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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 1-4 of 4 results.

A249730 Odd nonprime terms in A250251.

Original entry on oeis.org

1, 9, 15, 25, 35, 49, 77, 121, 143, 147, 153, 169, 221, 289, 323, 361, 437, 529, 667, 841, 899, 961, 1147, 1369, 1517, 1681, 1763, 1849, 2021, 2209, 2491, 2809, 3127, 3481, 3599
Offset: 1

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Author

Antti Karttunen, Nov 18 2014

Keywords

Comments

This sequence lists the numbers which together with all their multiples with the powers of 2 give the starting values for the prime recurrences whose union is A250251, the fixed points of permutations A250249 and A250250.
After 1, contains the terms from columns 2 and 3 of the Sieve of Eratosthenes: A083140 (A083221), but only from those rows r for which A055396(r) (the index of smallest dividing prime r) is fixed by A250249 and A250250, i.e., is in A250251. The first r for which this is not the case is 73, which is in A249729 instead. However, because there are infinitely many primes in A250251, and especially because 2 and 3 are among them, this sequence is infinite.

Crossrefs

Subsequence of A250251.
Cf. A083140, A083221, A249821, A250249, A250250.

A250249 Permutation of natural numbers: a(1) = 1, a(n) = A083221(a(A055396(n)), a(A246277(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 39, 34, 35, 36, 37, 38, 63, 40, 41, 54, 43, 44, 33, 46, 47, 48, 49, 50, 75, 52, 53, 42, 65, 56, 99, 58, 59, 60, 61, 62, 57, 64, 95, 78, 67, 68, 111, 70, 71, 72, 103, 74, 51, 76, 77, 126, 79, 80, 45, 82
Offset: 1

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Author

Antti Karttunen, Nov 17 2014

Keywords

Comments

This is a "doubly-recursed" version of A249817.
For primes p_n, a(p_n) = p_{a(n)}.
The first 7-cycle occurs at: (33 39 63 57 99 81 45), which is mirrored by the cycle (66 78 126 114 198 162 90) with terms double the size and also by the cycle (137 167 307 269 523 419 197), consisting of primes (p_33, p_39, p_63, ...).

Examples

			For n = 42 = 2*3*7, we see that it occurs as the 21st term on the top row of A246278 (A055396(42) = 1 and A246277(42) = 21), recursing on both yields a(1) = 1, a(21) = 27, thus we find A083221(1,27), the 27th term on A083221's topmost row (also A005843) which is 54, thus a(42) = 54.
Examples for cases where n is a prime:
a(3709) = a(p_518) = p_{a(518)} = A000040(1162) = 9397.
a(3719) = a(p_519) = p_{a(519)} = A000040(1839) = 15767.

Links

Crossrefs

Inverse: A250250.
Fixed points: A250251, their complement: A249729.
Cf. A000035, A000040, A055396, A078442, A083221, A049076.
Differs from A250250 for the first time at n=33, where a(33) = 39, while A250250(33) = 45.
Differs from the "vanilla version" A249817 for the first time at n=42, where a(42) = 54, while A249817(42) = 42.

Formula

a(1) = 1, a(n) = A083221(a(A055396(n)), a(A246277(n))).
Other identities. For all n >= 1:
a(2n) = 2*a(n), or equally, a(n) = a(2n)/2. [The even bisection halved gives the sequence back].
a(p_n) = p_{a(n)}, or equally, a(n) = A049084(a(A000040(n))). [Restriction to primes induces the same sequence].
A078442(a(n)) = A078442(n), A049076(a(n)) = A049076(n). [Preserves the "order of primeness of n"].
A000035(n) = A000035(a(n)). [Preserves the parity].

A250250 Permutation of natural numbers: a(1) = 1, a(n) = A246278(a(A055396(n)),a(A078898(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 45, 34, 35, 36, 37, 38, 33, 40, 41, 54, 43, 44, 81, 46, 47, 48, 49, 50, 75, 52, 53, 42, 125, 56, 63, 58, 59, 60, 61, 62, 39, 64, 55, 90, 67, 68, 135, 70, 71, 72, 103, 74, 51, 76, 77, 66, 79, 80, 99, 82, 83
Offset: 1

Views

Author

Antti Karttunen, Nov 17 2014

Keywords

Comments

This is a "doubly-recursed" version of A249818.

Links

Crossrefs

Inverse: A250249.
Fixed points: A250251, their complement: A249729.
Cf. A000035, A000040, A055396, A078442, A078898, A049076, A246278.
See also other (somewhat) similar permutations: A245821, A057505.
Differs from the "vanilla version" A249818 for the first time at n=42, where a(42) = 54, while A249818(42) = 42.

Formula

a(1) = 1, a(n) = A246278(a(A055396(n)), a(A078898(n))).
Other identities. For all n >= 1:
a(2n) = 2*a(n), or equally, a(n) = a(2n)/2. [The even bisection halved gives the sequence back].
a(p_n) = p_{a(n)}, or equally, a(n) = A049084(a(A000040(n))). [Restriction to primes induces the same sequence].
A078442(a(n)) = A078442(n), A049076(a(n)) = A049076(n). [Preserves the "order of primeness of n"].
A000035(n) = A000035(a(n)). [Preserves the parity].

A249729 Numbers not fixed by A250249 and A250250.

Original entry on oeis.org

21, 27, 33, 39, 42, 45, 51, 54, 55, 57, 63, 65, 66, 69, 73, 75, 78, 81, 84, 85, 87, 90, 91, 93, 95, 99, 102, 103, 105, 108, 110, 111, 114, 115, 117, 119, 123, 125, 126, 129, 130, 132, 133, 135, 137, 138, 141, 145, 146, 150, 155, 156, 159, 161, 162, 165, 167, 168, 170, 171, 174, 175, 177, 180, 181, 182, 183, 185, 186, 187, 189, 190, 195, 197, 198, 201
Offset: 1

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Author

Antti Karttunen, Nov 18 2014

Keywords

Comments

Numbers for which A250249(n) <> n (equally: A250250(n) <> n).
If n is a member, then 2n is also a member. If any 2n is a member, then n is also a member. If n is a member, then the n-th prime, p_n (= A000040(n)) is also a member. If p_n is a member, then its index n is also a member. Thus the sequence is completely determined by its odd nonprime terms: 21, 27, 33, 39, 45, ..., and is obtained as a union of their multiples with powers of 2, and all prime recurrences that start with those values. For example, because 21 is present, then 2*21 = 42 is also present. Furthermore, 73 = p_21 is also present, as well as 367 = p_73 as well as 181 = p_42. See also comments at A250251 and A250249.

Crossrefs

Complement: A250251.
Cf. A000040, A250249, A250250.
Showing 1-4 of 4 results.

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