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: Reinhard Zumkeller

Reinhard Zumkeller's wiki page.

Reinhard Zumkeller has authored 5209 sequences. Here are the ten most recent ones:

A358890 a(n) is the first term of the first maximal run of n consecutive numbers with increasing greatest prime factors.

Original entry on oeis.org

14, 4, 1, 8, 90, 168, 9352, 46189, 2515371, 721970, 6449639, 565062156, 11336460025, 37151747513, 256994754033, 14037913234203
Offset: 1

Views

Author

Reinhard Zumkeller, Jan 10 2003

Keywords

Comments

a(16) > 10^13. - Giovanni Resta, Jul 25 2013
The convention gpf(1) = A006530(1) = 1 is used (otherwise we would have a(2) = 2 and a(3) = 24). - Pontus von Brömssen, Dec 05 2022
a(17) > 10^14. - Martin Ehrenstein, Dec 10 2022

Examples

			a(7) = 9352 because the first sequence of seven consecutive numbers with increasing greatest prime factors is 9352=167*7*2^3, 9353=199*47, 9354=1559*3*2, 9355=1871*5, 9356=2339*2^2, 9357=3119*3, and 9358=4679*2. [Corrected by _Jon E. Schoenfield_, Sep 21 2022]

Crossrefs

Cf. A006530, A070087, A079748, A079749 (erroneous version), A100384.

Programs

  • Maple
    V:= Vector(11): count:= 0:
a:= 1: m:= 1: w:= 1:
for k from 2 while count < 11 do
  v:= max(numtheory:-factorset(k));
  if v > m then m:= v
  else
    if V[k-a] = 0 then V[k-a]:= a; count:= count+1; fi;
    a:= k; m:= v;
  fi
od:
convert(V,list); # Robert Israel, Dec 05 2022
  • Python
    from sympy import factorint
def A358890(n):
    m = 1
    gpf1 = 1
    k = 1
    while 1:
        while 1:
            gpf2 = max(factorint(m+k))
            if gpf2 < gpf1: break
            gpf1 = gpf2
            k += 1
        if k == n: return m
        m += k
        gpf1 = gpf2
        k = 1 # Pontus von Brömssen, Dec 05 2022

Formula

A079748(a(n)) = n-1.
From Pontus von Brömssen, Dec 05 2022: (Start)
A079748(a(n)-1) = 0 for n != 3.
For n != 3, a(n) = A070087(m)+1, where m is the smallest positive integer such that A070087(m+1) - A070087(m) = n.
(End)

Extensions

More terms from Don Reble, Jan 17 2003
Corrected by Jud McCranie, Feb 11 2003
a(14)-a(15) from Giovanni Resta, Jul 25 2013
Name edited, a(1) and a(2) corrected by Pontus von Brömssen, Dec 05 2022
a(16) from Martin Ehrenstein, Dec 07 2022

A316322 Sum of piles of first n primes: a(n) = Sum(prime(i)*(2*i-1): 1<=i<=n).

Original entry on oeis.org

2, 11, 36, 85, 184, 327, 548, 833, 1224, 1775, 2426, 3277, 4302, 5463, 6826, 8469, 10416, 12551, 15030, 17799, 20792, 24189, 27924, 32107, 36860, 42011, 47470, 53355, 59568, 66235, 73982, 82235, 91140, 100453, 110734, 121455, 132916, 145141, 158000, 171667, 186166, 201189, 217424, 234215, 251748
Offset: 1

Views

Author

N. J. A. Sloane, Jul 03 2018, based on Reinhard Zumkeller's A083215

Keywords

Examples

			............................................ 7
........................... 5 ............ 7 5 7
............ 3 .......... 5 3 5 ........ 7 5 3 5 7
2 ........ 3 2 3 ...... 5 3 2 3 5 .... 7 5 3 2 3 5 7
a(1)=2 ... a(2)=11 .... a(3)=36 ...... a(4)=85.

