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.

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A165496 Values k: A165495(k) is odd.

Original entry on oeis.org

1, 3, 6, 7, 10, 11, 13, 16, 24, 30, 32, 40, 55, 61, 62, 91, 115, 129, 139, 190, 218, 325, 330, 344, 359, 412, 499, 709, 762, 779, 791, 1392, 2230, 2440, 2947, 3355, 6008, 6124, 6718, 6899, 7563, 7872, 8070, 12529, 15204, 17582, 19313, 20706, 28825
Offset: 1

Views

Author

Hugo van der Sanden, Sep 21 2009

Keywords

Comments

If A064491 reaches 2(2k-1)^2, it will then start a run of odd numbers if k is in this sequence.

Crossrefs

Cf. A064491, A165495

A165930 a(1) = 1, for n > 1: a(n) = tau(sum of the previous terms) where tau(k) = number of the divisors of k.

Original entry on oeis.org

1, 1, 2, 3, 2, 3, 6, 6, 8, 6, 4, 8, 6, 8, 7, 2, 2, 6, 5, 4, 12, 8, 8, 4, 4, 12, 8, 4, 12, 10, 6, 4, 8, 8, 12, 16, 4, 8, 8, 8, 4, 8, 8, 4, 4, 8, 8, 4, 4, 12, 8, 4, 16, 4, 12, 4, 8, 8, 4, 4, 16, 8, 12, 8, 8, 8, 18, 18, 12, 8, 8, 4, 8, 4, 8, 4, 4, 16, 4, 4, 16, 4, 8, 8, 8, 4, 8, 4, 24, 8, 4, 12, 8, 8, 4, 4, 16
Offset: 1

Views

Author

Jaroslav Krizek, Sep 30 2009

Keywords

Comments

a(1) = 1, for n > 1: a(n) = tau(sum_(i=1...n-1) a(i)) = A000005(sum_(i=1...n-1) a(i)). a(n) = inverse of partial sums of A064491(n), i.e. a(1) = A064491(1), for n > 1: a(n) = A064491(n) - A064491(n-1), i.e. first differences of sequence A064491.

Examples

			a(4) = tau(a(1)+a(2)+a(3)) = tau(4) = 3.

Links

Programs

  • Haskell
    a165930 n = a165930_list !! (n-1)
a165930_list = 1 : zipWith (-) (tail a064491_list) a064491_list
-- Reinhard Zumkeller, Mar 29 2014
  • PARI
    print1(1);s=1;for(n=2,1e4,a=numdiv(s);print1(","a);s+=a)

Extensions

Program and editing by Charles R Greathouse IV, Nov 02 2009

A221181 Records in A096335 (values).

Original entry on oeis.org

0, 2, 3, 8, 13, 26, 39, 41, 47, 82, 85, 376, 402, 1190, 1193, 1261, 1353, 1425, 1426, 2217, 3840, 3842, 3846, 4129, 4131, 5922, 6250, 6252, 12203, 16807, 17081, 18257, 18258, 19039, 19043, 19070, 21185, 21187, 21397, 31602, 36888, 36889, 40422, 67383, 67384
Offset: 1

Views

Author

T. D. Noe and N. J. A. Sloane, Jan 15 2013

Keywords

Comments

At present it is not known whether this sequence is finite or infinite.

Links

Crossrefs

Cf. A064491, A096335, A221182.

Programs

  • C
    See Links section.

Extensions

More terms from Rémy Sigrist, Nov 06 2020

A221182 Records in A096335 (positions).

Original entry on oeis.org

1, 3, 6, 11, 27, 60, 83, 224, 227, 425, 815, 1641, 8267, 19305, 19317, 55167, 105600, 105611, 105617, 137637, 180621, 180627, 180643, 494175, 494197, 680579, 826245, 826279, 896775, 2246985, 2893319, 6754759, 6754763, 7144847, 7144881, 8208147, 8969895
Offset: 1

Views

Author

T. D. Noe and N. J. A. Sloane, Jan 15 2013

Keywords

Comments

At present it is not known whether this sequence is finite or infinite.

