A359098 -id:A359098 - cp's OEIS frontend

A358737 a(n) is the greatest prime number dividing A359098(n).

101, 139, 53, 557, 223, 31, 1117, 43, 373, 59, 17, 1123, 281, 5, 563, 23, 47, 1129, 29, 283, 103, 7, 227, 71, 379, 569, 67, 163, 571, 127, 13, 229, 191, 37, 41, 383, 1151, 3, 1153, 577, 11, 17, 89, 193, 61, 43, 83, 1163, 97, 233, 53, 389, 73, 167, 1171, 293

Offset: 1

Rémy Sigrist, Jan 04 2023

Bugeaud proves that a(n) tends to infinity as n tends to infinity.

			For n = 2:
- A359098(2) = 1112 = 2^3 * 139,
- hence a(2) = 139.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Yann Bugeaud, On the digital representation of integers with bounded prime factors, Osaka J. Math. 55 (2018), 315-324; arXiv:1609.07926 [math.NT], 2016.

Cf. A006530, A359098.

Map[FactorInteger[#][[-1, 1]] &, Select[Range[1111, 1172], And[Mod[#, 10] != 0, Total@ Most@ DigitCount[#] == 4] &]] (* Michael De Vlieger, Jan 04 2023 *)

{ for (n=1, 1172, if (n%10 && #select(d->d, digits(n))==4, f = factor(n); print1 (f[#f~, 1]", "))) }

from itertools import count, islice
from sympy import primefactors
def A358737_gen(): # generator of terms
    for a in count(3):
        a10 = 10**a
        for ad in range(1,10):
            for b in range(2,a):
                b10 = 10**b
                for bd in range(1,10):
                    for c in range(1,b):
                        c10 = 10**c
                        yield from (max(primefactors(ad*a10+bd*b10+cd*c10+dd)) for cd in range(1,10) for dd in range(1,10))
A358737_list = list(islice(A358737_gen(),30)) # Chai Wah Wu, Jan 04 2023

a(n) = A006530(A359098(n)).

A359651 Numbers with exactly three nonzero decimal digits and not ending with 0.

Original entry on oeis.org

111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 123, 124, 125, 126, 127, 128, 129, 131, 132, 133, 134, 135, 136, 137, 138, 139, 141, 142, 143, 144, 145, 146, 147, 148, 149, 151, 152, 153, 154, 155, 156, 157, 158, 159, 161, 162, 163, 164, 165, 166, 167, 168, 169, 171, 172, 173, 174, 175

Offset: 1

Charles R Greathouse IV, Jan 09 2023

Bugeaud proves that the largest prime factor in a(n) increases without bound; in particular, for any e > 0 and all large n, the largest prime factor in a(n) is (1-e) * log log a(n) * log log log a(n) / log log log log a(n). So the largest prime factor in a(n) is more than k log n log log n/log log log n for any k < 1/3 and large enough n.

It appears that a(1293) = 4096 is the largest power of 2 in the sequence, a(1349) = 4608 is the largest 3-smooth number in this sequence, a(1598) = 6075 is the largest 5-smooth number in this sequence, a(5746) = 500094 is the largest 7- and 11-smooth number in this sequence, a(9158) = 5010005 is the largest 13-smooth member in this sequence, etc.

Yann Bugeaud, On the digital representation of integers with bounded prime factors, Osaka J. Math. 55 (2018), 315-324; arXiv:1609.07926 [math.NT], 2016.

Cf. A359098.

Select[Range[111,175],Length[Select[IntegerDigits[#],Positive]]==3&&Mod[#,10]!=0 &] (* Stefano Spezia, Jan 15 2023 *)

list(lim)=my(v=List()); for(d=3, #Str(lim\=1), my(A=10^(d-1)); forstep(a=A, 9*A, A, for(i=1, d-2, my(B=10^i); forstep(b=a+B, a+9*B, B, for(n=b+1, b+9, if(n>lim, return(Vec(v))); listput(v, n)))))); Vec(v)

A359983 Numbers with exactly two nonzero decimal digits and not ending with 0.

Original entry on oeis.org

11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 102, 103, 104, 105, 106, 107, 108, 109, 201

Offset: 1

Charles R Greathouse IV, Jan 20 2023

It appears that a(49) = 64 is the largest power of 2 in the sequence, a(78) = 96 is the largest 3-smooth number in this sequence, a(113) = 405 is the largest 5-smooth number in this sequence, a(170) = 1008 is the largest 7- and 11-smooth number in this sequence, a(243) = 9009 is the largest 13-smooth number in this sequence, a(259) = 20007 is the largest 19-smooth number in this sequence, etc.

Yann Bugeaud, On the digital representation of integers with bounded prime factors, Osaka J. Math. 55 (2018), 315-324; arXiv:1609.07926 [math.NT], 2016.

Cf. A359651, A359098. Subsequence of A038687.

a(n)=my(t=divrem(n-1,81)); 10*(t[2]\9+1)*10^t[1]+t[2]%9+1

Numbers of the form a*10^b + c where 0 < a,c < 10 and b > 0.

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A358737 a(n) is the greatest prime number dividing A359098(n).

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A359651 Numbers with exactly three nonzero decimal digits and not ending with 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A359983 Numbers with exactly two nonzero decimal digits and not ending with 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula