A344980 -id:A344980 - cp's OEIS frontend

A344970 a(n) = A011772(n) / gcd(A011772(n), A344875(n)).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 7, 5, 1, 1, 1, 1, 15, 1, 11, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 11, 1, 7, 1, 1, 19, 1, 1, 1, 5, 1, 16, 9, 23, 1, 16, 1, 1, 17, 13, 1, 9, 1, 8, 1, 1, 1, 15, 1, 31, 9, 1, 25, 11, 1, 1, 23, 5, 1, 21, 1, 1, 1, 4, 7, 1, 1, 16, 1, 1, 1, 4, 17, 43, 29, 16, 1, 35, 13, 23, 1, 47, 19, 1, 1, 1, 11

Offset: 1

Antti Karttunen, Jun 04 2021

Denominator of the ratio A344875(n)/A011772(n): 1/1, 3/3, 2/2, 7/7, 4/4, 6/3, 6/6, 15/15, 8/8, 12/4, 10/10, 14/8, 12/12, 18/7, 8/5, 31/31, 16/16, 24/8, 18/18, 28/15, 12/6, 30/11, ..., = 1/1, 1/1, 1/1, 1/1, 1/1, 2/1, 1/1, 1/1, 1/1, 3/1, 1/1, 7/4, 1/1, 18/7, 8/5, 1/1, 1/1, 3/1, 1/1, 28/15, 2/1, 30/11, etc.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A011772, A344875, A344969, A344971 (numerators), A344972 (ratio floored down), A344974 (positions of ones), A344980 (of terms > 1).

A011772[n_] := Module[{m = 1}, While[Not[IntegerQ[m(m+1)/(2n)]], m++]; m];
A344875[n_] := Product[{p, e} = pe; If[p == 2, 2^(1+e)-1, p^e-1], {pe, FactorInteger[n]}];
a[n_] := A011772[n]/GCD[A011772[n], A344875[n]];
Array[a, 100] (* Jean-François Alcover, Jun 12 2021 *)

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
A344875(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^(f[2, i]+(2==f[1, i]))-1)); };
A344970(n) = { my(u=A011772(n)); (u/gcd(u, A344875(n))); };

a(n) = A011772(n) / A344969(n) = A011772(n) / gcd(A011772(n), A344875(n)).

A344974 Numbers k such that A011772(k) divides A344875(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 17, 18, 19, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 36, 37, 39, 40, 41, 43, 47, 49, 50, 53, 55, 57, 58, 59, 61, 64, 67, 68, 71, 73, 74, 75, 78, 79, 81, 82, 83, 89, 93, 96, 97, 98, 100, 101, 103, 106, 107, 109, 111, 113, 120, 121, 122, 125, 127, 128, 129, 131, 136, 137

Offset: 1

Antti Karttunen, Jun 04 2021

nonn

Cf. A011772, A344875, A344884 (characteristic function).
Positions of ones in A344970, of zeros in A344973.
Union of A000961 and A344975. Complement of A344980.
Cf. also A344595 (subsequence).

A011772[n_] := Module[{m = 1}, While[Not[IntegerQ[m(m+1)/(2n)]], m++]; m];
A344875[n_] := Product[{p, e} = pe; If[p == 2, 2^(1+e)-1, p^e-1], {pe, FactorInteger[n]}];
Select[Range[200], Divisible[A344875[#], A011772[#]]&] (* Jean-François Alcover, Jun 12 2021 *)

PARI
```
isA344974(n) = (0==A344973(n));
```

A344694 Numbers k, not powers of primes, for which A011772(k) divides A344875(k), and for all proper divisors d of k, A011772(d) < A011772(k).

Original entry on oeis.org

900, 1260, 1560, 3740, 6552, 6669, 9680, 18981, 19880, 35784, 36080, 59040, 62238, 62244, 81872, 103730, 108500, 118910, 134420, 160160, 171740, 185724, 211072, 217833, 222224, 225929, 227528, 259325, 351072, 384944, 404294, 414778, 422604, 425178, 446600, 456228, 463008, 488205, 490105, 527100, 574308, 581184, 598400

Offset: 1

Antti Karttunen, Jun 05 2021

nonn

Has many terms common with A344595.

Cf. A011772, A344880, A344973.
Intersection of A024619, A344881 and A344974. Intersection of A344881 and A344975.
Cf. also A344595, A344980.

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
A344875(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^(f[2, i]+(2==f[1, i]))-1)); };
A344880(n) = { my(x=A011772(n)); fordiv(n, d, if(A011772(d)==x, return(d==n))); };
A344973(n) = (A344875(n)%A011772(n));
isA344694(n) = ((n>1)&&!isprimepower(n)&&(0==A344973(n))&&A344880(n));

cp's OEIS Frontend

A344970 a(n) = A011772(n) / gcd(A011772(n), A344875(n)).

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A344974 Numbers k such that A011772(k) divides A344875(k).

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A344694 Numbers k, not powers of primes, for which A011772(k) divides A344875(k), and for all proper divisors d of k, A011772(d) < A011772(k).

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