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

1,2

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

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

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 *)

Prog

(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
(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

Crossrefs

Cf. A000217, A066561.

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

Cf. also A080982, A344005.

Sequence in context: A345947 A350001 A344969 * A060451 A220291 A129187

Adjacent sequences: A011769 A011770 A011771 * A011773 A011774 A011775

Keyword

nonn,easy,look,nice

Author

Kenichiro Kashihara (Univxiq(AT)aol.com)

Extensions

More terms from Stefan Steinerberger, Apr 03 2006

Status

approved