A328762 -id:A328762 - cp's OEIS frontend

A328587 Numbers n for which A257993(A276086(A276086(n))) is less than A257993(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.

12, 24, 30, 42, 54, 60, 72, 84, 90, 102, 114, 120, 132, 144, 150, 162, 174, 180, 192, 204, 210, 216, 228, 246, 258, 276, 288, 306, 318, 336, 348, 366, 378, 396, 408, 420, 432, 444, 450, 462, 474, 480, 492, 504, 510, 522, 534, 540, 552, 564, 570, 582, 594, 600, 612, 624, 636, 648, 666, 678, 696, 708, 726, 738, 756, 768, 786, 798, 816

Offset: 1

Antti Karttunen, Oct 21 2019

nonn

Numbers n for which A328578(n) is less than A257993(n).
All terms are multiples of 6. The final digit {0, 2, 4, 6, 8} of the decimal representation seems to be quite evenly distributed.
Other multiples of six are in A328586 and A328589.
210 is the first term not present in A328632.

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

Union of A328632 \ {0} and A328762.
Positions of negative terms in A328590.
Cf. A257993, A276086, A328578, A328585, A328586, A328588.

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)));
isA328587(n) = (A328578(n) < A257993(n));

A328632 Numbers k such that A276086(k) == 1 (mod 6), where A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 12, 24, 30, 42, 54, 60, 72, 84, 90, 102, 114, 120, 132, 144, 150, 162, 174, 180, 192, 204, 216, 228, 246, 258, 276, 288, 306, 318, 336, 348, 366, 378, 396, 408, 420, 432, 444, 450, 462, 474, 480, 492, 504, 510, 522, 534, 540, 552, 564, 570, 582, 594, 600, 612, 624, 636, 648, 666, 678, 696, 708, 726, 738, 756, 768, 786, 798, 816

Offset: 1

Antti Karttunen, Oct 27 2019

nonn

Numbers k >= 0 for which A328578(k) = A257993(A276086(A276086(k))) = 2, where A276086 converts the primorial base expansion of k into its prime product form, and A257993 returns the index of the least prime not present in its argument. - The original, equivalent definition.
Numbers k for which A276087(k) is an even number, but not a multiple of three.
All terms are multiples of 6, and thus apart from the initial zero, this is a subsequence of A328587, numbers k for which A257993(A276086(A276086(k))) is less than A257993(k).

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

Row 2 of A328631.
After the initial zero, setwise difference A328587 \ A328762. Also setwise difference A008588 \ A358843.
Positions of 1's in A358840 and A358841 (characteristic function), positions of 2's in A328578.
Cf. A257993, A276086, A328578, A358845 (= a(n)/6).
Cf. also A328317.

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)));
isA328632(n) = (2==A328578(n));

isA326832(n) = A358841(n);
A358841(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (1==(m%6)); }; \\ Antti Karttunen, Dec 12 2022

{k | A358840(k) == 1}. - Antti Karttunen, Dec 02 2022

Definition replaced with a simpler one and the original definition moved to the comments section by Antti Karttunen, Dec 03 2022

A328589 Numbers n that are multiples of 6 and for which A257993(A276086(A276086(n))) is larger than A257993(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

240, 270, 300, 330, 360, 390, 630, 660, 690, 720, 750, 780, 810, 1050, 1080, 1110, 1140, 1170, 1200, 1230, 1500, 1530, 1560, 1590, 1620, 1650, 1890, 1920, 1950, 1980, 2010, 2040, 2070, 2550, 2580, 2610, 2640, 2670, 2700, 2940, 2970, 3000, 3030, 3060, 3090, 3120, 3360, 3390, 3420, 3450, 3480, 3510, 3540, 3810, 3840, 3870, 3900, 3930, 3960, 4200

Offset: 1

Antti Karttunen, Oct 21 2019

nonn

Multiples of six such that the least nondivisor prime of the original n is less than the least nondivisor prime of the number obtained after two iterations of A276086 is.

All terms are multiples of 5 (and thus of 30), because when applied to any number which is a multiple of 6, but not of 5 (and thus not a multiple of 30, so the primorial expansion ends with "x00", where x <> 0, and A257993(n) = 3), A276086 will yield a number of the form 30k+5 or 30k+25 (A084967) whose primorial expansion ends either as "...021" or as "...401" (with the least significant zero either in position 2 or 3), thus then A328578(n) = A257993(A276086(A276086(n))) is definitely not larger than 3, while A257993(6k) >= 3 for all k >= 1.

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

Subsequence of A328588.
Other multiples of 6 are either in A328586 or in A328587.
Cf. A049345, A084967, A257993, A276086, A328578, A328585, A328588, A328590, A328762.

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)));
isA328589(n) = ((!(n%6))&&(A328578(n) > A257993(n)));

A328633 Numbers n for which A328578(n) = A257993(A276086(A276086(n))) = 3, 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

2, 6, 18, 34, 36, 48, 66, 78, 96, 108, 122, 126, 138, 154, 156, 168, 186, 198, 212, 222, 234, 244, 252, 264, 282, 294, 312, 324, 332, 342, 354, 364, 372, 384, 402, 414, 422, 426, 438, 454, 456, 468, 486, 498, 516, 528, 542, 546, 558, 574, 576, 588, 606, 618, 632, 642, 654, 664, 672, 684, 702, 714, 732, 744, 752, 762, 774, 784, 792, 804

Offset: 1

Antti Karttunen, Oct 27 2019

nonn

Numbers n for which A276087(n) is a multiple of 6, but not of 5.
Question: Is the even bisection of A328316, starting from A328316(4) as: 6, 18, 43218, ..., a subsequence of this sequence? See also A328317.
Subsequence such that both k and A276087(k) are in this sequence starts as: 2, 6, 18, 34, 36, 48, 66, 154, 156, 186, 234, 244, 294, 312, 324, 354, 364, 384, 426, 438, 454, 456, 542, 546, 558, 588, 606, ...
When A276086 is applied to any number which is a multiple of 6, but not of 5 (and thus not a multiple of 30, implying that the number's primorial expansion ends with "x00", where x <> 0, and A257993(n) = 3), the original number will be converted to a number of the form 30k+5 or 30k+25 (A084967) whose primorial expansion ends either as "...021" or as "...401", with the least significant zero in position A328578(n), which is seen to be always either 3 or 2.

			294 = 7^2 * 3 * 2 has primorial base expansion (A049345) "12400", which, when converted to a prime product form (A276086) yields 11^1 * 7^2 * 5^4 * 3^0 * 2^0 = 336875. This in turn has primorial base representation [11,2,9,1,0,2,1], which when converted to prime product form gives 17^11 * 13^2 * 11^9 * 7^1 * 5^0 * 3^2 * 2^1 = 1720796647657111567992931482, which has the required property of being a multiple of 6 but not of 5, thus 294 is included in this sequence.

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

Row 3 of A328631.
Cf. A049345, A257993, A276086, A276087, A328578.
Cf. also A328316, A328317, A328589, A328762.

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)));
isA328633(n) = (3==A328578(n));

isA328633(n) = { my(u=A276086(A276086(n))); ((u%5)&&!(u%6)); };

cp's OEIS Frontend

A328587 Numbers n for which A257993(A276086(A276086(n))) is less than A257993(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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A328632 Numbers k such that A276086(k) == 1 (mod 6), where A276086 is the primorial base exp-function.

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A328589 Numbers n that are multiples of 6 and for which A257993(A276086(A276086(n))) is larger than A257993(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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A328633 Numbers n for which A328578(n) = A257993(A276086(A276086(n))) = 3, 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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