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 1-9 of 9 results.

A154296 Primes of the form (1+2+3+...+m)/15 = A000217(m)/15, for some m.

Original entry on oeis.org

3, 7, 29, 31
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 06 2009

Keywords

Comments

Original definition: Primes of the form 1/x+2/x+3/x+4/x+5/x+6/x+7/x+..., x=15.
The corresponding m-values are m=9, 14, 29, 30. It is clear that for m > 30, T(m)/15 = m*(m+1)/30 cannot be a prime. - M. F. Hasler, Dec 31 2012
All of the sequences A154296, ..., A154304 could or should be grouped together in a single ("fuzzy"?) table. It would be more interesting to have the function f(n) which gives the *number* of primes of the form T(k)/n. - M. F. Hasler, Jan 06 2013
Also primes p such that 120*p+1 is a perfect square. - Lamine Ngom, Jul 22 2023

Crossrefs

Cf. A057570, A154293, A154297, A154298, A154299, A154300, A154301, A154302, A154303, A154304.

Programs

  • Mathematica
    lst={};s=0;Do[s+=n/15;If[Floor[s]==s,If[PrimeQ[s],AppendTo[lst,s]]],{n,0,9!}];lst
Select[(Accumulate[Range[200]])/15,PrimeQ] (* Harvey P. Dale, Oct 30 2011 *)
  • PARI
    select(x->denominator(x)==1 & isprime(x), vector(30,m,m^2+m)/30)  \\ M. F. Hasler, Dec 31 2012

Extensions

Edited by M. F. Hasler, Dec 31 2012

A154303 Primes of the form (1+2+...+m)/90 = A000217(m)/90.

Original entry on oeis.org

7, 11, 179, 181
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 06 2009

Keywords

Comments

Original definition: "Primes of the form : 1/x+2/x+3/x+4/x+5/x+6/x+7/x+..., x=90."
The corresponding m-values are m=35,44,179,180. It is clear that for m>180, T(m)/90 = m(m+1)/180 cannot be a prime, since then each factor in the numerator is larger than the denominator. All of the sequences A154296, ..., A154304 could or should be grouped together in a single ("fuzzy"?) table. It would be more interesting to have the function f(n) which gives the *number* of primes of the form T(k)/n. - M. F. Hasler, Jan 06 2013

Crossrefs

Cf. A057570, A154293, A154296 - A154304.

Programs

  • Mathematica
    lst={};s=0;Do[s+=n/90;If[Floor[s]==s,If[PrimeQ[s],AppendTo[lst,s]]],{n,0,5*9!}];lst
  • PARI
    d=90*2;for(m=1,999,(m^2+m)%d==0&isprime((m^2+m)/d)&print1(m",")) \\ print the m-values(!) - use A154304(90) to get A154303 as a list/vector. \\ - M. F. Hasler, Jan 06 2013

Extensions

Edited by M. F. Hasler, Jan 06 2013

A154297 Primes of the form (1+2+3+...+m)/21 = A000217(m)/21, for some m.

Original entry on oeis.org

5, 11, 41, 43
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 06 2009

Keywords

Comments

This asks for primes p which are a triangular number divided by 21, or, 2*3*7*p=k*(k+1) for some k. Matching factors shows that the sequence is complete [R. J. Mathar, Aug 15 2010]
Original definition: Primes of the form : 1/x+2/x+3/x+4/x+5/x+6/x+7/x+..., x=21.
The corresponding m-values are m=14, 21, 41, 42 (cf. A154296). It is clear that for m>42, A000217(m)/21 = m(m+1)/42 cannot be a prime. - M. F. Hasler, Dec 31 2012

Crossrefs

Cf. A057570, A154293, A154296, A154298, A154299, A154300, A154301, A154302, A154303, A154304.

Programs

  • Mathematica
    lst={};s=0;Do[s+=n/21;If[Floor[s]==s,If[PrimeQ[s],AppendTo[lst,s]]],{n,0,9!}];lst
#/21&/@Select[Accumulate[Range[100]],PrimeQ[#/21]&] (* Harvey P. Dale, Dec 17 2012 *)
  • PARI
    select(x->denominator(x)==1 & isprime(x), vector(42, m, m^2+m)/42)  \\ - M. F. Hasler, Dec 31 2012

Extensions

Added keywords fini,full - R. J. Mathar, Aug 15 2010
Edited by M. F. Hasler, Dec 31 2012

A154300 Primes of the form (1+2+...+m)/57 = A000217(m)/57.

Original entry on oeis.org

3, 13, 29, 113
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 06 2009

Keywords

Comments

Original definition: "Primes of the form : 1/x+2/x+3/x+4/x+5/x+6/x+7/x+..., x=57."
Primes which are some triangular number divided by 57. Finiteness of the sequence follows along the reasoning in A154297.
The corresponding m-values are m=18,38,57,113. It is clear that for m>2*57, T(m)/57 = m(m+1)/114 cannot be a prime, since then each factor in the numerator is larger than the denominator. See A154304 for further comments and PARI code. - M. F. Hasler, Jan 06 2013

Crossrefs

Cf. A057570, A154293, A154296 - A154304.

