A066561 -id:A066561 - 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

A345056 a(n) is the next larger number k > n for which A011772(k) = A011772(n), or n itself if no such larger number exists.

Original entry on oeis.org

1, 6, 3, 14, 10, 6, 21, 20, 12, 10, 55, 18, 26, 28, 15, 62, 34, 36, 57, 24, 21, 33, 253, 30, 50, 39, 117, 28, 58, 40, 93, 72, 66, 68, 105, 36, 74, 95, 78, 60, 82, 70, 129, 48, 45, 69, 1081, 88, 56, 75, 153, 130, 106, 63, 55, 84, 171, 203, 1711, 120, 122, 124, 126, 254, 325, 66, 201, 136, 92, 210, 355, 96, 146, 111, 100, 114

Offset: 1

Antti Karttunen, Jun 07 2021

nonn

Cf. A000217, A002024, A011772, A066561, A345057, A345058.

A011772(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))); }; \\ From A011772
A345056(n) = { my(x=A011772(n), y=binomial(x+1,2)); for(i=1+n,y,if(A011772(i)==x,return(i))); (n); };

a(n) = n + A345057(n).

A345057 Distance to the next larger number k > n for which A011772(k) = A011772(n), or 0 if no such number exists.

Original entry on oeis.org

0, 4, 0, 10, 5, 0, 14, 12, 3, 0, 44, 6, 13, 14, 0, 46, 17, 18, 38, 4, 0, 11, 230, 6, 25, 13, 90, 0, 29, 10, 62, 40, 33, 34, 70, 0, 37, 57, 39, 20, 41, 28, 86, 4, 0, 23, 1034, 40, 7, 25, 102, 78, 53, 9, 0, 28, 114, 145, 1652, 60, 61, 62, 63, 190, 260, 0, 134, 68, 23, 140, 284, 24, 73, 37, 25, 38, 154, 0

Offset: 1

Antti Karttunen, Jun 07 2021

nonn

Cf. A000217 (positions of zeros), A002024, A011772, A066561, A345056, A345058.

A011772(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))); }; \\ From A011772
A345057(n) = { my(x=A011772(n), y=binomial(x+1,2)); for(i=1+n,y,if(A011772(i)==x,return(i-n))); (0); };

a(n) = A345056(n) - n.

A345058 Number of distinct k > 0 for which A011772(k) = n.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 2, 7, 4, 2, 2, 4, 4, 3, 1, 5, 4, 4, 3, 7, 2, 3, 2, 7, 6, 4, 3, 3, 4, 7, 4, 13, 3, 4, 2, 4, 6, 3, 4, 7, 6, 3, 4, 7, 1, 5, 2, 7, 11, 6, 2, 7, 4, 5, 2, 11, 3, 4, 2, 7, 8, 5, 5, 7, 2, 3, 4, 4, 5, 3, 4, 13, 9, 6, 6, 9, 2, 4, 4, 8, 15, 6, 2, 11, 3, 3, 3, 7, 6, 3, 1, 5, 4, 7, 3, 13, 10, 11, 7, 6, 6, 5, 4, 8, 2

Offset: 1

Antti Karttunen, Jun 07 2021

nonn

Question: Are there any numbers n other than 1, 3, 15, 45, 91 for which a(n) = 1?

			A011772 obtains the value 6 only as A011772(7)=6 and A011772(36)=6, therefore a(6) = 2.

Cf. A000217, A002024, A011772, A066561, A345056, A345057.

A011772(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))); }; \\ From A011772
A345058(n) = { my(x=A011772(n), y=binomial(x+1,2)); sum(i=1,y,(A011772(i)==x)); };

a(n) = Sum_{i=1..A000217(n)} [A011772(i) = A011772(n)], where [ ] is the Iverson bracket.

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 *)

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

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A345056 a(n) is the next larger number k > n for which A011772(k) = A011772(n), or n itself if no such larger number exists.

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A345057 Distance to the next larger number k > n for which A011772(k) = A011772(n), or 0 if no such number exists.

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A345058 Number of distinct k > 0 for which A011772(k) = n.

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A080983 Smallest triangular number having n^2 as divisor.

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