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

Author

Kenichiro Kashihara (Univxiq(AT)aol.com)

Keywords

• nonn
• easy
• look
• nice

Comments

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

Links

• 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

Crossrefs

Cf. A000217, A066561.

Cf. A343995, A343996, A343997, A343998, A345984 (partial sums).

Cf. also A080982, A344005.

Programs

Haskell

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

Mathematica

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 *)

PARI

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

Python

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

Formula

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

Extensions

More terms from Stefan Steinerberger, Apr 03 2006