A328633 -id:A328633 - 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.

A328317 Smallest prime not dividing A328316(n), with a(0) = 1 by convention; Equally, for n > 0, smallest prime dividing A328316(1+n).

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2

Offset: 0

Antti Karttunen, Oct 14 2019

a(2n+1) = 2 for all n >= 0. Does the pattern of 5's in the even bisection also continue?

Cf. A020639, A053669, A276086, A326810, A328316, A328318, A328319, A328322, A328323, A328585, A328586, A328633.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328316(n) = if(!n,0,A276086(A328316(n-1)));
A053669(n) = forprime(p=2, , if(n%p, return(p))); \\ From A053669
A328317(n) = if(0==n,1,A053669(A328316(n)));
\\ Or alternatively as:
A020639(n)=if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1)
A328317(n) = A020639(A328316(1+n));

a(0) = 1; and for n > 0, a(n) = A053669(A328316(n)).
a(n) = A020639(A328316(1+n)).
For n >= 1, a(n) = A326810(A328316(n-1)). - Antti Karttunen, Nov 15 2019

a(12)-a(13) from Jinyuan Wang, Jul 20 2020

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];

A328586 Even numbers n for which A257993(n) is equal to A257993(A276086(A276086(n))), where A276086 converts the primorial base expansion of n into its prime product form, and A257993 returns the index of the least prime not present in its argument.

Original entry on oeis.org

6, 18, 36, 48, 66, 78, 96, 108, 126, 138, 156, 168, 186, 198, 222, 234, 252, 264, 282, 294, 312, 324, 342, 354, 372, 384, 402, 414, 426, 438, 456, 468, 486, 498, 516, 528, 546, 558, 576, 588, 606, 618, 642, 654, 672, 684, 702, 714, 732, 744, 762, 774, 792, 804, 822, 834, 846, 858, 876, 888, 906, 918, 936, 948, 966, 978, 996, 1008

Offset: 1

Antti Karttunen, Oct 21 2019

nonn

All terms are multiples of 6, but very few multiples of 5 (and thus of 10) are present: the first ones are at a(169) = 2520 and a(254) = 3780. Among the first 10000 terms, there are only 28 ending with decimal digit 0, while those that end with either 2 or 4 are 2450 both, and with either 6 or 8, both have 2536 each.

Other multiples of six are in A328587 and A328589.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primorial base

Cf. A257993, A276086, A328578, A328316, A328317, A328633.

Even terms in A328585. Cf. also A328587, A328588, A328589.

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)));
isA328586(n) = ((!(n%2))&&(A328578(n) == A257993(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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A328317 Smallest prime not dividing A328316(n), with a(0) = 1 by convention; Equally, for n > 0, smallest prime dividing A328316(1+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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A328586 Even numbers n for which A257993(n) is equal to A257993(A276086(A276086(n))), where A276086 converts the primorial base expansion of n into its prime product form, and A257993 returns the index of the least prime not present in its argument.

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