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 1-10 of 11 results. Next

A077678 Squarefree numbers beginning with 2.

Original entry on oeis.org

2, 21, 22, 23, 26, 29, 201, 202, 203, 205, 206, 209, 210, 211, 213, 214, 215, 217, 218, 219, 221, 222, 223, 226, 227, 229, 230, 231, 233, 235, 237, 238, 239, 241, 246, 247, 249, 251, 253, 254, 255, 257, 258, 259, 262, 263, 265, 266, 267, 269, 271, 273, 274
Offset: 1

Views

Author

Amarnath Murthy, Nov 16 2002

Keywords

Comments

Intersection of A005117 and A217394. - Michel Marcus, Sep 14 2013
Lower density is 3/(10*Pi^2), upper density is 20/(9*Pi^2). - Charles R Greathouse IV, Nov 05 2017

Links

Crossrefs

Cf. A077677, A077679, A077680, A077681, A077682, A077683, A077684, A077685.

Programs

  • Mathematica
    s2Q[n_]:=First[IntegerDigits[n]]==2 && 1==Max@@Last/@FactorInteger[n]; Select[Range[274],s2Q[#] &] (* Jayanta Basu, May 22 2013 *)
  • PARI
    is(n)=n>1 && digits(n)[1]==2 && issquarefree(n) \\ Charles R Greathouse IV, Nov 05 2017

Extensions

More terms from Sascha Kurz, Jan 28 2003

A077679 Squarefree numbers beginning with 3.

Original entry on oeis.org

3, 30, 31, 33, 34, 35, 37, 38, 39, 301, 302, 303, 305, 307, 309, 310, 311, 313, 314, 317, 318, 319, 321, 322, 323, 326, 327, 329, 330, 331, 334, 335, 337, 339, 341, 345, 346, 347, 349, 353, 354, 355, 357, 358, 359, 362, 365, 366, 367, 370, 371, 373, 374, 377
Offset: 1

Views

Author

Amarnath Murthy, Nov 16 2002

Keywords

Comments

Intersection of A005117 and A217395. - Michel Marcus, Sep 14 2013
Lower density is 1/(5*Pi^2), upper density is 5/(3*Pi^2). - Charles R Greathouse IV, Nov 05 2017

Links

Crossrefs

Cf. A077677, A077678, A077680, A077681, A077682, A077683, A077684, A077685.

Programs

  • Mathematica
    Select[Range[400],SquareFreeQ[#]&&IntegerDigits[#][[1]]==3&] (* Harvey P. Dale, Oct 10 2018 *)
  • PARI
    isok(n) = (issquarefree(n) && (digits(n, 10)[1] == 3)) \\ Michel Marcus, Jul 31 2013

Extensions

More terms from Sascha Kurz, Jan 28 2003

A077680 Squarefree numbers beginning with 4.

Original entry on oeis.org

41, 42, 43, 46, 47, 401, 402, 403, 406, 407, 409, 410, 411, 413, 415, 417, 418, 419, 421, 422, 426, 427, 429, 430, 431, 433, 434, 435, 437, 438, 439, 442, 443, 445, 446, 447, 449, 451, 453, 454, 455, 457, 458, 461, 462, 463, 465, 466, 467, 469, 470, 471, 473
Offset: 1

Views

Author

Amarnath Murthy, Nov 16 2002

Keywords

Comments

Lower density is 3/(20*Pi^2), upper density is 4/(3*Pi^2). - Charles R Greathouse IV, Nov 05 2017

Links

Crossrefs

Intersection of A005117 and A217397.
Cf. A077677, A077678, A077679, A077681, A077682, A077683, A077684, A077685.

Programs

  • Maple
    select(numtheory:-issqrfree, [seq(seq(i,i=4*10^d+1 .. 5*10^d-1),d=1..3)]); # Robert Israel, May 07 2025
  • Mathematica
    Select[Range[499],First[IntegerDigits[#]]==4&&SquareFreeQ[#]&] (* Harvey P. Dale, Apr 24 2018 *)
  • PARI
    is(n)=n>40 && digits(n)[1]==4 && issquarefree(n) \\ Charles R Greathouse IV, Nov 05 2017
  • Python
    from functools import lru_cache
from math import isqrt
from sympy import mobius
def A077680(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    @lru_cache(maxsize=None)
    def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
    def h(x): return 0 if x<4 else h(5*10**((l:=len(s:=str(x)))-2)-1)-g(4*10**(l-1)-1)+(g(x) if s[0]=='4' else g(5*10**(l-1)-1) if s[0]>'4' else 0)
    def f(x): return n+x-h(x)
    return bisection(f,n,n) # Chai Wah Wu, May 07 2025

Extensions

More terms from Sascha Kurz, Jan 28 2003

A077681 Squarefree numbers beginning with 5.

