A344595 -id:A344595 - cp's OEIS frontend

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

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));
```

A344975 Numbers k such that A011772(k) divides A344875(k), but k is not a power of prime (in A000961).

Original entry on oeis.org

6, 10, 18, 21, 24, 26, 28, 34, 36, 39, 40, 50, 55, 57, 58, 68, 74, 75, 78, 82, 93, 96, 98, 100, 106, 111, 120, 122, 129, 136, 146, 147, 150, 155, 162, 164, 171, 178, 183, 194, 196, 201, 202, 203, 205, 218, 219, 222, 224, 226, 237, 242, 250, 253, 274, 288, 291, 292, 294, 298, 300, 301, 305, 309, 314, 324, 327, 333

Offset: 1

Antti Karttunen, Jun 04 2021

nonn

This is not a subsequence of A344882. The first terms of this sequence that do not occur there are: 900, 1260, 1560, ... etc, see A344694. First terms of A344882 not present here are: 60, 66, 88, 92, 105, etc.

Cf. A000961, A011772, A344875, A344882, A344973.
Intersection of A024619 and A344974.
Cf. also A344595, A344694, A344978 (subsequences).

isA344975(n) = ((n>1)&&!isprimepower(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));

A344977 Numbers k for which A011772(k) exceeds A344878(k).

Original entry on oeis.org

14, 15, 30, 33, 35, 38, 42, 45, 51, 56, 62, 63, 65, 69, 70, 84, 85, 86, 87, 88, 90, 91, 99, 102, 104, 105, 114, 115, 116, 117, 123, 126, 130, 133, 134, 135, 141, 143, 145, 152, 153, 154, 158, 159, 161, 165, 168, 170, 172, 174, 175, 177, 180, 182, 185, 186, 195, 200, 204, 206, 210, 213, 215, 217, 221, 225, 234, 235

Offset: 1

Antti Karttunen, Jun 04 2021

nonn

Cf. A011772, A344878.
Positions of negative terms in A344976.
Cf. A344595 (subsequence).

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
A344878(n) = if(1==n,n, my(f=factor(n)~); lcm(vector(#f, i, (f[1, i]^(f[2, i]+(2==f[1, i]))-1))));
isA344977(n) = (A011772(n)>A344878(n));

{k such that A344976(k) < 0}.

cp's OEIS Frontend

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

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A344975 Numbers k such that A011772(k) divides A344875(k), but k is not a power of prime (in A000961).

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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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A344977 Numbers k for which A011772(k) exceeds A344878(k).

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