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.

A145296 Smallest k such that k^2 + 1 is divisible by A002144(n)^3.

Original entry on oeis.org

57, 239, 1985, 10133, 9466, 11389, 27590, 51412, 153765, 344464, 107551, 296344, 172078, 432436, 931837, 753090, 676541, 2321221, 2027724, 3394758, 1706203, 4841182, 1438398, 2947125, 398366, 5657795, 4942017, 9400802, 11906503
Offset: 1

Views

Author

Klaus Brockhaus, Oct 08 2008

Keywords

Examples

			a(3) = 1985 since A002144(3) = 17, 1985^2 + 1 = 3940226 = 2*17^3*401 and for no k < 1985 does 17^3 divide k^2+1.

Links

Crossrefs

Cf. A002144 (primes of form 4n+1), A002313 (-1 is a square mod p), A059321, A145297, A145298, A145299.

Programs

  • PARI
    {m=12000000; pmax=300; z=70; v=vector(z); for(n=1, m, fac=factor(n^2+1); for(j=1, #fac[, 1], if(fac[j, 2]>=3&&fac[j, 1]<=pmax, q=primepi(fac[j, 1]); if(q<=z&&v[q]==0, v[q]=n)))); t=1; j=0; while(t&&j
  • PARI
    {e=3; forprime(p=2, 300, if(p%4==1, q=p^e; m=q; while(!ispower(m-1,2,&n), m=m+q); print1(n, ",")))} \\ Klaus Brockhaus, Oct 09 2008
  • Python
    from itertools import islice
from sympy import nextprime, sqrt_mod_iter
def A145296_gen(): # generator of terms
    p = 1
    while (p:=nextprime(p)):
        if p&3==1:
            yield min(sqrt_mod_iter(-1,p**3))
A145296_list = list(islice(A145296_gen(),20)) # Chai Wah Wu, May 04 2024

A145297 Smallest k such that k^2+1 is divisible by A002144(n)^4.

Original entry on oeis.org

182, 239, 27493, 34522, 800982, 1251967, 623098, 6304056, 6459524, 20099637, 22709274, 35764191, 40317977, 54397650, 166206108, 187800003, 165728858, 152475014, 282599844, 312923750, 154613663, 485200742, 912190662, 548850444
Offset: 1

Views

Author

Klaus Brockhaus, Oct 11 2008

Keywords

Examples

			a(1) = 182 since A002144(1) = 5, 182^2+1 = 33125 = 5^4*53 and for no k < 182 does 5^4 divide k^2+1.

Links

Crossrefs

Cf. A002144 (primes of form 4n+1), A002313 (-1 is a square mod p), A059321, A145296, A145298, A145299.

Programs

  • PARI
    {e=4; forprime(p=2, 250, if(p%4==1, q=p^e; m=q; while(!ispower(m-1,2,&n), m=m+q); print1(n, ",")))}
  • Python
    from itertools import islice
from sympy import nextprime, sqrt_mod_iter
def A145297_gen(): # generator of terms
    p = 1
    while (p:=nextprime(p)):
        if p&3==1:
            yield min(sqrt_mod_iter(-1,p**4))
A145297_list = list(islice(A145297_gen(),20)) # Chai Wah Wu, May 04 2024

A145298 Smallest k such that k^2+1 is divisible by A002144(n)^5.

Original entry on oeis.org

1068, 143044, 390112, 7745569, 6423465, 46464143, 23048345, 144762466, 404034898, 2153335831, 331407850, 1108900220, 2581164875, 760839155, 10734466938, 6595297216, 773302059, 61063137802, 31915893786, 112699451831
Offset: 1

Views

Author

Klaus Brockhaus, Oct 14 2008

Keywords

Examples

			a(4) = 7745569 since A002144(4) = 29, 7745569^2+1 = 59993839133762 = 2*29^5*97*15077 and for no k < 7745569 does 29^5 divide k^2+1.

Links

Crossrefs

Cf. A002144 (primes of form 4n+1), A002313 (-1 is a square mod p), A059321, A145296, A145297, A145299.

Programs

  • PARI
    {e=5; forprime(p=2, 200, if(p%4==1, q=p^e; m=q; while(!ispower(m-1,2,&n), m=m+q); print1(n, ",")))}
  • Python
    from itertools import islice
from sympy import nextprime, sqrt_mod_iter
def A145298_gen(): # generator of terms
    p = 1
    while (p:=nextprime(p)):
        if p&3==1:
            yield min(sqrt_mod_iter(-1,p**5))
A145298_list = list(islice(A145298_gen(),20)) # Chai Wah Wu, May 04 2024

A143082 Longer side (k) of bordered rectangle described in A168339.

