

Original entry on oeis.org

10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 23, 25, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263
Offset: 1

Views

Author

Omar E. Pol, Jul 13 2009

Keywords

Comments

This sequence consists of all primes > 10, together with the composites 10, 12, 14, 15, 16, 20, and 25. - Franklin T. Adams-Watters, May 14 2010

Crossrefs

Cf. A161344, A162530, A162789, A162903.

Programs

Extensions

Edited by Omar E. Pol, Jul 18 2009
Edited by Omar E. Pol, Jul 19 2009
More terms from Franklin T. Adams-Watters, May 14 2010
Previous Showing 31-40 of 52 results. Next

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