A354930 -id:A354930 - cp's OEIS frontend

A354940 Square array A(n, k) = A354930(n, k)/n, read by falling antidiagonals.

1, 2, 3, 3, 5, 4, 4, 7, 7, 5, 5, 9, 13, 9, 3, 7, 11, 16, 13, 6, 7, 8, 13, 19, 17, 8, 13, 4, 9, 17, 25, 21, 11, 19, 5, 3, 11, 19, 31, 25, 13, 25, 8, 9, 5, 13, 23, 37, 29, 16, 31, 11, 11, 7, 11, 16, 25, 43, 37, 23, 37, 15, 17, 10, 31, 3, 17, 27, 49, 41, 26, 43, 19, 19, 14, 41, 4, 13, 19, 29, 61, 49, 31, 49, 22, 25, 16, 51, 6, 25, 7

Offset: 1

Antti Karttunen, Jun 15 2022

Array is read by descending antidiagonals with (n,k) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), etc.

			The top left 15x16 corner of the array:
n\k  |  1   2   3   4   5   6   7   8    9   10   11   12   13   14   15
-----+---------------------------------------------------------------------
   1 |  1,  2,  3,  4,  5,  7,  8,  9,  11,  13,  16,  17,  19,  23,  25,
   2 |  3,  5,  7,  9, 11, 13, 17, 19,  23,  25,  27,  29,  31,  37,  41,
   3 |  4,  7, 13, 16, 19, 25, 31, 37,  43,  49,  61,  64,  67,  73,  79,
   4 |  5,  9, 13, 17, 21, 25, 29, 37,  41,  49,  53,  57,  61,  73,  81,
   5 |  3,  6,  8, 11, 13, 16, 23, 26,  31,  36,  41,  43,  46,  51,  53,
   6 |  7, 13, 19, 25, 31, 37, 43, 49,  61,  67,  73,  79,  97, 103, 109,
   7 |  4,  5,  8, 11, 15, 19, 22, 25,  29,  32,  39,  43,  47,  50,  53,
   8 |  3,  9, 11, 17, 19, 25, 27, 33,  41,  43,  49,  57,  59,  67,  73,
   9 |  5,  7, 10, 14, 16, 19, 23, 25,  28,  32,  37,  41,  43,  46,  50,
  10 | 11, 31, 41, 51, 61, 71, 81, 91, 101, 121, 131, 141, 151, 171, 181,
  11 |  3,  4,  6,  9, 12, 15, 17, 23,  25,  28,  31,  34,  37,  45,  47,
  12 | 13, 25, 37, 49, 61, 73, 85, 97, 109, 121, 145, 157, 169, 181, 193,
  13 |  7,  8,  9, 11, 14, 20, 22, 23,  27,  33,  37,  40,  46,  47,  48,
  14 |  5, 15, 19, 29, 43, 47, 57, 61,  71,  89,  99, 103, 113, 127, 131,
  15 |  4,  8, 16, 19, 23, 31, 38, 46,  49,  53,  61,  64,  76,  79,  83,
  16 |  7, 11, 13, 17, 23, 27, 29, 33,  43,  49,  59,  61,  65,  71,  75,

Cf. A345992, A354930.
Cf. A354932 (column 1).
Rows 1 .. 7 (some of these are conjectural): A000961, A061345 (without its initial 1), A137827, A354934, A354935, A354936, A354937, A354938, A354939.

up_to = 105;
A345992(n) = for(m=1, oo, if((m*(m+1))%n==0, return(gcd(n,m))));
memoA354930sq = Map();
A354930sq(n, k) = { my(v=0); if(!mapisdefined(memoA354930sq,[n,k-1],&v),if(1==k, v=0, v = A354930sq(n, k-1))); for(i=1+v,oo,if(A345992(i)==n,mapput(memoA354930sq,[n,k],i); return(i))); };
A354940sq(n, k) = (A354930sq(n, k)/n);
A354940list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A354940sq(col,(a-(col-1))))); (v); };
v354940 = A354940list(up_to);
A354940(n) = v354940[n];

A345992 Let m = A344005(n) = smallest m such that n divides m*(m+1); a(n) = gcd(n,m).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 5, 1, 1, 2, 1, 4, 3, 2, 1, 8, 1, 2, 1, 7, 1, 5, 1, 1, 11, 2, 7, 4, 1, 2, 3, 5, 1, 6, 1, 11, 9, 2, 1, 3, 1, 2, 17, 4, 1, 2, 5, 7, 3, 2, 1, 15, 1, 2, 9, 1, 5, 11, 1, 4, 23, 14, 1, 8, 1, 2, 3, 19, 7, 6, 1, 5, 1, 2, 1, 4, 17, 2, 29, 8, 1, 9, 13, 23, 3, 2, 19, 32, 1, 2, 11, 4, 1, 17

