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.

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A364554 a(n) = number of primes of the form T(k)/n, for some k, where T(k)=A000217(k) is a triangular number.

Original entry on oeis.org

1, 2, 3, 1, 3, 2, 2, 1, 3, 1, 2, 2, 1, 2, 4, 1, 0, 2, 1, 1, 4, 2, 2, 2, 1, 2, 2, 0, 1, 3, 1, 0, 4, 1, 3, 2, 2, 1, 3, 2, 1, 2, 0, 1, 2, 0, 1, 2, 1, 1, 4, 1, 1, 3, 1, 1, 4, 1, 1, 3, 1, 0, 2, 1, 2, 1, 0, 2, 2, 2, 0, 0, 1, 1, 4, 1, 1, 2, 1, 0, 2, 1, 2, 2, 2, 2, 4, 0, 1, 4
Offset: 1

Views

Author

Lamine Ngom, Jul 28 2023

Keywords

Comments

Implementing a suggestion in the comment section of sequences A154296, ..., A154304, this sequence computes the number of primes of the form T(k)/n.
Equivalently, number of primes p such that 8*n*p+1 is a perfect square.
Let's consider, for all primes p, the set of linear recurrences {b(m)} defined as follows:
If p = 2, then {b(m)} = A074378 (numbers of the form x*(4*x -+ 1)); otherwise, b(m) = b(m-1) + 2*b(m-2) - 2*b(m-3) - b(m-4) + b(m-5) with initial terms b(0) = 0, b(1) = (p-1)/2, b(2) = (p+1)/2, b(3) = 2*p-1 and b(4) = 2*p+1. Numbers of the form x*(p*x -+ 1)/2.
Then a(n) = number of sequences {b(m)} in which n is a term.
This implies that:
i) for any n, the largest prime of the form T(k)/n is at most 2*n+1;
ii) if n is prime, then a(n) < 4. (3 and 5 are the only primes p such that a(p) = 3; primes p such that a(p) = 0 are A109998.)
Deeper in the examination of these results, we notice that the set of primes p of the form T(k)/n arises from the factorization of n. This set is exactly all primes p of the form (2*r -+ 1)/d or (r -+ 1)/(2*d), where d is some divisor of n and r is the ratio n/d. (Proof is welcome.)
Indices k of corresponding triangular numbers T(k) such that T(k) = n*p are then:
2*r if p = (2*r + 1)/d,
2*r - 1 if p = (2*r - 1)/d,
r if p = (r + 1)/(2*d),
r - 1 if p = (r - 1)/(2*d).
And pluging the value of p in the equivalent definition, the expression 8*n*p+1 yields respectively to following perfect squares: (4*r+1)^2, (4*r-1)^2, (2*r+1)^2 and (2*r-1)^2.

Examples

			a(15) = 4 since there are exactly 4 triangular numbers T(k) such that T(k) = 15*p, with p prime.
T(9)/15 = 45/15 = 3, T(14)/15 = 105/15 = 7, T(29)/15 = 435/15 = 29 and T(30)/15 = 465/15 = 31.
a(17) = 0 since there is no triangular number T(k) such that T(k) = 17*p, with p prime.

Crossrefs

Cf. A000217, A154296, ..., A154304, A074378, A001318, A057569, A057570, A109998.
Cf. A364555 (indices of 0's).

Formula

Conjecture: a(n) = number of primes in the union of sets {(2*r -+ 1)/d; (r -+ 1)/(2*d)}, with d divisor of n and r = n/d.

A364555 Numbers m such that no triangular number is m times a prime.

Original entry on oeis.org

17, 28, 32, 43, 46, 62, 67, 71, 72, 80, 88, 94, 101, 103, 104, 108, 109, 110, 118, 122, 124, 127, 130, 137, 144, 148, 149, 151, 152, 161, 162, 163, 170, 171, 172, 178, 181, 185, 188, 196, 197, 202, 206, 208, 212, 214, 218, 223, 226, 236, 238, 241, 242, 256, 257, 258
Offset: 1

Views

Author

Lamine Ngom, Jul 28 2023

Keywords

Comments

Numbers m such that A364554(m) = 0.
Primes in sequence are A109998.
Conjecture: Numbers m such that there is no prime number in the union of all sets {(2*r -+ 1)/d; (r -+ 1)/(2*d)}, where d is some divisor of m and r = m/d.

Examples

			17 is a term since there isn't any triangular number T(k) such that T(k) = 17*p, with p prime.
28 is a term since there isn't any triangular number T(k) such that T(k) = 28*p, with p prime.