Links

Crossrefs

Cf. A064491, A096335, A221181.

Programs

  • C
    See Links section.

Extensions

More terms from Rémy Sigrist, Nov 06 2020

A085865 a(1) = 2, a(n+1) = a(n)*{sigma(a(n))}, where sigma(n) is the sum of the divisors function.

Original entry on oeis.org

2, 6, 72, 14040, 707616000, 2299654269739008000, 25353370781704805143366427867873280000, 3361021040447648920437074194752848938805829494939344230020451929790873600000
Offset: 1

Views

Author

Amarnath Murthy, Jul 06 2003

Keywords

Comments

a(1) = 1, a(n+1) = a(n) + sigma(a(n)) gives A081973.

Examples

			a(3) = 72 and sigma(72)= 195, hence a(4) = 72*195 = 14040.

Crossrefs

Cf. A064491, A081973, A085864.

Programs

  • PARI
    for(n=1,9,if(n==1,a=2,a*=sigma(a)); print(a); )

Extensions

More terms from Ray Chandler, Jul 16 2003

A139422 a(1)=a(2)=1. For n >= 3, a(n) = a(n-1) + d(a(n-1)) + d(a(n-2)), where d(m) is the number of positive divisors of m.

Original entry on oeis.org

1, 1, 3, 6, 12, 22, 32, 42, 56, 72, 92, 110, 124, 138, 152, 168, 192, 222, 244, 258, 272, 290, 308, 328, 348, 368, 390, 416, 444, 468, 498, 524, 538, 548, 558, 576, 609, 638, 654, 670, 686, 702, 726, 754, 774, 794, 810, 834, 862, 874
Offset: 1

Views

Author

Leroy Quet, Apr 21 2008

Keywords

Crossrefs

Cf. A139423, A064491.

Programs

  • Maple
    with(numtheory): a[1]:=1: a[2]:=1: for n from 3 to 50 do a[n]:=a[n-1]+tau(a[n-1])+tau(a[n-2]) end do: seq(a[n],n=1..50); # Emeric Deutsch, Apr 26 2008
  • Mathematica
    Join[{a = 1, b = 1}, Table[c = b + Total[DivisorSigma[0, {a, b}]]; a = b; b = c; c, {n, 48}]] (* Jayanta Basu, Jun 06 2013 *)

Extensions

More terms from Emeric Deutsch, Apr 26 2008

A139423 a(1)=1, a(2)=2. For n >= 3, a(n) = a(n-1) + d(a(n-1)) + d(a(n-2)), where d(m) is the number of positive divisors of m.

Original entry on oeis.org

1, 2, 5, 9, 14, 21, 29, 35, 41, 47, 51, 57, 65, 73, 79, 83, 87, 93, 101, 107, 111, 117, 127, 135, 145, 157, 163, 167, 171, 179, 187, 193, 199, 203, 209, 217, 225, 238, 255, 271, 281, 285, 295, 307, 313, 317, 321, 327, 335, 343
Offset: 1

Views

Author

Leroy Quet, Apr 21 2008

Keywords

Crossrefs

Cf. A139422, A064491.

Programs

  • Maple
    with(numtheory): a[1]:=1: a[2]:=2: for n from 3 to 50 do a[n]:=a[n-1]+tau(a[n-1])+tau(a[n-2]) end do: seq(a[n],n=1..50); # Emeric Deutsch, Apr 30 2008
  • Mathematica
    Join[{a = 1, b = 2}, Table[c = b + Total[DivisorSigma[0, {a, b}]]; a = b; b = c; c, {n, 48}]] (* Jayanta Basu, Jun 06 2013 *)

Extensions

More terms from Emeric Deutsch, Apr 30 2008

A140481 a(1) = 1; for n >= 1, a(n+1) is obtained by adding to a(n) the a(n)-th smallest number not dividing a(n).

