A360119 -id:A360119 - cp's OEIS frontend

A360129 Numbers k such that A360119(k) > 1, but which have no divisors d > 1 such that d+1 is also a divisor.

572, 1144, 1292, 1768, 2288, 2584, 2590, 3496, 3596, 4030, 4576, 4930, 6292, 6992, 7052, 7192, 7436, 8050, 8170, 8246, 9044, 9152, 9212, 9430, 9724, 9758, 10868, 11408, 12136, 12584, 12950, 13984, 14104, 14212, 14384, 14396, 14872, 15370, 16168, 16492, 16588, 16796, 17066, 17168, 17732, 18130, 18304

Offset: 1

Antti Karttunen, Feb 20 2023

nonn

Subsequence of A088725.

Cf. A088722, A360119, A360128.

A360119(n) = { my(d=divisors(n), erot=vecsort(vector(#d-1, k, d[k+1] - d[k])), s=#d); for(i=1,#erot,if(n%erot[i], s--, if(1==i || erot[i]!=erot[i-1], s--))); (s); };
A360128(n) = !sumdiv(n,d,(d>1)&&!(n%(d+1)));
isA360129(n) = (A360128(n)&&(A360119(n)>1));

{k | A088722(k) = 0 and A360119(k) > 1}.

A360118 Number of differences (not all necessarily distinct) between consecutive divisors of n which are not also divisors of n.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 1, 3, 0, 1, 0, 1, 0, 3, 1, 1, 0, 2, 1, 3, 1, 1, 1, 1, 0, 3, 1, 3, 0, 1, 1, 3, 1, 1, 0, 1, 1, 5, 1, 1, 0, 2, 2, 3, 1, 1, 0, 3, 2, 3, 1, 1, 0, 1, 1, 5, 0, 3, 1, 1, 1, 3, 4, 1, 0, 1, 1, 5, 1, 3, 1, 1, 2, 4, 1, 1, 1, 3, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 0, 1, 2, 5, 0, 1, 1, 1, 1, 7

Offset: 1

Antti Karttunen, Feb 20 2023

nonn

			For n=70, its divisors are {1, 2, 5, 7, 10, 14, 35, 70} and their first differences are {1, 3, 2, 3, 4, 21, 35}, of which the differences {3, 3, 4, 21} are not divisors, so a(70) = 4. Note that in contrast to A060763, here the difference 3 is counted twice because there are two copies of it among the differences.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. also A060763, A060764, A138652, A360119.

A360118(n) = { my(d=divisors(n), erot = vector(#d-1, k, d[k+1] - d[k])); sum(i=1,#erot,!!(n%erot[i])); };

For all n >= 1, A060763(n) <= a(n) < A060764(n).

a(n) = A060764(n) - A360119(n).

cp's OEIS Frontend

A360129 Numbers k such that A360119(k) > 1, but which have no divisors d > 1 such that d+1 is also a divisor.

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