A080982 -id:A080982 - cp's OEIS frontend

A011772 Smallest number m such that m(m+1)/2 is divisible by n.

1, 3, 2, 7, 4, 3, 6, 15, 8, 4, 10, 8, 12, 7, 5, 31, 16, 8, 18, 15, 6, 11, 22, 15, 24, 12, 26, 7, 28, 15, 30, 63, 11, 16, 14, 8, 36, 19, 12, 15, 40, 20, 42, 32, 9, 23, 46, 32, 48, 24, 17, 39, 52, 27, 10, 48, 18, 28, 58, 15, 60, 31, 27, 127, 25, 11, 66, 16, 23, 20, 70, 63, 72, 36, 24

Offset: 1

Kenichiro Kashihara (Univxiq(AT)aol.com)

The graph of the function is split into rays of which the densest ones are y(n) = n-1 = a(n) iff n is an odd prime power, and y(n) = n/2 = a(n) or a(n)+1 if n = 8k-2 (except for k = 9, 10, 14, 16, 19, 24, ...) or 8k+2 (except for k = 8, 11, 16, 17, 19, 26, 33, ...). The next most-frequent rays are similar: y(n) = n/r for r = 3, 4, 5, ... and r = 4/3, etc. - M. F. Hasler, May 30 2021

Alois P. Heinz, Table of n, a(n) for n = 1..16383 (first 1000 terms from T. D. Noe)
C. Ashbacher, The Pseudo-Smarandache Function and the Classical Functions of Number Theory, Smarandache Notions Journal, Vol. 9, No. 1-2, 1998, 79-82.
Jason Earls, The Smarandache sum of composites between factors function, in Smarandache Notions Journal (2004), Vol. 14.1, page 246.
K. Kashihara, Comments and Topics on Smarandache Notions and Problems, Erhus University Press, 1996, 50 pages. See p. 35.
K. Kashihara, Comments and Topics on Smarandache Notions and Problems, Erhus University Press, 1996, 50 pages. [Cached copy] See p. 35.
Eric Weisstein's World of Mathematics, Pseudosmarandache Function

Cf. A000217, A066561.
Cf. A343995, A343996, A343997, A343998, A345984 (partial sums).
Cf. also A080982, A344005.

import Data.List (findIndex)
import Data.Maybe (fromJust)
a011772 n = (+ 1) $ fromJust $
   findIndex ((== 0) . (`mod` n)) $ tail a000217_list
-- Reinhard Zumkeller, Mar 23 2013

Table[m := 1; While[Not[IntegerQ[(m*(m + 1))/(2n)]], m++ ]; m, {n, 1, 90}] (* Stefan Steinerberger, Apr 03 2006 *)
(Sqrt[1+8#]-1)/2&/@Flatten[With[{r=Accumulate[Range[300]]},Table[ Select[r, Divisible[#,n]&,1],{n,80}]]] (* Harvey P. Dale, Feb 05 2012 *)

a(n)=if(n==1,return(1)); my(f=factor(if(n%2,n,2*n)), step=vecmax(vector(#f~, i, f[i,1]^f[i,2]))); forstep(m=step,2*n,step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))) \\ Charles R Greathouse IV, Jun 25 2017

from math import isqrt
def A011772(n):
    m = (isqrt(8*n+1)-1)//2
    while (m*(m+1)) % (2*n):
        m += 1
    return m # Chai Wah Wu, May 30 2021

A000217(a(n)) = A066561(n).
a(2^k) = 2^(k+1)-1; a(m) = m-1 for odd prime powers m. - Reinhard Zumkeller, Feb 26 2003
a(n) <= 2n-1 for all numbers n; a(n) <= n-1 for odd n. - Stefan Steinerberger, Apr 03 2006
a(n) >= (sqrt(8n+1)-1)/2 for all n. - Charles R Greathouse IV, Jun 25 2017
a(n) < n-1 for all n except the prime powers where a(n) = n-1 (n odd) or 2n-1 (n = 2^k). - M. F. Hasler, May 30 2021
a(n) = A344005(2*n). - N. J. A. Sloane, Jul 06 2021
a(n) = 2*n-1 iff n is a power of 2. - Shu Shang, Aug 01 2022

More terms from Stefan Steinerberger, Apr 03 2006

A110353 Value of k pertaining to A110351(n). Least k such that the sum (n+1) + (n+2) + ...+(n+k) is a multiple of the n-th triangular number n(n+1)/2.

Original entry on oeis.org

1, 1, 5, 11, 4, 8, 41, 55, 26, 34, 21, 27, 64, 6, 65, 239, 118, 134, 56, 15, 56, 208, 160, 176, 274, 298, 161, 175, 115, 125, 929, 319, 120, 50, 189, 260, 628, 208, 65, 575, 204, 216, 429, 55, 369, 988, 657, 687, 1126, 324, 221, 584, 1324, 485, 120, 343, 494, 1594

Offset: 1

Amarnath Murthy, Jul 21 2005

nonn

For many values of n, a(n) = A110352(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A110351, A110352.

Cf. A000217, A080982.

import Data.List (findIndex)
import Data.Maybe (fromJust)
a110353 n = (+ 1) $ fromJust $
   findIndex ((== 0) . (`mod` t)) $ dropWhile (<= t) a000217_list
   where t = a000217 n
-- Reinhard Zumkeller, Mar 23 2013

More terms from Joshua Zucker, May 08 2006

A080983 Smallest triangular number having n^2 as divisor.

Original entry on oeis.org

1, 28, 36, 496, 300, 36, 1176, 8128, 3240, 300, 7260, 2016, 14196, 1176, 4950, 130816, 41616, 3240, 64980, 25200, 4851, 7260, 139656, 131328, 195000, 14196, 265356, 270480, 353220, 25200, 461280, 2096128, 29403, 41616, 1225, 738720, 936396

Offset: 1

Reinhard Zumkeller, Feb 26 2003

nonn

a(n)=A000217(A080982(n)).

Donovan Johnson, Table of n, a(n) for n = 1..5000

Cf. A066561.

a080983 = a000217 . a080982  -- Reinhard Zumkeller, Mar 23 2013

With[{trnos=Accumulate[Range[2500]]},Flatten[Table[Select[trnos, Divisible[ #,n^2]&,1],{n,40}]]] (* Harvey P. Dale, Jun 01 2014 *)

cp's OEIS Frontend

A011772 Smallest number m such that m(m+1)/2 is divisible by n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Formula

Extensions

A110353 Value of k pertaining to A110351(n). Least k such that the sum (n+1) + (n+2) + ...+(n+k) is a multiple of the n-th triangular number n(n+1)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Extensions

A080983 Smallest triangular number having n^2 as divisor.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica