A346509 -id:A346509 - cp's OEIS frontend

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

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

Offset: 1

Stefano Spezia, Sep 27 2020

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

Index entries for sequences related to digits.

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

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

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

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

Stefano Spezia, Aug 08 2021

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

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

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 *)

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

a(n) < A052268(n).

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

a(6)-a(11) from Michael S. Branicky, Aug 09 2021

a(12)-a(14) from Martin Ehrenstein, Aug 22 2021

A346507 Positive integers k that are the product of two integers greater than 1 and ending with 1.

Original entry on oeis.org

121, 231, 341, 441, 451, 561, 651, 671, 781, 861, 891, 961, 1001, 1071, 1111, 1221, 1271, 1281, 1331, 1441, 1491, 1551, 1581, 1661, 1681, 1701, 1771, 1881, 1891, 1911, 1991, 2091, 2101, 2121, 2201, 2211, 2321, 2331, 2431, 2501, 2511, 2541, 2601, 2651, 2751, 2761

Offset: 1

Stefano Spezia, Jul 21 2021

All the terms end with 1 (A017281).

			121 = 11*11, 231 = 11*21, 341 = 11*31, 441 = 21*21, 451 = 11*41, ...

Stefano Spezia, Table of n, a(n) for n = 1..10000

Cf. A017281 (supersequence), A053742 (ending with 5), A324297 (ending with 6), A346508, A346509, A346510.

a={}; For[n=1, n<=300, n++, For[k=1, kMax[a], AppendTo[a, 10n+1]]]]; a

isok(k) = fordiv(k, d, if ((d>1) && (dMichel Marcus, Jul 28 2021

def aupto(lim): return sorted(set(a*b for a in range(11, lim//11+1, 10) for b in range(a, lim//a+1, 10)))
print(aupto(2761)) # Michael S. Branicky, Jul 22 2021

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

The conjecture is true since it can be proved that a(n) = (sqrt(a(n-1)) + g(n-1))^2 where [g(n): n > 1] is a bounded sequence of positive real numbers. - Stefano Spezia, Aug 21 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

Stefano Spezia, Aug 24 2021

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

Cf. A017341, A052268, A347253.

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

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

a(n) < A052268(n).

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

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

Stefano Spezia, Sep 26 2021

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

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).

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

a(n) < A052268(n).

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

a(9)-a(11) from Michael S. Branicky, Sep 26 2021

a(12)-a(15) from Martin Ehrenstein, Oct 25 2021

A346508 Positive integers k such that 10*k+1 is equal to the product of two integers greater than 1 and ending with 1 (A346507).

Original entry on oeis.org

12, 23, 34, 44, 45, 56, 65, 67, 78, 86, 89, 96, 100, 107, 111, 122, 127, 128, 133, 144, 149, 155, 158, 166, 168, 170, 177, 188, 189, 191, 199, 209, 210, 212, 220, 221, 232, 233, 243, 250, 251, 254, 260, 265, 275, 276, 282, 287, 291, 296, 298, 309, 311, 313, 317

Offset: 1

Stefano Spezia, Jul 21 2021

			107 is a term because 21*51 = 1071 = 107*10 + 1.

Stefano Spezia, Table of n, a(n) for n = 1..10000

Cf. A016873 (ending with 5), A017281, A324298 (ending with 6), A346507, A346509, A346510.

a={}; For[n=1, n<=350, n++, For[k=1, kMax[10a+1], AppendTo[a, n]]]]; a

def aupto(lim): return sorted(set(a*b//10 for a in range(11, 10*lim//11+2, 10) for b in range(a, 10*lim//a+2, 10) if a*b//10 <= lim))
print(aupto(318)) # Michael S. Branicky, Aug 21 2021

a(n) = (A346507(n) - 1)/10.
Conjecture: lim_{n->infinity} a(n)/a(n-1) = 1.
The conjecture is true since a(n) = (A346507(n) - 1)/10 and lim_{n->infinity} A346507(n)/A346507(n-1) = 1. - Stefano Spezia, Aug 21 2021

A346510 a(n) is the number of nontrivial divisors of A346507(n) ending with 1.

Original entry on oeis.org

1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 4, 2, 4, 2

Offset: 1

Stefano Spezia, Jul 21 2021

			a(42) = 4 since there are 4 nontrivial divisors of A346507(42) = 2541 ending with 1: 11, 21, 121 and 231.

Cf. A017281, A070824, A346388 (ending with 5), A346389 (ending with 6), A346392, A346507, A346508, A346509.

b={}; For[n=1, n<=500, n++, For[k=1, kMax[b], AppendTo[b, 10n+1]]]]; (* A346507 *) a={}; For[i =1, i<=Length[b], i++, AppendTo[a, Length[Drop[Select[Divisors[Part[b, i]], (Mod[#, 10]==1&)], -1]]-1]]; a

f(n) = sumdiv(n, d, (d>1) && (d(f(x)), [1..5000])) \\ Michel Marcus, Jul 28 2021

from sympy import divisors
def f(n): return sum(d%10 == 1 for d in divisors(n)[1:-1])
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)))
print(list(map(f, A346507upto(5000)))) # Michael S. Branicky, Jul 31 2021

a(n) = A346392(A346507(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

Stefano Spezia, Oct 22 2021

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

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).

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

a(n) < A052268(n).

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

a(9)-a(10) from Michael S. Branicky, Oct 22 2021

a(11)-a(15) from Martin Ehrenstein, Nov 06 2021

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

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

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A346507 Positive integers k that are the product of two integers greater than 1 and ending with 1.

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

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

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A346508 Positive integers k such that 10*k+1 is equal to the product of two integers greater than 1 and ending with 1 (A346507).

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A346510 a(n) is the number of nontrivial divisors of A346507(n) ending with 1.

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

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