A346950 -id:A346950 - cp's OEIS frontend

A346951 Positive integers k such that 10*k+9 is equal to the product of two integers ending with 3 (A346950).

0, 3, 6, 9, 12, 15, 16, 18, 21, 24, 27, 29, 30, 33, 36, 39, 42, 45, 48, 51, 52, 54, 55, 57, 60, 63, 66, 68, 69, 72, 75, 78, 81, 84, 87, 90, 93, 94, 96, 98, 99, 102, 105, 107, 108, 111, 114, 117, 120, 121, 123, 126, 129, 132, 133, 135, 138, 141, 144, 146, 147, 150

Offset: 1

Stefano Spezia, Aug 08 2021

			15 is a term because 3*53 = 159 = 15*10 + 9.

Cf. A016873 (ending with 5), A017377, A324298 (ending with 6), A346950, A346952, A346953.

a={}; For[n=0, n<=150, n++, For[k=0, k<=n, k++, If[Mod[10*n+9, 10*k+3]==0 && Mod[(10*n+9)/(10*k+3), 10]==3&& 10*n+9>Max[10a+9], AppendTo[a, n]]]]; a

def aupto(lim): return sorted(set(a*b//10 for a in range(3, 10*lim//3+4, 10) for b in range(a, 10*lim//a+4, 10) if a*b//10 <= lim))
print(aupto(150)) # Michael S. Branicky, Aug 11 2021

a(n) = (A346950(n) - 9)/10.

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

A346953 a(n) is the number of divisors of A346950(n) ending with 3.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 4, 2, 2, 4, 2, 2

Offset: 1

Stefano Spezia, Aug 08 2021

a(n) = 1 if A346950(n) = k^2 where k is either a prime ending with 3 or the product of a prime ending with 7 and a prime ending with 9. - Robert Israel, Nov 03 2024

			a(17) = 4 since there are 4 divisors of A346950(17) = 429 ending with 3: 3, 13, 33 and 143.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A017377, A346388 (ending with 5), A346389 (ending with 6), A346950, A346951, A346952.

N:= 10000: # for a(1) .. a(M) where the last term of A346950 less than N is A346950(M)
S:= {}:
for n from 3 to floor(sqrt(N)) by 10 do
  S:= S union map(`*`, {seq(i,i= n .. floor(N/n), 10)},n)
od:
S:= sort(convert(S,list)):
map(t -> nops(select(t -> t mod 10 = 3, numtheory:-divisors(t))), S); # Robert Israel, Nov 03 2024

b={}; For[n=0, n<=450, n++, For[k=0, k<=n, k++, If[Mod[10*n+9, 10*k+3]==0 && Mod[(10*n+9)/(10*k+3), 10]==3 && 10*n+9>Max[b], AppendTo[b, 10*n+9]]]]; (* A346950 *) a={}; For[i =1, i<=Length[b], i++, AppendTo[a, Length[Drop[Select[Divisors[Part[b, i]], (Mod[#, 10]==3&)]]]]]; a

from sympy import divisors
def f(n): return sum(d%10 == 3 for d in divisors(n)[1:-1])
def A346950upto(lim): return sorted(set(a*b for a in range(3, lim//3+1, 10) for b in range(a, lim//a+1, 10)))
print(list(map(f, A346950upto(2129)))) # Michael S. Branicky, Aug 11 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

A347253 Positive integers that are the product of two integers ending with 4.

Original entry on oeis.org

16, 56, 96, 136, 176, 196, 216, 256, 296, 336, 376, 416, 456, 476, 496, 536, 576, 616, 656, 696, 736, 756, 776, 816, 856, 896, 936, 976, 1016, 1036, 1056, 1096, 1136, 1156, 1176, 1216, 1256, 1296, 1316, 1336, 1376, 1416, 1456, 1496, 1536, 1576, 1596, 1616, 1656

Offset: 1

Stefano Spezia, Aug 24 2021

			16 = 4*4, 56 = 4*14, 96 = 4*24, 136 = 4*34, 176 = 4*44, 196 = 14*14, 216 = 4*54, ...

Cf. A017341 (supersequence), A053742 (ending with 5), A139245 (ending with 2), A324297 (ending with 6), A346950 (ending with 3), A347254, A347255.

a={}; For[n=0, n<=200, n++, For[k=0, k<=n, k++, If[Mod[10*n+6, 10*k+4]==0 && Mod[(10*n+6)/(10*k+4), 10]==4 && 10*n+6>Max[a], AppendTo[a, 10*n+6]]]]; a

def aupto(lim): return sorted(set(a*b for a in range(4, lim//4+1, 10) for b in range(a, lim//a+1, 10)))
print(aupto(1660)) # Michael S. Branicky, Aug 24 2021

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

A348054 Positive integers that are the product of two integers ending with 7.

