A328634 -id:A328634 - cp's OEIS frontend

A328578 Index of the least prime not dividing A276086(A276086(n)): a(n) = A257993(A276087(n)).

2, 1, 3, 1, 4, 1, 3, 1, 4, 1, 5, 1, 2, 1, 5, 1, 4, 1, 3, 1, 6, 1, 6, 1, 2, 1, 6, 1, 7, 1, 2, 1, 4, 1, 3, 1, 3, 1, 5, 1, 6, 1, 2, 1, 6, 1, 6, 1, 3, 1, 7, 1, 7, 1, 2, 1, 7, 1, 5, 1, 2, 1, 5, 1, 4, 1, 3, 1, 6, 1, 6, 1, 2, 1, 7, 1, 7, 1, 3, 1, 7, 1, 8, 1, 2, 1, 6, 1, 8, 1, 2, 1, 6, 1, 7, 1, 3, 1, 7, 1, 7, 1, 2, 1, 7, 1

Offset: 0

Antti Karttunen, Oct 20 2019

Index of the least significant zero digit in the primorial base expansion of A276086(n), when the rightmost digit is in the position 1.

The scatter plot shows both regular looking as well as more chaotic regions. This can be more clearly seen in related A328579. See also A328839.

Antti Karttunen, Table of n, a(n) for n = 0..32768
Rémy Sigrist, Colored logarithmic scatterplot of the ordinal transform of the sequence for n = 0..510510
Rémy Sigrist, Density plot of the sequence for n = 0..510510
Index entries for sequences related to primorial base

Cf. A000720, A055396, A276086, A276087, A328403, A328570, A328579 (the corresponding prime), A328590, A328763, A328766, A328839.
Cf. A328585 (where equal with A257993), A328587 (less than), A328588 (greater than).
Cf. A328761 (the first occurrence of each n).
Cf. also array A328631 and its rows A005408, A328632, A328633, A328634, A328635, A328636 (positions of terms 1 .. 6 in this sequence).

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328570(n) = { my(i=1, p=2); while(n && (n%p), i++; n = n\p; p = nextprime(1+p)); (i); };
A328578(n) = A328570(A276086(n));

A257993(n) = { for(i=1,oo,if(n%prime(i),return(i))); }
A328578(n) = A257993(A276086(A276086(n)));

a(n) = A328570(A276086(n)) = A257993(A276087(n)) = A055396(A328403(n)).
a(n) = A000720(A328579(n)).
a(n) = A257993(n) + A328590(n).
a(n) = A055396(A328763(n)).
For all n >= 0, a(A328761(n)) = n.

A328631 Square array where the row n lists all nonnegative numbers k for which A328578(k) = n, read by falling antidiagonals.

Original entry on oeis.org

1, 3, 0, 5, 12, 2, 7, 24, 6, 4, 9, 30, 18, 8, 10, 11, 42, 34, 16, 14, 20, 13, 54, 36, 32, 38, 22, 28, 15, 60, 48, 64, 58, 26, 50, 82, 17, 72, 66, 152, 62, 40, 52, 88, 116, 19, 84, 78, 184, 112, 44, 56, 106, 118, 148, 21, 90, 96, 210, 166, 46, 74, 110, 140, 178, 208, 23, 102, 108, 242, 176, 68, 76, 128, 142, 196, 412, 418

Offset: 1

Antti Karttunen, Oct 27 2019

Array is read by descending antidiagonals with (n,k) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ... where A(n,k) is the k-th solution x to A328578(x) = n.

			The top left 12 X 12 corner of the array:
   n +------------------------------------------------------
   1 |   1,   3,   5,   7,   9,  11,  13,  15,  17,  19, ...
   2 |   0,  12,  24,  30,  42,  54,  60,  72,  84,  90, ...
   3 |   2,   6,  18,  34,  36,  48,  66,  78,  96, 108, ...
   4 |   4,   8,  16,  32,  64, 152, 184, 210, 242, 274, ...
   5 |  10,  14,  38,  58,  62, 112, 166, 176, 214, 218, ...
   6 |  20,  22,  26,  40,  44,  46,  68,  70,  86,  92, ...
   7 |  28,  50,  52,  56,  74,  76,  80,  94,  98, 100, ...
   8 |  82,  88, 106, 110, 128, 130, 134, 158, 182, 262, ...
   9 | 116, 118, 140, 142, 146, 160, 164, 170, 188, 190, ...
  10 | 148, 178, 196, 200, 202, 206, 328, 352, 374, 376, ...
  11 | 208, 412, 416, 562, 568, 586, 590, 592, 596, 614, ...
  12 | 418, 598, 626, 628, 778, 800, 802, 826, 830, 832, ...

Rows 1-6: A005408, A328632, A328633, A328634, A328635, A328636.
Column 1: A328761.
Cf. A257993, A276086, A328578.

up_to = 78;
A257993(n) = { for(i=1,oo,if(n%prime(i),return(i))); }
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328578(n) = A257993(A276086(A276086(n)));
memoA328631sq = Map();
A328631sq(n, k) = { my(v=0); if(!mapisdefined(memoA328631sq,[n,k-1],&v),if(1==k, v=-1, v = A328631sq(n, k-1))); for(i=1+v,oo,if(A328578(i)==n,mapput(memoA328631sq,[n,k],i); return(i))); };
A328631list(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] = A328631sq(col,(a-(col-1))))); (v); };
v328631 = A328631list(up_to);
A328631(n) = v328631[n];

cp's OEIS Frontend

A328578 Index of the least prime not dividing A276086(A276086(n)): a(n) = A257993(A276087(n)).

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A328631 Square array where the row n lists all nonnegative numbers k for which A328578(k) = n, read by falling antidiagonals.

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