Crossrefs

Cf. A000217, A109998, A364554.

A110089 Smallest prime beginning (through <*2+1> or/and <*2-1>) a complete Cunningham chain (of the first or the second kind) of length n.

Original entry on oeis.org

11, 3, 2, 509, 2, 89, 16651, 15514861, 85864769, 26089808579, 665043081119, 554688278429, 758083947856951, 95405042230542329, 69257563144280941
Offset: 1

Views

Author

Alexandre Wajnberg, Sep 04 2005

Keywords

Comments

The word "complete" indicates each chain is exactly n primes long for the operator in function (i.e. the chain cannot be a subchain of another one); and the first and/or last term may be involved in a chain of the other kind (i.e. the chain may be connected to another one). a(1)-a(8) computed by Gilles Sadowski.

Examples

			a(1)=11 because 2, 3, 5 and 7 are included in longer chains than one prime long; and 11 (although included in a <2p+1> chain) has no prime connection through <2p-1>.
a(2)=3 because 3 begins (through 2p+1>) the first complete two primes chain: 3-> 7 (even if 3 and 7 are also part of two others chains, but through <2p-1>).
a(3)=2 because (although 2 begins also a five primes chain through <2p+1>) it begins, through <2p-1>, the first complete three primes chain encountered: 2->3->5.

Links

Crossrefs

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059455, A007700, A059759, A059760, A059761, A059762, A059763, A059764, A059765, A038397, A104349, A091314, A069362, A016093, A014937, A057326, A110059, A110056, A110038, A059766, A110027, A059764, A110025, A110024, A059763, A110022, A109998, A109946, A109927, A109835, A005603.

Formula

a(n) = min(A005602(n), A005603(n)). - R. J. Mathar, Jul 23 2008

Extensions

a(8)-a(13) via A005602, A005603 from R. J. Mathar, Jul 23 2008
a(14)-a(15) via A005602, A005603 from Jason Yuen, Sep 03 2024

A110092 Smallest prime ending (through <*2+1> or <*2-1> separately) a complete Cunningham chain (of the first or the second kind) of length n.

Original entry on oeis.org

17, 59, 73, 4079, 47, 2879, 1065601
Offset: 1

Views

Author

Alexandre Wajnberg, Sep 04 2005

Keywords

Comments

The word "complete" indicates each chain is exactly n primes long for the operator in function (i.e. the chain cannot be a subchain of another one); and the first and/or last term may not be involved in a chain of the other kind (i.e. the chain may not be connected to another one).

Examples

			a(1)=17 because 2, 3, 5, 7, 11 and 13 are part of longer chains whatever the operator; 17 is the first completely isolated prime.
a(2)=59 because it ends the first two primes chain not connected to another one: 29->59.

Links

Crossrefs

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059455, A007700, Cf. A059759, A059760, A059761, A059762, A059763, A059764, A059765, A038397, A104349, A091314, A069362, A016093, A014937, A057326, A110059, A110056, A110038, A059766, A110027, A059764, A110025, A110024, A059763, A110022, A109998, A109946, A109927, A109835, A005603.

Extensions

Terms computed by Gilles Sadowski.

A110093 Smallest prime ending (through <*2+1> or/and <*2-1>) a complete Cunningham chain (of the first or the second kind) of length n.

Original entry on oeis.org

11, 7, 5, 4079, 47, 2879, 1065601
Offset: 1

Views

Author

Alexandre Wajnberg, Sep 04 2005

Keywords

Comments

The word "complete" indicates each chain is exactly n primes long for the operator in function (i.e. the chain cannot be a subchain of another one); but the first and/or last term may be involved in a chain of the other kind (i.e. the chain may be connected to another one).

Examples

			a(1)=11 because 2, 3, 5 and 7 are not ending chains; or are part of chains longer than one prime; 11, although is part of a five primes <2p+1> chain, is isolated through <2p-1>.
a(2)=7 because 7 ends through <2p+1> the first two primes chain: 3->7 (even if both primes are also part of <2p-1> chains).

Links

Crossrefs

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059455, A007700, Cf. A059759, A059760, A059761, A059762, A059763, A059764, A059765, A038397, A104349, A091314, A069362, A016093, A014937, A057326, A110059, A110056, A110038, A059766, A110027, A059764, A110025, A110024, A059763, A110022, A109998, A109946, A109927, A109835, A005603.

Extensions

Terms computed by Gilles Sadowski.
Showing 1-5 of 5 results.

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