A337856 -id:A337856 - cp's OEIS frontend

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

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

Offset: 1

Stefano Spezia, Jul 21 2021

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

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

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

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

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

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

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

A324298 Positive integers k such that 10*k+6 is equal to the product of two integers ending with 6 (A324297).

Original entry on oeis.org

3, 9, 15, 21, 25, 27, 33, 39, 41, 45, 51, 57, 63, 67, 69, 73, 75, 81, 87, 89, 93, 99, 105, 111, 117, 119, 121, 123, 129, 135, 137, 141, 145, 147, 153, 159, 165, 169, 171, 177, 183, 185, 189, 195, 197, 201, 207, 211, 213, 217, 219, 223, 225, 231, 233, 237, 243, 249

Offset: 1

Stefano Spezia, Mar 16 2019

All the terms of this sequence are odd.

Why? If an integer 10*k+6 = (10*a+6) * (10*b+6), then k = 10*a*b + 6*(a+b) + 3, so k is odd. - Bernard Schott, May 13 2019

			145 is a term because 26*56 = 1456 = 145*10 + 6. - _Bernard Schott_, May 13 2019

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

Cf. A017341, A053742 (ending with 5), A324297, A337856, A346389.

a={}; For[n=0,n<=250,n++,For[k=0,k<=n,k++,If[Mod[10*n+6,10*k+6]==0 && Mod[(10*n+6)/(10*k+6),10]==6 && 10*n+6>Max[10*a+6],AppendTo[a,n]]]]; a

isok6(n) = (n%10) == 6; \\ A017341
isok(k) = {my(n=10*k+6, d=divisors(n)); fordiv(n, d, if (isok6(d) && isok6(n/d), return(1))); return (0);} \\ Michel Marcus, Apr 14 2019

def aupto(lim): return sorted(set(a*b//10 for a in range(6, 10*lim//6+2, 10) for b in range(a, 10*lim//a+2, 10) if a*b//10 <= lim))
print(aupto(249)) # Michael S. Branicky, Aug 21 2021

a(n) = (A324297(n) - 6)/10.
Conjecture: lim_{n->infinity} a(n)/a(n-1) = 1.
The conjecture is true since a(n) = (A324297(n) - 6)/10 and lim_{n->infinity} A324297(n)/A324297(n-1) = 1. - Stefano Spezia, Aug 21 2021

A337855 Number of n-digit positive integers that are the product of two integers ending with 5.

Original entry on oeis.org

0, 2, 18, 180, 1800, 18000, 180000, 1800000, 18000000, 180000000, 1800000000, 18000000000, 180000000000, 1800000000000, 18000000000000, 180000000000000, 1800000000000000, 18000000000000000, 180000000000000000, 1800000000000000000, 18000000000000000000, 180000000000000000000

Offset: 1

Stefano Spezia, Sep 27 2020

a(n) is the number of n-digit numbers in A053742.

Christian Krause, LODA, an assembly language, a computational model and a tool for mining integer sequences
Index entries for linear recurrences with constant coefficients, signature (10).
Index entries for sequences related to digits.

Cf. A011557 (powers of 10), A052268 (number of n-digit integers), A053742 (product of two integers ending with 5), A093136, A337856.

Mathematica
```
LinearRecurrence[{10},{0,0,2,18},22]
```

O.g.f.: 2*(1 - x)*x^2/(1 - 10*x).
E.g.f.: (9*exp(10*x) - 9 - 90*x + 50*x^2)/500.
a(n) = 10*a(n-1) for n > 3 , with a(1) = 0, a(2) = 2 and a(3) = 18.
a(n) = 18*10^(n-3) for n > 2.
a(n) = 18*A011557(n - 3) for n > 2.
a(n) = 2*A052268(n - 2) for n > 2.
Sum_{i=2..n} a(n) = A093136(n - 1) for n > 1.
a(n) = 2*floor((k + 27*10^(n-2))/30), with 2 < k < 28. [This formula was found in the form k = 7 by Christian Krause's LODA miner] - Stefano Spezia, Dec 06 2021

A346389 a(n) is the number of proper divisors of A324297(n) ending with 6.

