cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A108352 a(n) = primal code characteristic of n, which is the least positive integer, if any, such that (n o)^k = 1, otherwise equal to 0. Here "o" denotes the primal composition operator, as illustrated in A106177 and A108371 and (n o)^k = n o ... o n, with k occurrences of n.

Original entry on oeis.org

1, 0, 2, 2, 2, 0, 2, 2, 0, 0, 2, 0, 2, 0, 2, 2, 2, 0, 2, 3, 2, 0, 2, 3, 2, 0, 2, 3, 2, 0, 2, 2, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 3, 0, 0, 2, 3, 2, 0, 2, 3, 2, 0, 2, 3, 2, 0, 2, 0, 2, 0, 0, 2, 2, 0, 2, 3, 2, 0, 2, 0, 2, 0, 3, 3, 2, 0, 2, 3, 2, 0, 2, 0, 2, 0, 2, 3, 2, 0, 2, 3, 2, 0, 2, 3, 2, 0, 0, 2, 2, 0, 2, 3, 2
Offset: 1

Views

Author

Jon Awbrey, May 31 2005, revised Jun 01 2005

Keywords

Examples

			a(1) = 1 because (1 o)^1 = ({ } o)^1 = 1.
a(2) = 0 because (2 o)^k = (1:1 o)^k = 2, for all positive k.
a(3) = 2 because (3 o)^2 = (2:1 o)^2 = 1.
a(4) = 2 because (4 o)^2 = (1:2 o)^2 = 1.
a(5) = 2 because (5 o)^2 = (3:1 o)^2 = 1.
a(6) = 0 because (6 o)^k = (1:1 2:1 o)^k = 6, for all positive k.
a(7) = 2 because (7 o)^2 = (4:1 o)^1 = 1.
a(8) = 2 because (8 o)^2 = (1:3 o)^1 = 1.
a(9) = 0 because (9 o)^k = (2:2 o)^k = 9, for all positive k.
a(10) = 0 because (10 o)^k = (1:1 3:1 o)^k = 10, for all positive k.
Detail of calculation for compositional powers of 12:
(12 o)^2 = (1:2 2:1) o (1:2 2:1) = (1:1 2:2) = 18
(12 o)^3 = (1:1 2:2) o (1:2 2:1) = (1:2 2:1) = 12
Detail of calculation for compositional powers of 20:
(20 o)^2 = (1:2 3:1) o (1:2 3:1) = (3:2) = 25
(20 o)^3 = (3:2) o (1:2 3:1) = 1.
From _Antti Karttunen_, Nov 20 2019: (Start)
For n=718, because 718 = prime(1)^1 * prime(72)^1, its partial function primal code is (1:1 72:1), which, when composed with itself stays same (that is, A106177(718,718) = 718), thus, as 1 is never reached, a(718) = 0, like is true for all even nonsquare semiprimes.
For n=1804, as 1804 = prime(1)^2 * prime(5)^1 * prime(13)^1, its primal code is (1:2 5:1 13:1), which, when composed with itself yields 203401 = prime(5)^2 * prime(13)^2, i.e., primal code (5:2 13:2), which when composed with (1:2 5:1 13:1) yields 1, which happened on the second iteration, thus a(1804) = 2+1 = 3.
(End)

Links

Crossrefs

Cf. A001747, A046315, A055231, A061396, A062504, A062537, A062860, A065091, A106177, A106178.
Cf. A108353, A108370, A108371, A108372, A108373, A108374, A111801.

Programs

  • PARI
    A106177sq(n,k) = { my(f = factor(k)); prod(i=1,#f~,f[i, 1]^valuation(n, prime(f[i, 2]))); }; \\ As in A106177.
A108352(n) = { my(orgn=n,xs=Set([]), k=1); while(n>1, if(vecsearch(xs,n), return(0)); xs = setunion([n],xs); n = A106177sq(n,orgn); k++); (k); }; \\ Antti Karttunen, Nov 20 2019

Formula

a(A065091(n)) = 2 for all n, a(A001747(n)) = 0 for all n, except n=2, and a(A046315(n)) = 2 for n > 1. - Antti Karttunen, Nov 20 2019

Extensions

Links and cross-references added, Aug 19 2005
Term a(63) corrected and five more terms added (up to a(105)) by Antti Karttunen, Nov 20 2019

A108370 Numbers whose primal code characteristic = 0, that is, positive n for which A108352(n) = 0.