Original entry on oeis.org

49, 119, 189, 259, 289, 329, 399, 459, 469, 539, 609, 629, 679, 729, 749, 799, 819, 889, 959, 969, 999, 1029, 1099, 1139, 1169, 1239, 1269, 1309, 1369, 1379, 1449, 1479, 1519, 1539, 1589, 1649, 1659, 1729, 1739, 1799, 1809, 1819, 1869, 1939, 1989, 2009, 2079, 2109

Offset: 1

Stefano Spezia, Sep 26 2021

			49 = 7*7, 119 = 7*17, 189 = 7*27, 259 = 7*37, 289 = 17*17, 329 = 7*47, 399 = 7*57, ...

Cf. A017377 (supersequence), A053742 (ending with 5), A139245 (ending with 2), A324297 (ending with 6), A346950 (ending with 3), A347253 (ending with 4), A348055.

a={}; For[n=0, n<=210, n++, For[k=0, k<=n, k++, If[Mod[10*n+9, 10*k+7]==0 && Mod[(10*n+9)/(10*k+7), 10]==7 && 10*n+9>Max[a], AppendTo[a, 10*n+9]]]]; a

def aupto(lim): return sorted(set(a*b for a in range(7, lim//7+1, 10) for b in range(a, lim//a+1, 10)))
print(aupto(2110)) # Michael S. Branicky, Sep 26 2021

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

A348546 Number of positive integers with n digits that are equal both to the product of two integers ending with 3 and to that of two integers ending with 7.

Original entry on oeis.org

0, 0, 8, 129, 1771, 21802, 252793, 2826973, 30872783

Offset: 1

Stefano Spezia, Oct 22 2021

a(n) is the number of n-digit numbers in A348544.

Cf. A052268, A346950, A346952, A348054, A348055, A348544, A348547.

Table[{lo, hi}={10^(n-1), 10^n}; Length@Select[Intersection[Union@Flatten@Table[a*b, {a, 3, Floor[hi/3], 10}, {b, a, Floor[hi/a], 10}], Union@Flatten@Table[a*b, {a, 7, Floor[hi/7], 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(3, hi//3+1, 10) for b in range(a, hi//a+1, 10) if lo <= a*b < hi) & 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, Oct 22 2021

a(n) < A052268(n).
a(n) = A346952(n) + A348055(n) - A348547(n).
Conjecture: lim_{n->infinity} a(n)/a(n-1) = 10.

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

A348547 Number of positive integers with n digits and final digit 9 that are equal to the product of two integers ending with the same digit.

Original entry on oeis.org

1, 4, 49, 524, 5596, 58706, 608886, 6267854, 64180304, 654605898, 6656849267, 67539297095, 683989985496, 6916722312963, 69859080168037

Offset: 1

Stefano Spezia, Oct 22 2021

a(n) is the number of n-digit numbers in A348545.

Cf. A052268, A346950, A346952, A348054, A348055, A348545, A348546.

Table[{lo, hi}={10^(n-1), 10^n}; Length@Select[Union[Union@Flatten@Table[a*b, {a, 3, Floor[hi/3], 10}, {b, a, Floor[hi/a], 10}], Union@Flatten@Table[a*b, {a, 7, Floor[hi/7], 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(3, hi//3+1, 10) for b in range(a, hi//a+1, 10) if lo <= a*b < hi) | 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, Oct 22 2021

a(n) < A052268(n).
a(n) = A346952(n) + A348055(n) - A348546(n).
Conjecture: lim_{n->infinity} a(n)/a(n-1) = 10.

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

a(10)-a(15) from Martin Ehrenstein, Nov 06 2021

A348544 Positive integers that are equal both to the product of two integers ending with 3 and to that of two integers ending with 7.

Original entry on oeis.org

189, 399, 459, 609, 729, 819, 969, 999, 1029, 1239, 1269, 1449, 1479, 1539, 1659, 1729, 1809, 1869, 1989, 2079, 2109, 2289, 2349, 2499, 2619, 2639, 2679, 2709, 2889, 2919, 3009, 3059, 3129, 3159, 3219, 3249, 3339, 3429, 3519, 3549, 3699, 3759, 3819, 3969, 4029

Offset: 1

Stefano Spezia, Oct 22 2021

Intersection of A346950 and A348054.

			189 = 7*27 = 3*63, 399 = 3*133 = 7*57, 459 = 3*153 = 17*27, 609 = 3*203 = 7*87, ...

