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.

A337856 Number of positive integers with n digits that are the product of two integers ending with 6.

Original entry on oeis.org

0, 2, 20, 230, 2515, 26889, 282211, 2930013, 30196730, 309564822, 3161099901, 32182595954, 326874672928, 3313770788984
Offset: 1

Views

Author

Stefano Spezia, Sep 27 2020

Keywords

Comments

a(n) is the number of n-digit numbers in A324297.

Links

Crossrefs

Cf. A017341, A324297, A337855, A346509.

Programs

  • Python
    def A337856(n):
    k, n1, n2, pset = 0, 10**(n-1)//2-18, 10**n//2-18, set()
    while 50*k**2+60*k < n2:
        a, b = divmod(n1-30*k,50*k+30)
        m = max(k,a+int(b>0))
        r = 50*k*m+30*(k+m)
        while r < n2:
            pset.add(r)
            m += 1
            r += 50*k+30
        k += 1
    return len(pset) # Chai Wah Wu, Sep 26 2021

Formula

Conjecture: Lim_{n->infinity} a(n)/a(n-1) = 10.

Extensions

a(5) corrected by and a(6)-a(9) from Jinyuan Wang, Oct 01 2020
a(10)-a(13) from Bert Dobbelaere, Oct 20 2020
a(14) from Martin Ehrenstein, Aug 06 2021

A346509 Number of positive integers with n digits that are the product of two integers greater than 1 and ending with 1.

Original entry on oeis.org

0, 0, 12, 200, 2660, 31850, 361985, 3982799, 42914655, 455727689, 4788989458, 49930700093, 517443017072, 5336861879564
Offset: 1

Views

Author

Stefano Spezia, Jul 21 2021

Keywords

Comments

a(n) is the number of n-digit numbers in A346507.

Crossrefs

Cf. A017281, A052268, A087630, A337855 (ending with 5), A337856 (ending with 6), A346507.

Programs

  • PARI
    a(n) = {my(res = 0); forstep(i = 10^(n-1) + 1, 10^n, 10, f = factor(i); if(bigomega(f) == 1, next); d = divisors(f); for(j = 2, (#d~ + 1)>>1, if(d[j]%10 == 1 && d[#d + 1 - j]%10 == 1, res++; next(2) ) ) ); res } \\ David A. Corneth, Jul 22 2021
  • Python
    def A346507upto(lim): return sorted(set(a*b for a in range(11, lim//11+1, 10) for b in range(a, lim//a+1, 10)))
def a(n): return len(A346507upto(10**n)) - len(A346507upto(10**(n-1)))
print([a(n) for n in range(1, 9)]) # Michael S. Branicky, Jul 22 2021

Formula

Conjecture: Lim_{n->infinity} a(n)/a(n-1) = 10.

Extensions

a(6)-a(9) from Michael S. Branicky, Jul 22 2021
a(10) from David A. Corneth, Jul 22 2021
a(11) from Michael S. Branicky, Jul 23 2021
a(11) corrected and extended with a(12) by Martin Ehrenstein, Aug 03 2021
a(13)-a(14) from Martin Ehrenstein, Aug 05 2021

A346952 Number of positive integers with n digits that are the product of two integers ending with 3.

Original entry on oeis.org

1, 3, 37, 398, 4303, 45765, 480740, 5005328, 51770770, 532790460, 5461696481, 55814395421, 568944166801, 5787517297675
Offset: 1

Views

Author

Stefano Spezia, Aug 08 2021

Keywords

Comments

a(n) is the number of n-digit numbers in A346950.

Crossrefs

Cf. A017377, A052268, A346509 (ending with 1), A337855 (ending with 5), A337856 (ending with 6), A346950.

Programs

  • Mathematica
    Table[{lo,hi}={10^(n-1),10^n};Length@Select[Union@Flatten@Table[a*b,{a,3,Floor[hi/3],10},{b,a,Floor[hi/a],10}],lo<#Giorgos Kalogeropoulos, Aug 16 2021 *)
  • Python
    def a(n):
  lo, hi = 10**(n-1), 10**n
  return len(set(a*b for a in range(3, hi//3+1, 10) for b in range(a, hi//a+1, 10) if lo <= a*b < hi))
print([a(n) for n in range(1, 9)]) # Michael S. Branicky, Aug 09 2021

Formula

a(n) < A052268(n).
Conjecture: Lim_{n->infinity} a(n)/a(n-1) = 10.

Extensions

a(6)-a(11) from Michael S. Branicky, Aug 09 2021
a(12)-a(14) from Martin Ehrenstein, Aug 22 2021

A347255 Number of positive integers with n digits that are the product of two integers ending with 4.

Original entry on oeis.org

0, 3, 25, 281, 2941, 30596, 315385, 3231664, 32972224, 335346193, 3402373313, 34454358909, 348373701706, 3518101287286, 35491654274101
Offset: 1

Views

Author

Stefano Spezia, Aug 24 2021

Keywords

Comments

a(n) is the number of n-digit numbers in A347253.