Original entry on oeis.org

3, 5, 5, 6, 5, 6, 7, 7, 8, 7, 9, 8, 7, 9, 8, 10, 9, 9, 10, 10, 9, 11, 11, 10, 9, 11, 11, 10, 12, 12, 11, 11, 13, 12, 12, 11, 13, 13, 13, 12, 11, 14, 13, 13, 12, 15, 14, 14, 13, 13, 15, 15, 17, 14, 13, 16, 15, 15, 15, 14, 13, 16, 16, 15, 15, 14, 17, 17, 16, 16, 15, 15, 17, 17, 17, 16
Offset: 1

Views

Author

R. H. Hardin and N. J. A. Sloane, Nov 27 2009

Keywords

Crossrefs

Cf. A145299, A168339. Equals A161357 - 2.

Formula

a(n) = A161357(n)+2. [From R. J. Mathar, Mar 06 2010]

Extensions

Extended with output of the program in A168339, variable dimn+2, by R. J. Mathar, Mar 06 2010

A145871 Smallest k such that k^2+1 is divisible by A002144(n)^7.

Original entry on oeis.org

32318, 6826318, 96940388, 7986582530, 24900904028, 92615568742, 416081467190, 988322434636, 3219884218827, 4867146503697, 26457926739667, 47023298541694, 26661771973542, 90980209992989, 257680081342861, 283410689912607
Offset: 1

Views

Author

Klaus Brockhaus, Oct 22 2008

Keywords

Examples

			a(2) = 6826318 since A002144(2) = 13, 6826318^2+1 = 46598617437125 = 5^3*13^7*13*457 and for no k < 6826318 does 13^7 divide k^2+1. a(4) = 7986582530 since A002144(4) = 29, 7986582530^2+1 = 63785500508501200901 = 29^7*197*409*45893 and for no k < 7986582530 does 29^7 divide k^2+1.

Crossrefs

Cf. A002144 (primes of form 4n+1), A002313 (-1 is a square mod p), A059321, A145296, A145297, A145298, A145299, A145872, A145873.

Programs

  • PARI
    {e=7; forprime(p=2, 40, if(p%4==1, q=p^e; m=q; while(!issquare(m-1, &n), m=m+q); print1(n, ",")))}

Extensions

More terms from Klaus Brockhaus, Nov 12 2008

A145872 Smallest k such that k^2+1 is divisible by A002144(n)^8.

Original entry on oeis.org

110443, 6826318, 3379649772, 61012922706, 1019349744435, 287369842623, 11331029931180, 71294762793847, 239822883201307, 923990886302412, 2369608176604944, 3156215819652023, 521749964271465, 2026364722410364
Offset: 1

Views

Author

Klaus Brockhaus, Oct 22 2008

Keywords

Examples

			a(1) = 110443 since A002144(1) = 5, 110443^2+1 = 12197656250 = 2*5^8*13*1201 and for no k < 110443 does 5^8 divide k^2+1. a(3) = 3379649772 since A002144(3) = 17, 3379649772^2+1 = 11422032581379651985 = 5*13*17^8*97*259697 and for no k < 3379649772 does 17^8 divide k^2+1.

Crossrefs

Cf. A002144 (primes of form 4n+1), A002313 (-1 is a square mod p), A059321, A145296, A145297, A145298, A145299, A145871, A145873.

Programs

  • PARI
    {e=8; forprime(p=2, 40, if(p%4==1, q=p^e; m=q; while(!issquare(m-1, &n), m=m+q); print1(n, ",")))}

Extensions

More terms from Klaus Brockhaus, Nov 12 2008

A145873 Smallest k such that k^2+1 is divisible by A002144(n)^9.

Original entry on oeis.org

280182, 822557039, 24306922095, 4563230639355, 15069267560119, 112076323050317, 50928660480181, 3138611770750343, 9110883894036198, 50251663587824641, 76004727767164666, 310872228812491206, 521749964271465
Offset: 1

Views

Author

Klaus Brockhaus, Oct 30 2008

Keywords

Examples

			a(1) = 280182 since A002144(1) = 5, 280182^2+1 = 78501953125 = 5^9*40193 and for no k < 280182 does 5^9 divide k^2+1. a(3) = 24306922095 since A002144(3) = 17, 24306922095^2+1 = 590826461732399189026 = 2*17^9*29*673*127637 and for no k < 24306922095 does 17^9 divide k^2+1.

Crossrefs

Cf. A002144 (primes of form 4n+1), A002313 (-1 is a square mod p), A059321, A145296, A145297, A145298, A145299, A145871, A145872.
Showing 1-7 of 7 results.

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