Original entry on oeis.org
4, 6, 10, 12, 14, 15, 16, 20, 21, 22, 26, 27, 28, 30, 33, 34, 35, 36, 38, 39, 42, 44, 46, 48, 50, 51, 52, 54, 55, 57, 58, 60, 62, 64, 65, 66, 68, 69, 70, 74, 76, 77, 78, 80, 82, 84, 85, 86, 87, 91, 92, 93, 94, 95, 99, 100, 102, 105, 106, 108, 110, 111, 112, 114, 115, 116, 118, 119, 122, 123, 124, 129, 130, 132, 133
Offset: 1
10 = 2*5 has maximal exponent (A051903) 1, and its arithmetic derivative A003415(10) = 2+5 = 7 also has maximal exponent 1, thus 10 is included in this sequence.
15 = 3*5 has maximal exponent 1, and its arithmetic derivative A003415(15) = 3+5 = 8 = 2^3 has maximal exponent 3, thus 15 is included in this sequence.
For 8 = 2^3, its arithmetic derivative A003415(8) = 12 = 2^2 * 3, and as 2 < 3 (highest exponent of 12 is less than that of 8), 8 is NOT included here, and from this we also see that A100716 is not a subsequence of this sequence.
-
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A328311(n) = if(n<=1,0,1+(A051903(A003415(n)) - A051903(n)));
isA328321(n) = (A328311(n)>0);
A328320
Numbers for which A328311(n) = 1 + A051903(A003415(n)) - A051903(n) is zero (including 1 as the initial term).
Original entry on oeis.org
1, 2, 3, 5, 7, 8, 9, 11, 13, 17, 18, 19, 23, 24, 25, 29, 31, 32, 37, 40, 41, 43, 45, 47, 49, 53, 56, 59, 61, 63, 67, 71, 72, 73, 75, 79, 81, 83, 88, 89, 90, 96, 97, 98, 101, 103, 104, 107, 109, 113, 117, 120, 121, 125, 126, 127, 128, 131, 136, 137, 139, 147, 149, 150, 151, 152, 153, 157, 160, 162, 163, 167, 168, 169
Offset: 1
-
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A328311(n) = if(n<=1,0,1+(A051903(A003415(n)) - A051903(n)));
isA328320(n) = (0==A328311(n));
A328250
Square array A(n,k) read by descending antidiagonals where A(n,k) is the k-th solution x to A328248(x) = n-1.
Original entry on oeis.org
4, 8, 1, 12, 2, 9, 16, 3, 18, 50, 20, 5, 25, 99, 306, 24, 6, 45, 125, 549, 5831, 27, 7, 49, 207, 1611, 6849, 20230, 28, 10, 63, 343, 2662, 14225, 33026, 52283, 32, 11, 75, 375, 2842, 16299, 47107, 225998, 286891, 36, 13, 90, 531, 2891, 19431, 49806, 1336047, 1292750, 10820131, 40, 14, 98, 686, 4575, 21231, 117649, 1422275, 2886982, 21628098, 38452606
Offset: 1
The upper left corner of the array:
4, 8, 12, 16, 20, 24, 27, 28,
1, 2, 3, 5, 6, 7, 10, 11,
9, 18, 25, 45, 49, 63, 75, 90,
50, 99, 125, 207, 343, 375, 531, 686,
306, 549, 1611, 2662, 2842, 2891, 4575, 4802,
5831, 6849, 14225, 16299, 19431, 21231, 22638, 24010,
20230, 33026, 47107, 49806, 117649, 121671, 145386, 162707,
52283, 225998, 1336047, 1422275, 1500759, 1576899, 2309503, 3023398,
286891, 1292750, 2886982, 3137526, 6882453, 8703459, 15358457, 16777114,
10820131, 21628098, 23934105, 24332763, 46295435, 51320698, 52320191, 56199375,
38452606, ...
...
-
up_to = 45; \\ 10585 = binomial(145+1,2)
A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i,2]>=f[i,1],return(0), s += f[i, 2]/f[i, 1])); (n*s));
A328248(n) = { my(k=1); while(n && !issquarefree(n), k++; n = A003415checked(n)); (!!n*k); };
memoA328250sq = Map();
A328250sq(n, k) = { my(v=0); if(!mapisdefined(memoA328250sq,[n,k-1],&v),if(1==k, v=0, v = A328250sq(n, k-1))); for(i=1+v,oo,if((1+A328248(i))==n,mapput(memoA328250sq,[n,k],i); return(i))); };
A328250list(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] = A328250sq(col,(a-(col-1))))); (v); };
v328250 = A328250list(up_to);
A328250(n) = v328250[n];
Showing 1-3 of 3 results.
Comments