Offset: 1

Robert Dougherty-Bliss and N. J. A. Sloane, Jul 15 2021

nonn

By definition, a(n)*A345993(n) = n.
a(n) is even iff n/2 is in A344000. This is true, but essentially trivial, and does not provide any insight into either sequence.
Empirical: For n >= 3, a(n) <= n/3, and a(n) = n/3 iff n is in 3*{2^odd, primes == -1 mod 6}.
If n = 2*p^k where p is an odd prime then m = A344005(n) = p^k - 1 and a(n) = 2. Conversely, it appears that if a(n) = 2 then n is twice an odd prime power. (Corrected by Antti Karttunen, Jun 14 2022)
a(n) = 1 if n is a prime power. - Chai Wah Wu, Jun 01 2022
From Antti Karttunen, Jun 14 2022: (Start)
Conversely, if a(n) = 1 [i.e., A345993(n) = n] then n is a power of prime. (This follows from N. J. A. Sloane's Jul 11 2021 theorem given in A344005).
Apparently, a(n) = 3 iff n = A354984(k) = 3*A137827(k), for some k >= 1.
(End)

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A011772, A137827, A182665, A344000, A344005, A345993, A345994, A345995, A354930, A354931 (the least occurrence of each n=1..), A354984.

Cf. also A007528, A051119, A284600.

# load Findm from A344005
ans:=[];
for n from 1 to 40 do t1:=Findm(n)[1]; ans:=[op(ans), igcd(n,t1)]; od:
ans;

smd[n_]:=Module[{m=1},While[Mod[m(m+1),n]!=0,m++];GCD[n,m]]; Array[smd,110] (* Harvey P. Dale, Jan 07 2022 *)

f(n) = my(m=1); while ((m*(m+1)) % n, m++); m; \\ A344005
a(n) = gcd(n,f(n)); \\ Michel Marcus, Aug 06 2021
(Python 3.8+)
from math import gcd, prod
from itertools import combinations
from sympy import factorint
from sympy.ntheory.modular import crt
def A345992(n):
    if n == 1:
        return 1
    plist = tuple(p**q for p, q in factorint(n).items())
    return 1 if len(plist) == 1 else gcd(n,int(min(min(crt((m, n//m), (0, -1))[0], crt((n//m, m), (0, -1))[0]) for m in (prod(d) for l in range(1, len(plist)//2+1) for d in combinations(plist, l))))) # Chai Wah Wu, Jun 01 2022

a(n) = gcd(n, A182665(n)) = gcd(A182665(n), A344005(n)). - Antti Karttunen, Jun 13 2022

A354931 a(n) = the least k for which A345992(k) = n.

Original entry on oeis.org

1, 6, 12, 20, 15, 42, 28, 24, 45, 110, 33, 156, 91, 70, 60, 112, 51, 342, 76, 140, 231, 198, 69, 168, 325, 234, 108, 476, 87, 930, 124, 96, 561, 170, 315, 1116, 185, 418, 195, 360, 123, 714, 172, 220, 585, 1426, 141, 336, 245, 850, 204, 988, 159, 270, 385, 1064, 1311, 522, 177, 3660, 488, 434, 504, 320, 715, 1254

Offset: 1

Antti Karttunen, Jun 14 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..3349 (computed using million term data file provided for A344005 by _N. J. A. Sloane_)

Column 1 of A354930.

Cf. A344005, A345992, A354932.

s[n_] := Module[{m = 1}, While[!Divisible[m*(m+1), n], m++]; GCD[n, m]]; a[n_] := Module[{k = n}, While[s[k] != n, k+=n]; k]; Array[a,60] (* Amiram Eldar, Jun 15 2022 *)

A345992(n) = gcd(n,A344005(n));
A354931(n) = for(k=1,oo,if(A345992(k)==n,return(k)));

a(n) = n * A354932(n).

A354984 Numbers that are 3 * prime powers congruent to 1 (mod 3).

Original entry on oeis.org

12, 21, 39, 48, 57, 75, 93, 111, 129, 147, 183, 192, 201, 219, 237, 291, 309, 327, 363, 381, 417, 453, 471, 489, 507, 543, 579, 597, 633, 669, 687, 723, 768, 813, 831, 849, 867, 921, 939, 993, 1011, 1029, 1047, 1083, 1101, 1119, 1137, 1191, 1227, 1263, 1299, 1317, 1371, 1389, 1461, 1497, 1569, 1587, 1623, 1641, 1713

Offset: 1

Antti Karttunen, Jun 15 2022

nonn

Numbers k of the form 9m+3 such that k/3 = p^k, with p a prime and k >= 1.

Intersection of A017197 and 3*A246655.
Cf. A137827, A354983 (characteristic function).
Row 3 of A354930 (conjectured).

Select[Range[2000], Mod[#, 9] == 3 && PrimePowerQ[#/3] &] (* Amiram Eldar, Jun 15 2022 *)

A354983(n) = ((3==(n%9)) && isprimepower(n/3));
isA354984(n) = A354983(n);

a(n) = 3 * A137827(n).

cp's OEIS Frontend

A354940 Square array A(n, k) = A354930(n, k)/n, read by falling antidiagonals.

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A345992 Let m = A344005(n) = smallest m such that n divides m*(m+1); a(n) = gcd(n,m).

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A354931 a(n) = the least k for which A345992(k) = n.

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A354984 Numbers that are 3 * prime powers congruent to 1 (mod 3).

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