Programs

  • Mathematica
    lst={};s=0;Do[s+=n/57;If[Floor[s]==s,If[PrimeQ[s],AppendTo[lst,s]]],{n,0,2*9!}];lst
Select[Accumulate[Range[1000]]/57,PrimeQ] (* Harvey P. Dale, Jun 24 2015 *)
  • PARI
    d=57*2;for(m=1,999,(m^2+m)%d==0&isprime((m^2+m)/d)&print1(m",")) \\ print the m-values(!) - use A154304(57) to get A154300 as a vector. \\ - M. F. Hasler, Jan 06 2013

Extensions

Keywords fini,full added by R. J. Mathar, Aug 15 2010
Edited by M. F. Hasler, Jan 06 2013

A154301 Primes of the form (1+2+...+m)/75 = A000217(m)/75.

Original entry on oeis.org

17, 37, 149, 151
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 06 2009

Keywords

Comments

Original definition: "Primes of the form : 1/x+2/x+3/x+4/x+5/x+6/x+7/x+..., x=75."
The corresponding m-values are m=50,74,149,150. It is clear that for m>2*75, T(m)/75 = m(m+1)/150 cannot be a prime, since then each factor in the numerator is larger than the denominator. See A154304 for further comments and PARI code. - M. F. Hasler, Jan 06 2013

Crossrefs

Cf. A057570, A154293, A154296 - A154304.

Programs

  • Mathematica
    lst={};s=0;Do[s+=n/75;If[Floor[s]==s,If[PrimeQ[s],AppendTo[lst,s]]],{n,0,3*9!}];lst
  • PARI
    d=75*2;for(m=1,999,(m^2+m)%d==0&isprime((m^2+m)/d)&print1(m",")) \\ print the m-values(!) - use A154304(75) to get A154301 as a vector. \\ - M. F. Hasler, Jan 06 2013

Extensions

Keywords fini,full added by R. J. Mathar, Aug 15 2010
Edited by M. F. Hasler, Jan 06 2013

A154302 Primes of the form (1+2+...+m)/87 = A000217(m)/87.

Original entry on oeis.org

5, 19, 43, 173
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 06 2009

Keywords

Comments

Original definition: "Primes of the form : 1/x+2/x+3/x+4/x+5/x+6/x+7/x+..., x=87."
The corresponding m-values are m=29,57,86,173. It is clear that for m>2*87, T(m)/87 = m(m+1)/174 cannot be a prime, since then each factor in the numerator is larger than the denominator. See A154304 for further comments and PARI code. - M. F. Hasler, Jan 06 2013

Crossrefs

Cf. A057570, A154293, A154296 - A154304.

Programs

  • Mathematica
    lst={};s=0;Do[s+=n/87;If[Floor[s]==s,If[PrimeQ[s],AppendTo[lst,s]]],{n,0,4*9!}];lst
  • PARI
    d=87*2;for(m=1,999,(m^2+m)%d==0&isprime((m^2+m)/d)&print1(m",")) \\ print the m-values(!) - use A154304(87) to get A154302 as a vector. \\ - M. F. Hasler, Jan 06 2013

Extensions

Edited by M. F. Hasler, Jan 06 2013

A154298 Primes of the form (1+...+m)/33 = A000217(m)/33, for some m.

Original entry on oeis.org

2, 7, 17, 67
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 06 2009

Keywords

Comments

Primes which are some triangular number A000217 divided by 33. Finiteness is shown with the same strategy as in A154297.
Original definition: Primes of the form : 1/x+2/x+3/x+4/x+5/x+6/x+7/x+..., x=33.
The corresponding m-values are m=11, 21, 33, 66 (cf. A154296). It is clear that for m>66, A000217(m)/33 = m(m+1)/66 cannot be a prime. - M. F. Hasler, Dec 31 2012

Crossrefs

Cf. A057570, A154293, A154296 - A154304.

Programs

  • Mathematica
    lst={};s=0;Do[s+=n/33;If[Floor[s]==s,If[PrimeQ[s],AppendTo[lst,s]]],{n,0,9!}];lst
Select[Accumulate[Range[200]]/33,PrimeQ] (* Harvey P. Dale, Aug 11 2025 *)
  • PARI
    select(x->denominator(x)==1 & isprime(x), vector(66, m, m^2+m)/66)  \\ - M. F. Hasler, Dec 31 2012

Extensions

Edited by M. F. Hasler, Dec 31 2012

A154299 Primes of the form (1+...+m)/51 = A000217(m)/51.

Original entry on oeis.org

3, 11, 101, 103
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 06 2009

Keywords

Comments

Primes p of the form k*(k+1)/(2*51): There is a finite set of solutions to 2*3*17*p=k*(k+1). - R. J. Mathar, Aug 15 2010
Original definition: "Primes of the form : 1/x+2/x+3/x+4/x+5/x+6/x+7/x+..., x=51."
The corresponding m-values are m=17, 33, 101, 102 (cf. A154296-A154304, see the latter for more comments and PARI code). It is clear that for m>102, A000217(m)/51 = m(m+1)/102 has at least 2 factors and hence cannot be prime. - M. F. Hasler, Dec 31 2012

Crossrefs

Cf. A057570, A154293, A154296 - A154304.

Programs

  • Mathematica
    lst={};s=0;Do[s+=n/51;If[Floor[s]==s,If[PrimeQ[s],AppendTo[lst,s]]],{n,0,9!}];lst
Select[Accumulate[Range[1000]/51],PrimeQ] (* Harvey P. Dale, Jun 21 2012 *)
  • PARI
    M=51*2;select(x->denominator(x)==1 & isprime(x), vector(M, m, m^2+m)/M)  \\ M. F. Hasler, Dec 31 2012

Extensions

Keywords fini,full added by R. J. Mathar, Aug 15 2010

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.
Showing 1-9 of 9 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.