Original entry on oeis.org

2, 6, 9, 10, 12, 14, 18, 22, 26, 30, 34, 36, 38, 40, 42, 45, 46, 50, 54, 58, 60, 62, 66, 70, 72, 74, 78, 82, 84, 86, 90, 94, 98, 99, 102, 106, 110, 112, 114, 117, 118, 120, 122, 125, 126, 130, 132, 134, 138, 142, 144, 146, 150, 153, 154, 156, 158, 162, 166, 170, 171
Offset: 1

Views

Author

Jon Awbrey, Jun 01 2005

Keywords

Links

Crossrefs

Cf. A108352, A108371, A108372, A108373, A108374.

A108374 Numbers whose primal code characteristic = 4, that is, positive n for which A108352(n) = 4.

Original entry on oeis.org

756, 1176, 1188, 1200, 1400, 1404, 1620, 1836, 2052, 2160, 2200, 2400, 2484, 2600, 2904
Offset: 1

Views

Author

Jon Awbrey, Jun 16 2005, extended Jul 10 2005

Keywords

Examples

			Writing (prime(i))^j as i:j, we have the following table:
Primal Functions and Functional Digraphs for a(1) to a(15)
0756 = 1:2 2:3 4:1 || 4 -> 1 -> 2 -> 3
1176 = 1:3 2:1 4:2 || 4 -> 2 -> 1 -> 3
1188 = 1:2 2:3 5:1 || 5 -> 1 -> 2 -> 3
1200 = 1:4 2:1 3:2 || 3 -> 2 -> 1 -> 4
1400 = 1:3 3:2 4:1 || 4 -> 1 -> 3 -> 2
1404 = 1:2 2:3 6:1 || 6 -> 1 -> 2 -> 3
1620 = 1:2 2:4 3:1 || 3 -> 1 -> 2 -> 4
1836 = 1:2 2:3 7:1 || 7 -> 1 -> 2 -> 3
2052 = 1:2 2:3 8:1 || 8 -> 1 -> 2 -> 3
2160 = 1:4 2:3 3:1 || 2 -> 3 -> 1 -> 4
2200 = 1:3 3:2 5:1 || 5 -> 1 -> 3 -> 2
2400 = 1:5 2:1 3:2 || 3 -> 2 -> 1 -> 5
2484 = 1:2 2:3 9:1 || 9 -> 1 -> 2 -> 3
2600 = 1:3 3:2 6:1 || 6 -> 1 -> 3 -> 2
2904 = 1:3 2:1 5:2 || 5 -> 2 -> 1 -> 3

Links

Crossrefs

Cf. A108352, A108353, A108370, A108371, A108372, A108373.

A108371 Table of primal compositional powers (n o)^k, where "o" denotes the primal composition operator, as illustrated in sequence A106177 and where (n o)^k = n o ... o n, with k occurrences of n.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 1, 2, 1, 5, 1, 1, 2, 1, 6, 1, 1, 1, 2, 1, 7, 6, 1, 1, 1, 2, 1, 8, 1, 6, 1, 1, 1, 2, 1, 9, 1, 1, 6, 1, 1, 1, 2, 1, 10, 9, 1, 1, 6, 1, 1, 1, 2, 1, 11, 10, 9, 1, 1, 6, 1, 1, 1, 2, 1, 12, 1, 10, 9, 1, 1, 6, 1, 1, 1, 2, 1, 13, 18, 1, 10, 9, 1, 1, 6, 1, 1, 1, 2, 1, 14, 1, 12, 1, 10, 9, 1, 1, 6
Offset: 1