Original entry on oeis.org

1, 3, 8, 20, 46, 96, 204, 420, 864, 1752, 3520, 7068, 14160, 28360, 56736, 113508, 227040, 454176, 908424, 1816944, 3633908, 7267828, 14535662, 29071328, 58142704, 116285418, 232570884, 465141864, 930283760, 1860567600
Offset: 1

Views

Author

Eric Angelini, Jun 25 2008

Keywords

Examples

			The smallest number not dividing 1 is 2, so a(2) = 1+2 = 3.
The numbers not dividing 3 are 2, 4, 5, 6, ..., so a(3) = 3+5 = 8.
The numbers not dividing 8 are 3, 5, 6, 7, 9, 10, 11, 12, ..., so a(4) = 8+12 = 20.

Crossrefs

Cf. A000005, A064491, A140482.

Formula

a(n+1) = 2*a(n) + tau(a(n)) (cf. A000005). - Hans Havermann and Franklin T. Adams-Watters, Jun 25 2008. Using a(n+1) = a(n) + tau(a(n)) would give A064491.

Extensions

More terms from Hans Havermann, Jun 25 2008

A140484 Square roots of terms of A140483.

Original entry on oeis.org

1, 2, 3, 8, 9, 28, 39, 172, 185, 364, 369, 2236, 2253, 3936, 3943, 17320, 17323, 43002, 43005
Offset: 1

Views

Author

Franklin T. Adams-Watters, Jun 25 2008

Keywords

Crossrefs

Cf. A064491, A140483.

Extensions

a(16)-a(19) from Donovan Johnson, Oct 06 2009

A377539 The number of iterations of the map x -> x + A000005(x), starting from n, until reaching an even number, and always at least one iteration taken.

Original entry on oeis.org

1, 1, 4, 3, 3, 1, 2, 1, 1, 1, 6, 1, 5, 1, 4, 3, 4, 1, 3, 1, 2, 1, 2, 1, 1, 1, 16, 1, 16, 1, 15, 1, 14, 1, 13, 11, 13, 1, 12, 1, 12, 1, 11, 1, 10, 1, 2, 1, 1, 1, 9, 1, 9, 1, 8, 1, 7, 1, 7, 1, 6, 1, 5, 5, 5, 1, 5, 1, 4, 1, 4, 1, 3, 1, 2, 1, 2, 1, 2, 1, 1, 1, 38, 1, 37, 1, 36, 1, 36, 1, 35, 1, 35
Offset: 1

Views

Author

Ctibor O. Zizka, Oct 31 2024

Keywords

Comments

The iteration step is x -> A062249(x).
a(n) = 1 if and only if n is an odd square (A016754) or an even nonsquare (A157502). - Robert Israel, Oct 31 2024
Therefore, a(n) = 1 <=> A323158(n) = 0. - Antti Karttunen, Jan 15 2025

Examples

			For n = 2, there is a(2) = 1 iteration to an even number: 2 -> 4 (with at least one iteration so 2 itself is not the even number target).
For n = 3 there are a(3) = 4 iterations to reach an even number: 3 -> 5 -> 7 -> 9 -> 12.

Links

Crossrefs

Cf. A000005, A062249 (step), A064491 (trajectory of 1), A016754, A157502, A323158.

Programs

  • Maple
    f:= proc(n) local x,i;
  x:= n;
  for i from 1 do x:= x + numtheory:-tau(x); if x::even then return i fi od
end proc:
map(f, [$1..200]); # Robert Israel, Oct 31 2024
  • Mathematica
    a[n_] := -1 + Length@ NestWhileList[# + DivisorSigma[0, #] &, n, OddQ, {2, 1}]; Array[a, 100] (* Amiram Eldar, Oct 31 2024 *)
  • PARI
    A377539(n) = for(i=1,oo,if(!((n=(n+numdiv(n)))%2),return(i))); \\ Antti Karttunen, Jan 15 2025
Previous Showing 11-20 of 21 results. Next

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

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