Links

Crossrefs

Cf. A083215, A014285, A007504, A167214, A023662, A062020.

Programs

  • Maple
    seq(add((2*i-1)*ithprime(i),i=1..n), n=1..80); # Ridouane Oudra, Feb 19 2025
  • Mathematica
    nxt[{n_, a_}] := {n + 1, a + Prime[n + 1] (2 n + 1)}; NestList[nxt,{1,2},50][[All,2]] (* Harvey P. Dale, Jul 05 2018 *)
  • PARI
    a(n) = sum(i=1, n, prime(i)*(2*i-1)); \\ Michel Marcus, Jan 22 2022

Formula

From Ridouane Oudra, Feb 19 2025: (Start)
a(n) = Sum_{i=1..n} Sum_{j=1..n} max(prime(i), prime(j)).
a(n) = 2*A014285(n) - A007504(n).
a(n) = 2*A167214(n) - A023662(n).
a(n) = A167214(n) + A062020(n). (End)

A270571 Numbers with at least one arithmetic progression of four consecutive divisors.

Original entry on oeis.org

12, 24, 36, 48, 60, 72, 84, 96, 105, 108, 120, 132, 144, 156, 168, 180, 192, 204, 216, 228, 240, 252, 264, 276, 288, 300, 312, 315, 324, 336, 348, 360, 372, 384, 396, 408, 420, 432, 444, 456, 468, 480, 492, 504, 516, 525, 528, 540, 552, 564, 576, 588, 600
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 19 2016

Keywords

Comments

Contrast A094529 where the divisors in arithmetic progression do not have to be consecutive.

Examples

			348 is included because its divisors are 1, 2, 3, 4, 6, 12, 29, 58, 87, 116, 174, and 348, and the first four are in arithmetic progression.

Links

Crossrefs

Cf. A091011, A094529, A094530.

Programs

  • Mathematica
    ap4dQ[n_]:=Count[Partition[Divisors[n],4,1],_?(Length[ Union[ Differences[ #]]] == 1&)]>0; Select[ Range[700],ap4dQ]

Extensions

Edited by Harvey P. Dale and Alois P. Heinz, Mar 19 2016

A263110 n is the a(n)-th positive integer having its digitsum in base-16 representation.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 2, 1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4, 3, 2, 1, 6, 6, 6, 6, 6, 6, 6
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 09 2015

Keywords

Comments

Ordinal transform of A053836. - Alois P. Heinz, Dec 23 2018

Links

Crossrefs

Cf. A053836, A254524, A263017, A263109.

Programs

  • Haskell
    import Data.IntMap (empty, findWithDefault, insert)
a263110 n = a263110_list !! (n-1)
a263110_list = f 1 empty where
   f x m = y : f (x + 1) (insert q (y + 1) m) where
           y = findWithDefault 1 q m; q = a053836 x

A263109 n is the a(n)-th positive integer having its digitsum in base-12 representation.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 2, 1, 5, 5, 5, 5, 5, 5, 5, 5, 4, 3, 2, 1, 6, 6, 6, 6, 6, 6, 6, 5, 4, 3, 2, 1, 7, 7, 7, 7, 7, 7, 6, 5, 4, 3, 2, 1, 8, 8, 8
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 09 2015

Keywords

Comments

Ordinal transform of A053832. - Alois P. Heinz, Dec 23 2018

Links

Crossrefs

Cf. A053832, A254524, A263017, A263110.

Programs

  • Haskell
    import Data.IntMap (empty, findWithDefault, insert)
a263109 n = a263109_list !! (n-1)
a263109_list = f 1 empty where
   f x m = y : f (x + 1) (insert q (y + 1) m) where
           y = findWithDefault 1 q m; q = a053832 x
  • Mathematica
    b[_] = 1;
a[n_] := a[n] = With[{t = Total[IntegerDigits[n, 12]]}, b[t]++];
Array[a, 100] (* Jean-François Alcover, Dec 18 2021 *)

A261795 First differences of A261793.