Views

Author

Jon Awbrey, Jun 07 2005

Keywords

Examples

			Table: T(n,k) = (n o)^k
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `T(n,k)
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\ /
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 1 . 1
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\ / \ /
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 2 . 1 . 2
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\ / \ / \ /
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 3 . 2 . 1 . 3
` ` ` ` ` ` ` ` ` ` ` ` ` ` `\ / \ / \ / \ /
` ` ` ` ` ` ` ` ` ` ` ` ` ` 4 . 3 . 2 . 1 . 4
` ` ` ` ` ` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ /
` ` ` ` ` ` ` ` ` ` ` ` ` 5 . 4 . 1 . 2 . 1 . 5
` ` ` ` ` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ /
` ` ` ` ` ` ` ` ` ` ` ` 6 . 5 . 1 . 1 . 2 . 1 . 6
` ` ` ` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` ` ` ` ` ` ` 7 . 6 . 1 . 1 . 1 . 2 . 1 . 7
` ` ` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` ` ` ` ` ` 8 . 7 . 6 . 1 . 1 . 1 . 2 . 1 . 8
` ` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` ` ` ` ` 9 . 8 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 9
` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` ` ` `10 . 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 10
` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` ` `11 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 11
` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` `12 . 11. 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 12
` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` `13 . 12. 1 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 13
` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` `14 . 13. 18. 1 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 14
` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` `15 . 14. 1 . 12. 1 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 15
` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` `16 . 15. 14. 1 . 18. 1 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 16

Links

Crossrefs

Cf. A061396, A062537, A106177, A106178, A108352, A108370, A108372, A108373.

A108353 For each nonnegative integer n, a(n) is the smallest positive integer j whose primal code characteristic is n, that is, the smallest j such that A108352(j) = n.

Original entry on oeis.org

2, 1, 3, 20, 756, 178200
Offset: 0

Views

Author

Jon Awbrey, Jun 17 2005, extended Aug 20 2005

Keywords

Comments

Suggested by Antti Karttunen.

Examples

			Writing (prime(i))^j as i:j, we have the following table:
Primal Functions and Functional Digraphs for a(0) to a(5)
` ` ` 2 = 1:1 ` ` ` ` ` ` || 1 -> 1 (infinite loop)
` ` ` 1 = { } ` ` ` ` ` ` || 1
` ` ` 3 = 2:1 ` ` ` ` ` ` || 2 -> 1
` ` `20 = 1:2 3:1 ` ` ` ` || 3 -> 1 -> 2
` ` 756 = 1:2 2:3 4:1 ` ` || 4 -> 1 -> 2 -> 3
`178200 = 1:3 2:4 3:2 5:1 || 5 -> 1 -> 3 -> 2 -> 4

Links

Crossrefs

Cf. A106177, A108352, A108370, A108371, A108372, A108373, A108374, A111801.

A111801 Numbers that have a positive primal code characteristic, that is, positive integers j for which A108352(j) > 0.

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 11, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 37, 39, 41, 43, 44, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 61, 63, 64, 65, 67, 68, 69, 71, 73, 75, 76, 77, 79, 80, 81, 83, 85, 87, 88, 89, 91, 92, 93, 95, 96, 97, 100, 101, 103, 104, 105
Offset: 1

Views

Author

Jon Awbrey, Aug 19 2005

Keywords

Comments

This set of numbers is the complement of the set in A108370.

Links

Crossrefs

Cf. A106177, A108352, A108370, A108371, A108372, A108373, A108374.

A108372 Numbers whose primal code characteristic = 2, that is, positive n for which A108352(n) = 2.

Original entry on oeis.org

3, 4, 5, 7, 8, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 63, 64, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 100, 101, 103, 105, 107, 109, 111, 113, 115, 119, 121, 123, 127, 128, 129, 131
Offset: 1

Views

Author

Jon Awbrey, Jun 01 2005

Keywords

Links

Crossrefs

Cf. A108352, A108370, A108371, A108373, A108374.
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.