Original entry on oeis.org

1, 2, 2, 2, 1, 2, 3, 3, 2, 2, 2, 4, 2, 1, 2, 2, 3, 3, 2, 2, 4, 2, 5, 3, 3, 2, 2, 2, 4, 2, 2, 2, 3, 3, 4, 3, 4, 2, 5, 3, 3, 2, 2, 2, 2, 7, 2, 1, 2, 2, 3, 2, 3, 2, 2, 5, 3, 6, 3, 3, 2, 2, 2, 5, 2, 2, 3, 4, 3, 5, 2, 5, 4, 3, 2, 3, 6, 2, 2, 2, 6, 2, 2, 3, 2, 2, 3, 7

Offset: 1

Stefano Spezia, Jul 15 2021

			a(12) = 4 since there are 4 proper divisors of A324297(12) = 576 ending with 6: 6, 16, 36 and 96.

Cf. A017341, A032741, A324297, A324298, A337856, A346388 (ending with 5), A346392.

b={}; For[n=0, n<=450, n++, For[k=0, k<=n, k++, If[Mod[10*n+6, 10*k+6]==0 && Mod[(10*n+6)/(10*k+6), 10]==6 && 10*n+6>Max[b], AppendTo[b, 10*n+6]]]]; (* A324297 *) a={}; For[i =1, i<=Length[b], i++, AppendTo[a, Length[Drop[Select[Divisors[Part[b, i]], (Mod[#,10]==6&)], -1]]]]; a

a(n) = A346392(A324297(n)).

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

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

Stefano Spezia, Jul 25 2021

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

Index entries for linear recurrences with constant coefficients, signature (10).
Index entries for sequences related to digits.

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.

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

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).

A347748 Number of positive integers with n digits that are equal both to the product of two integers ending with 4 and to that of two integers ending with 6.

Original entry on oeis.org

0, 1, 12, 159, 1859, 20704, 223525, 2370684, 24842265, 258128126, 2665475963

Offset: 1

Stefano Spezia, Sep 12 2021

a(n) is the number of n-digit numbers in A347746.

Cf. A017341, A052268, A324297, A337856, A347253, A347255, A347746, A347749.

Table[{lo, hi}={10^(n-1), 10^n}; Length@Select[Intersection[Union@Flatten@Table[a*b, {a, 4, Floor[hi/4], 10}, {b, a, Floor[hi/a], 10}],Union@Flatten@Table[a*b, {a, 6, Floor[hi/6], 10}, {b, a, Floor[hi/a], 10}]], lo<#

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) & set(a*b for a in range(6, hi//6+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 06 2021

a(n) < A052268(n).
a(n) = A337856(n) + A347255(n) - A347749(n).
Conjecture: lim_{n->infinity} a(n)/a(n-1) = 10.

a(9)-a(10) from Michael S. Branicky, Oct 06 2021

a(11) from Frank A. Stevenson, Jan 06 2024

A347749 Number of positive integers with n digits and final digit 6 that are equal to the product of two integers ending with the same digit.

Original entry on oeis.org

0, 4, 33, 352, 3597, 36781, 374071, 3790993, 38326689, 386782889

Offset: 1

Stefano Spezia, Sep 12 2021

a(n) is the number of n-digit numbers in A347747.

Cf. A017341, A052268, A324297, A337856, A347253, A347255, A347747, A347748.

Table[{lo, hi}={10^(n-1), 10^n}; Length@Select[Union[Union@Flatten@Table[a*b, {a, 4, Floor[hi/4], 10}, {b, a, Floor[hi/a], 10}],Union@Flatten@Table[a*b, {a, 6, Floor[hi/6], 10}, {b, a, Floor[hi/a], 10}]], lo<#

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) | set(a*b for a in range(6, hi//6+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 06 2021

a(n) < A052268(n).
a(n) = A337856(n) + A347255(n) - A347748(n).
Conjecture: lim_{n->infinity} a(n)/a(n-1) = 10.

a(9)-a(10) from Michael S. Branicky, Oct 06 2021

cp's OEIS Frontend

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

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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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A324298 Positive integers k such that 10*k+6 is equal to the product of two integers ending with 6 (A324297).

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A337855 Number of n-digit positive integers that are the product of two integers ending with 5.

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A346389 a(n) is the number of proper divisors of A324297(n) ending with 6.

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

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