Cf. A017377 (supersequence), A346950, A348054, A348546.

max=4050; Select[Intersection[Union@Flatten@Table[a*b, {a, 3, Floor[max/3], 10}, {b, a, Floor[max/a], 10}], Union@Flatten@Table[a*b, {a, 7, Floor[max/7], 10}, {b, a, Floor[max/a], 10}]], 0<#

isok(m) = my(ok3=0, ok7=0); fordiv(m, d, if (((d % 10) == 3) && ((m/d % 10) == 3), ok3++); if (((d % 10) == 7) && ((m/d % 10) == 7), ok7++); if (ok3 && ok7, return(1))); \\ Michel Marcus, Oct 22 2021

def aupto(lim): return sorted(set(a*b for a in range(3, lim//3+1, 10) for b in range(a, lim//a+1, 10)) & set(a*b for a in range(7, lim//7+1, 10) for b in range(a, lim//a+1, 10)))
print(aupto(4029)) # Michael S. Branicky, Oct 22 2021

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

A348545 Positive integers with final digit 9 that are equal to the product of two integers ending with the same digit.

Original entry on oeis.org

9, 39, 49, 69, 99, 119, 129, 159, 169, 189, 219, 249, 259, 279, 289, 299, 309, 329, 339, 369, 399, 429, 459, 469, 489, 519, 529, 539, 549, 559, 579, 609, 629, 639, 669, 679, 689, 699, 729, 749, 759, 789, 799, 819, 849, 879, 889, 909, 939, 949, 959, 969, 989, 999

Offset: 1

Stefano Spezia, Oct 22 2021

Union of A346950 and A348054.

			9 = 3*3, 39 = 3*13, 49 = 7*7, 69 = 3*23, 99 = 3*33, 119 = 7*17, 129 = 3*43, 159 = 3*53, 169 = 13*13, 189 = 3*63 = 7*27, ...

Cf. A017377 (supersequence), A346950, A348054, A348547.

a={}; For[n=0, n<=100, n++, For[k=0, k<=n, k++, If[Mod[10*n+9, 10*k+3]==0 && Mod[(10*n+9)/(10*k+3), 10]==3 && 10*n+9>Max[a] || Mod[10*n+9, 10*k+7]==0 && Mod[(10*n+9)/(10*k+7), 10]==7 && 10*n+9>Max[a], AppendTo[a, 10*n+9]]]]; a

isok(m) = ((m%10) == 9) && sumdiv(m, d, (d % 10) == (m/d % 10)); \\ Michel Marcus, Oct 22 2021

def aupto(lim): return sorted(set(a*b for a in range(3, lim//3+1, 10) for b in range(a, lim//a+1, 10)) | set(a*b for a in range(7, lim//7+1, 10) for b in range(a, lim//a+1, 10)))
print(aupto(999)) # Michael S. Branicky, Oct 22 2021

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

A348548 Positive integers that are the product of two integers ending with 8.

Original entry on oeis.org

64, 144, 224, 304, 324, 384, 464, 504, 544, 624, 684, 704, 784, 864, 944, 1024, 1044, 1064, 1104, 1184, 1224, 1264, 1344, 1404, 1424, 1444, 1504, 1584, 1624, 1664, 1744, 1764, 1824, 1904, 1944, 1984, 2064, 2124, 2144, 2184, 2204, 2224, 2304, 2384, 2464, 2484, 2544

Offset: 1

Stefano Spezia, Oct 22 2021

			64 = 8*8, 144 = 8*18, 224 = 8*28, 304 = 8*38, 324 = 18*18, 384 = 8*48, ...

Cf. A017317 (supersequence), A053742 (ending with 5), A139245 (ending with 2), A324297 (ending with 6), A346950 (ending with 3), A347253 (ending with 4), A348054 (ending with 7), A348549.

a={}; For[n=0, n<=260, n++, For[k=0, k<=n, k++, If[Mod[10*n+4, 10*k+8]==0 && Mod[(10*n+4)/(10*k+8), 10]==8 && 10*n+4>Max[a], AppendTo[a, 10*n+4]]]]; a

def aupto(lim): return sorted(set(a*b for a in range(8, lim//8+1, 10) for b in range(a, lim//a+1, 10)))
print(aupto(2550)) # Michael S. Branicky, Oct 22 2021

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

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A346951 Positive integers k such that 10*k+9 is equal to the product of two integers ending with 3 (A346950).

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A346953 a(n) is the number of divisors of A346950(n) ending with 3.

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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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A347253 Positive integers that are the product of two integers ending with 4.

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A348054 Positive integers that are the product of two integers ending with 7.

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A348546 Number of positive integers with n digits that are equal both to the product of two integers ending with 3 and to that of two integers ending with 7.

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A348547 Number of positive integers with n digits and final digit 9 that are equal to the product of two integers ending with the same digit.

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A348544 Positive integers that are equal both to the product of two integers ending with 3 and to that of two integers ending with 7.

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