A344978 -id:A344978 - cp's OEIS frontend

A344976 a(n) = A344878(n) - A011772(n).

0, 0, 0, 0, 0, 3, 0, 0, 0, 8, 0, 6, 0, -1, -1, 0, 0, 16, 0, 13, 0, 19, 0, 15, 0, 0, 0, 35, 0, -3, 0, 0, -1, 32, -2, 48, 0, -1, 0, 45, 0, -14, 0, 38, -1, 43, 0, 30, 0, 0, -1, 45, 0, 51, 10, -18, 0, 56, 0, 13, 0, -1, -3, 0, -13, 19, 0, 96, -1, -8, 0, 57, 0, 0, 0, 70, 9, 0, 0, 60, 0, 80, 0, -6, -18, -1, -1, -2, 0, -11, -1, 131

Offset: 1

Antti Karttunen, Jun 04 2021

sign

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, A344876, A344878, A344977 (positions of negative terms), A344978, A344979 (positions of zeros), A344885 (and their characteristic function).

A011772[n_] := Module[{m = 1}, While[Not[IntegerQ[m(m+1)/(2n)]], m++]; m];
A344878[n_] := If[n == 1, 1, Module[{p, e}, LCM @@ Table[{p, e} = pe; (p^(e + If[p == 2, 1, 0])) - 1, {pe, FactorInteger[n]}]]];
a[n_] := A344878[n] - A011772[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
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))));
A344976(n) = (A344878(n)-A011772(n));

a(n) = A344878(n) - A011772(n).

a(n) <= A344876(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)));

A344979 Numbers k such that A011772(k) is equal to A344878(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, 23, 25, 26, 27, 29, 31, 32, 37, 39, 41, 43, 47, 49, 50, 53, 57, 59, 61, 64, 67, 71, 73, 74, 75, 78, 79, 81, 83, 89, 93, 97, 98, 101, 103, 107, 109, 111, 113, 121, 122, 125, 127, 128, 129, 131, 137, 139, 146, 147, 149, 150, 151, 157, 163, 167, 169, 173, 179, 181, 183, 191, 193, 194

Offset: 1

Antti Karttunen, Jun 05 2021

nonn

Cf. A011772, A344878, A344885 (characteristic function).
Positions of zeros in A344976.
Union of A000961 and A344978. Subsequence of A344974.

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))));
isA344979(n) = (A344878(n)==A011772(n));

cp's OEIS Frontend

A344976 a(n) = A344878(n) - A011772(n).

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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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A344979 Numbers k such that A011772(k) is equal to A344878(k).

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