Original entry on oeis.org

2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 2, 2, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2
Offset: 1

Views

Author

Reinhard Zumkeller, Sep 01 2015

Keywords

Comments

a(n) = A261794(A261793(n)).

Links

Crossrefs

Cf. A261794, A261793, A261018, A261789.

Programs

  • Haskell
    a261795 n = a261795_list !! (n-1)
a261795_list = zipWith (-) (tail a261793_list') a261793_list'

A261794 a(n) is the smallest nonzero number that is not a substring of n in decimal representation.

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1
Offset: 0

Views

Author

Reinhard Zumkeller, Sep 01 2015

Keywords

Comments

A261795(n) = a(A261793(n)).

Links

Crossrefs

Cf. A007089, A031298, A261795, A261793, A261461, A261787.

Programs

  • Haskell
    import Data.List (isInfixOf)
a261794 x = f $ tail a031298_tabf where
   f (cs:css) = if isInfixOf cs (a031298_row x)
                   then f css else foldr (\d v -> 10 * v + d) 0 cs

Extensions

Name corrected by Álvar Ibeas, Sep 08 2020

A261793 Successively add the smallest number that is not a substring in decimal representation.

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 17, 19, 21, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81
Offset: 1

Views

Author

Reinhard Zumkeller, Sep 01 2015

Keywords

Links

Crossrefs

Cf. A031298, A261794, A261795 (first differences), A261806, A260273, A261786

Programs

  • Haskell
    a261793 n = a261793_list !! (n-1)
a261793_list = iterate (\x -> x + a261794 x) 1

A261789 First differences of A261786.

Original entry on oeis.org

2, 2, 3, 1, 2, 4, 3, 1, 3, 3, 3, 2, 2, 4, 2, 5, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 1, 4, 3, 2, 2, 4, 2, 4, 4, 4, 4, 2, 5, 5, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3, 1, 4, 3, 4, 4, 5, 3, 3, 3, 3, 3, 1, 4, 3, 4, 3, 3, 1, 4, 3, 2, 2, 4, 2, 4, 4
Offset: 1

Views

Author

Reinhard Zumkeller, Sep 01 2015

Keywords

Comments

a(n) = A261787(A261786(n)).

Links

Crossrefs

Cf. A261645, A261461, A261786, A261018, A261795.

Programs

  • Haskell
    a261789 n = a261789_list !! (n-1)
a261789_list = zipWith (-) (tail a261786_list') a261786_list'

A261788 a(n) is the smallest k such that A261786(k) >= 3^n.

Original entry on oeis.org

1, 2, 5, 12, 30, 81, 224, 626, 1747, 4909, 13811, 38934, 109889, 310666, 880125, 2500221, 7125406, 20376598, 58472481, 168349612, 486198698, 1408140693, 4088769215, 11899761717, 34703682407
Offset: 0

Views

Author

Reinhard Zumkeller, Sep 01 2015

Keywords

Crossrefs

Cf. A000244, A261396, A261786, A261806.

Programs

  • Haskell
    a261788 n = a261788_list !! (n-1)
a261788_list = f 1 1 a261786_list' where
   f z k (x:xs) | x >= z    = k : f (3 * z) (k + 1) xs
                | otherwise = f z (k + 1) xs
  • PARI
    ts(n) = Str(fromdigits(digits(n, 3))); \\ A007089
f(n) = my(s=ts(n), k=1); while (#strsplit(s, ts(k)) != 1, k++); k; \\ A261787
lista(nn) = my(last=1, k=0, p=3^k); for (n=1, nn, if (last >= p, print1(n, ", "); k++; p = 3^k); last += f(last);); \\ Michel Marcus, Feb 06 2022

Extensions

a(11)-a(20) from Michel Marcus, Feb 06 2022
a(21)-a(24) from Jinyuan Wang, Dec 13 2024

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

Content is available under The OEIS End-User License Agreement.