Crossrefs

Cf. A017341, A052268, A347253.
Cf. A346509 (ending with 1), A346952 (ending with 3), A337855 (ending with 5), A337856 (ending with 6).

Programs

  • Python
    def a(n):
  lo, hi = 10**(n-1), 10**n
  return len(set(a*b for a in range(4, hi//4+1, 10) for b in range(a, hi//a+1, 10) if lo <= a*b < hi))
print([a(n) for n in range(1, 9)]) # Michael S. Branicky, Aug 24 2021

Formula

a(n) < A052268(n).
Conjecture: lim_{n->infinity} a(n)/a(n-1) = 10.

Extensions

a(9)-a(11) from Michael S. Branicky, Aug 25 2021
a(12)-a(15) from Martin Ehrenstein, Sep 29 2021

A348055 Number of positive integers with n digits that are the product of two integers ending with 7.

Original entry on oeis.org

0, 1, 20, 255, 3064, 34743, 380939, 4089499, 43282317, 453472867, 4715695283, 48760330737, 501941505404, 5148657883067, 52659616820819
Offset: 1

Views

Author

Stefano Spezia, Sep 26 2021

Keywords

Comments

a(n) is the number of n-digit numbers in A348054.

Crossrefs

Cf. A017377, A052268, A348054.
Cf. A346509 (ending with 1), A346629 (ending with 2), A346952 (ending with 3), A347255 (ending with 4), A337855 (ending with 5), A337856 (ending with 6), A348549 (ending with 8).

Programs

  • Python
    def a(n):
  lo, hi = 10**(n-1), 10**n
  return len(set(a*b for a in range(7, hi//7+1, 10) for b in range(a, hi//a+1, 10) if lo <= a*b < hi))
print([a(n) for n in range(1, 9)]) # Michael S. Branicky, Sep 26 2021

Formula

a(n) < A052268(n).
Conjecture: lim_{n->infinity} a(n)/a(n-1) = 10.

Extensions

a(9)-a(11) from Michael S. Branicky, Sep 26 2021
a(12)-a(15) from Martin Ehrenstein, Oct 25 2021

A346629 Number of n-digit positive integers that are the product of two integers ending with 2.

Original entry on oeis.org

1, 4, 45, 450, 4500, 45000, 450000, 4500000, 45000000, 450000000, 4500000000, 45000000000, 450000000000, 4500000000000, 45000000000000, 450000000000000, 4500000000000000, 45000000000000000, 450000000000000000, 4500000000000000000, 45000000000000000000, 450000000000000000000
Offset: 1

Views

Author

Stefano Spezia, Jul 25 2021

Keywords

Comments

a(n) is the number of n-digit numbers in A139245.
After initial 1 or 2 values the same as A137233. - R. J. Mathar, Aug 23 2021

Links

Crossrefs

Cf. A011557 (powers of 10), A017293 (positive integers ending with 2), A052268 (number of n-digit integers), A139245 (product of two integers ending with 2), A093143, A337855, A337856.
Cf. A137233.

Programs

  • Mathematica
    LinearRecurrence[{10},{1,4,45},25]

Formula

O.g.f.: x*(1 - 6*x + 5*x^2)/(1 - 10*x).
E.g.f.: (9*exp(10*x) - 9 + 110*x - 50*x^2)/200.
a(n) = 10*a(n-1) for n > 3, with a(1) = 1, a(2) = 4 and a(3) = 45.
a(n) = 45*10^(n-3) for n > 2.
a(n) = 45*A011557(n-3) for n > 2.
Sum_{i=1..n} a(n) = A093143(n-1).

A348549 Number of positive integers with n digits that are the product of two integers ending with 8.

Original entry on oeis.org

0, 1, 14, 195, 2200, 24013, 255969, 2687317, 27934809, 288342379, 2960920297, 30285890402, 308834717932, 3141625339760, 31895159990436
Offset: 1

Views

Author

Stefano Spezia, Oct 22 2021

Keywords

Comments

a(n) is the number of n-digit numbers in A348548.

Crossrefs

Cf. A052268, A348548.
Cf. A346509 (ending with 1), A346629 (ending with 2), A346952 (ending with 3), A347255 (ending with 4), A337855 (ending with 5), A337856 (ending with 6), A348055 (ending with 7).

Programs

  • Python
    def a(n):
  lo, hi = 10**(n-1), 10**n
  return len(set(a*b for a in range(8, hi//8+1, 10) for b in range(a, hi//a+1, 10) if lo <= a*b < hi))
print([a(n) for n in range(1, 9)]) # Michael S. Branicky, Oct 22 2021

Formula

a(n) < A052268(n).
Conjecture: lim_{n->infinity} a(n)/a(n-1) = 10.

Extensions

a(9)-a(10) from Michael S. Branicky, Oct 22 2021
a(11)-a(15) from Martin Ehrenstein, Nov 06 2021
Showing 1-7 of 7 results.

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