Original entry on oeis.org

5, 51, 53, 55, 57, 58, 59, 501, 502, 503, 505, 506, 509, 510, 511, 514, 515, 517, 518, 519, 521, 523, 526, 527, 530, 533, 534, 535, 537, 538, 541, 542, 543, 545, 546, 547, 551, 553, 554, 555, 557, 559, 561, 562, 563, 565, 566, 569, 570, 571, 573, 574, 577
Offset: 1

Views

Author

Amarnath Murthy, Nov 16 2002

Keywords

Comments

Lower density is 3/(25*Pi^2), upper density is 10/(9*Pi^2). - Charles R Greathouse IV, Nov 05 2017

Links

Crossrefs

Cf. A077677, A077678, A077679, A077680, A077682, A077683, A077684, A077685.

Programs

  • Mathematica
    Table[Select[Range[5*10^n,6*10^n-1],SquareFreeQ],{n,0,2}]//Flatten (* Harvey P. Dale, Apr 15 2017 *)
  • PARI
    is(n)=n>4 && digits(n)[1]==5 && issquarefree(n) \\ Charles R Greathouse IV, Nov 05 2017

Extensions

More terms from Sascha Kurz, Jan 28 2003

A077682 Squarefree numbers beginning with 6.

Original entry on oeis.org

6, 61, 62, 65, 66, 67, 69, 601, 602, 606, 607, 609, 610, 611, 613, 614, 615, 617, 618, 619, 622, 623, 626, 627, 629, 631, 633, 634, 635, 638, 641, 642, 643, 645, 646, 647, 649, 651, 653, 654, 655, 658, 659, 661, 662, 663, 665, 667, 669, 670, 671, 673, 674
Offset: 1

Views

Author

Amarnath Murthy, Nov 16 2002

Keywords

Comments

Intersection of A005117 and A217399. - Michel Marcus, Sep 14 2013
Lower density is 1/(10*Pi^2), upper density is 20/(21*Pi^2). - Charles R Greathouse IV, Nov 05 2017

Links

Crossrefs

Cf. A077677, A077678, A077679, A077680, A077681, A077683, A077684, A077685.
Cf. A005117, A217399.

Programs

  • Maple
    with(numtheory): select(issqrfree,[seq(seq(6*10^n+k,k=0..10^n-1),n=0..3)]); # Muniru A Asiru, Nov 22 2018
  • Mathematica
    Select[Range[1000],SquareFreeQ[#]&&First[IntegerDigits[#]]==6&] (* Harvey P. Dale, May 19 2012 *)
  • PARI
    is(n)=n>5 && digits(n)[1]==6 && issquarefree(n) \\ Charles R Greathouse IV, Nov 05 2017

Extensions

More terms from Sascha Kurz, Jan 28 2003

A077685 Squarefree numbers beginning with 9.

Original entry on oeis.org

91, 93, 94, 95, 97, 901, 902, 903, 905, 906, 907, 910, 911, 913, 914, 915, 917, 919, 921, 922, 923, 926, 929, 930, 933, 934, 935, 937, 938, 939, 941, 942, 943, 946, 947, 949, 951, 953, 955, 957, 958, 959, 962, 965, 966, 967, 969, 970, 971, 973, 974, 977, 978
Offset: 1

Views

Author

Amarnath Murthy, Nov 16 2002

Keywords

Comments

Intersection of A005117 and A217402. - Michel Marcus, Sep 14 2013
Lower density is 1/(15*Pi^2), upper density is 2/(3*Pi^2). - Charles R Greathouse IV, Nov 05 2017

Links

Crossrefs

Cf. A077677, A077678, A077679, A077680, A077681, A077682, A077683, A077684.

Programs

  • Mathematica
    Select[Range[1000],SquareFreeQ[#]&&First[IntegerDigits[#]]==9&] (* Harvey P. Dale, Dec 15 2013 *)
  • PARI
    isok(n) = issquarefree(n) && (digits(n)[1] == 9); \\ Michel Marcus, Sep 14 2013
  • Python
    from math import isqrt
from sympy import mobius
def A077685(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
    def h(x): return 0 if x<9 else h(10**(len(s:=str(x))-1)-1)+(g(x)-g(9*10**(len(s)-1)-1) if s[0]=='9' else 0)
    def f(x): return n+x-h(x)
    return bisection(f,n,n) # Chai Wah Wu, May 06 2025

Extensions

More terms from Sascha Kurz, Jan 28 2003

A217400 Numbers starting with 7.

Original entry on oeis.org

7, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 700, 701, 702, 703, 704, 705, 706, 707, 708, 709, 710, 711, 712, 713, 714, 715, 716, 717, 718, 719, 720, 721, 722, 723, 724, 725, 726, 727, 728, 729, 730, 731, 732, 733, 734, 735, 736, 737, 738, 739, 740, 741, 742
Offset: 1

Views

Author

Jeremy Gardiner, Oct 02 2012

Keywords

Comments

The lower and upper asymptotic densities of this sequence are 1/63 and 5/36, respectively. - Amiram Eldar, Feb 27 2021

Links

Crossrefs

Cf. A000870, A011537.
Subsequences include: A045713, A077332, A077683, A106417, A106427.
Cf. A131835, A217394, A217395, A217397, A217398, A217399, A217401, A217402.

Programs

  • Mathematica
    Select[Range[1000], IntegerDigits[#][[1]] == 7 &] (* T. D. Noe, Oct 02 2012 *)
  • Python
    def A217400(n): return n+(62*10**(len(str(9*n-8))-1))//9 # Chai Wah Wu, Dec 07 2024

Formula

a(n) = n + (62*10^floor(log_10(9*n-8))-8)/9. - Alan Michael Gómez Calderón, May 17 2023

A077677 Squarefree numbers beginning with 1.

Original entry on oeis.org

1, 10, 11, 13, 14, 15, 17, 19, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 118, 119, 122, 123, 127, 129, 130, 131, 133, 134, 137, 138, 139, 141, 142, 143, 145, 146, 149, 151, 154, 155, 157, 158, 159, 161, 163, 165, 166, 167, 170, 173, 174, 177
Offset: 1

Views

Author

Amarnath Murthy, Nov 16 2002

Keywords

Comments

Intersection of A005117 and A131835. - Michel Marcus, Sep 14 2013
Lower density is 3/(5*Pi^2), upper density is 10/(3*Pi^2). - Charles R Greathouse IV, Nov 05 2017

Links

Crossrefs

Cf. A077678, A077679, A077680, A077681, A077682, A077683, A077684, A077685.

Programs

  • Mathematica
    Select[Range[177], First[IntegerDigits[#]]==1 && SquareFreeQ[#] &] (* Jayanta Basu, May 23 2013 *)
  • PARI
    is(n)=n>0 && digits(n)[1]==1 && issquarefree(n) \\ Charles R Greathouse IV, Nov 05 2017
  • PARI
    list(lim)=my(v=List([1])); for(d=1,#Str(lim\=1)-1, my(D=10^d); forsquarefree(n=D,min(2*D,lim), listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Jan 10 2023
  • Python
    from math import isqrt
from sympy import mobius
def A077677(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
    def h(x): return 0 if x<1 else h(2*10**((l:=len(s:=str(x)))-2)-1)-g((m:=10**(l-1))-1)+(g(x) if s[0]=='1' else g((m<<1)-1))
    def f(x): return n+x-h(x)
    return bisection(f,n,n) # Chai Wah Wu, May 06 2025

Extensions

Corrected and extended by Sascha Kurz, Jan 28 2003

A077684 Squarefree numbers beginning with 8.

Original entry on oeis.org

82, 83, 85, 86, 87, 89, 802, 803, 805, 806, 807, 809, 811, 813, 814, 815, 817, 818, 821, 822, 823, 826, 827, 829, 830, 831, 834, 835, 838, 839, 842, 843, 849, 851, 853, 854, 857, 858, 859, 861, 862, 863, 865, 866, 869, 870, 871, 874, 877, 878, 879, 881, 883
Offset: 1

Views

Author

Amarnath Murthy, Nov 16 2002

Keywords

Comments

Intersection of A005117 and A217401. - Michel Marcus, Sep 14 2013
Lower density is 3/(40*Pi^2), upper density is 20/(27*Pi^2). - Charles R Greathouse IV, Nov 05 2017

Links

Crossrefs

Cf. A077677, A077678, A077679, A077680, A077681, A077682, A077683, A077685.

Programs

  • Mathematica
    Select[Range[900],SquareFreeQ[#]&&First[IntegerDigits[#]]==8&]  (* Harvey P. Dale, Feb 22 2011 *)
  • PARI
    is(n)=n>81 && digits(n)[1]==8 && issquarefree(n) \\ Charles R Greathouse IV, Nov 05 2017

Extensions

More terms from Sascha Kurz, Jan 28 2003
Definition clarified by Harvey P. Dale, Feb 22 2011

A077336 Smallest squarefree number beginning with n.

Original entry on oeis.org

1, 2, 3, 41, 5, 6, 7, 82, 91, 10, 11, 122, 13, 14, 15, 161, 17, 181, 19, 201, 21, 22, 23, 241, 251, 26, 271, 281, 29, 30, 31, 321, 33, 34, 35, 362, 37, 38, 39, 401, 41, 42, 43, 442, 451, 46, 47, 481, 491, 501, 51, 521, 53, 541, 55, 561, 57, 58, 59, 601, 61, 62, 631, 641, 65
Offset: 1

Views

Author

Amarnath Murthy, Nov 04 2002

Keywords

Crossrefs

Cf. A005117, A077687, A077677, A077678, A077679, A077680, A077681, A077682, A077683, A077684, A077685.

Extensions

More terms from Sascha Kurz, Jan 28 2003
Showing 1-10 of 11 results. Next

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