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.

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A175252 Numbers whose digit representation is equal to the digit representation of the initial terms of their sets of divisors in increasing order.

Original entry on oeis.org

1, 12, 124, 135, 1525, 13515, 124816, 12356910, 1243162124, 1525125625, 12478141928, 12510254150, 1234689111216, 1351553159265, 1597717414885, 12356910151830, 13791121336377, 123561015253050, 124510202550100, 135152575125375, 1236103206309618, 123456101215203060, 123569101518304590
Offset: 1

Views

Author

Jaroslav Krizek, Mar 14 2010

Keywords

Comments

From Michel Marcus, Sep 25 2022: (Start)
The term 124 (2^2*31) corresponds to the term of A077352 that is a prime.
The terms 135 (5*3^3), 1525 (5^2*61) and 1525125625 (5^4*2440201) correspond to the terms of A077353 that are powers of primes. (End)
The term 1597717414885 = 5 * 977 * 1741 * 187861, found by David A. Corneth, is especially remarkable for the magnitude of its 2nd smallest prime factor (counting repetitions). - Peter Munn, Oct 10 2022
Define g(n) to be the LCM of the divisors of a(n) that appear in the digit string of a(n) as specified in the definition, and let f(n) = log(g(n))/log(a(n)). Are there are only finitely many n for which f(n) >= f(4) = log(15)/log(135) = 0.55206901...? - Peter Munn, Oct 19 2022
a(26) > 10^23 (there are no terms with 23 digits). - Tim Peters, Dec 21 2022

Examples

			a(1) = 1: d(1) = {1}.
a(2) = 12: d(12) = {1, 2, 3, 4, 6, 12}.
a(3) = 124: d(124) = {1, 2, 4, 31, 62, 124}.
a(4) = 135: d(135) = {1, 3, 5, 9, 15, 27, 45, 135}.

Links

Crossrefs

Cf. A037278, A357692. Subsequence of A131835.
Cf. A077352, A077353.

Programs

  • PARI
    isok(k) = my(s=""); fordiv(k, d, s=concat(s, Str(d)); if (eval(s)==k, return(1)); if (eval(s)> k, return(0))); \\ Michel Marcus, Sep 22 2022
  • PARI
    is(n, {u = 10^5}) = { my(oldu = u, s, d, fe); s = ""; u = min(u, n); fe = factor(n, u); d = divisors(fe); if(#fe~ > 0 && fe[#fe~, 1] > u, d = select(x -> x < fe[#fe~, 1], d); ); for(i = 1, #d, if(d[i] > u, return(is(n, 10*oldu)); ); s=concat(s, Str(d[i])); if(eval(s) == n, return(1)); if(eval(s) > n, return(0)); ); is(n, 10*oldu); } \\ David A. Corneth, Oct 12 2022, Nov 07 2022
  • Python
    from sympy import divisors
def ok(n):
    target, s = str(n), ""
    if target[0] != "1": return False
    for d in divisors(n):
        s += str(d)
        if len(s) >= len(target): return s == target
        elif not target.startswith(s): return False
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Sep 22 2022

Extensions

a(9)-a(10) from Michel Marcus, Sep 22 2022
a(11)-a(12) from Michel Marcus, Oct 02 2022
a(13)-a(15) from Tim Peters, Oct 17 2022
a(16)-a(17) from Giovanni Resta, Oct 20 2022
a(18)-a(20) from Tim Peters, Oct 27 2022
a(21) from Tim Peters, Oct 30 2022
a(22)-a(23) from Tim Peters, Nov 04 2022

A077351 Smallest multiple of n using all the digits of all its divisors (a permutation of the concatenation of its divisors), or 0 if no such number exists.

Original entry on oeis.org

1, 12, 0, 124, 15, 1236, 0, 1248, 0, 11250, 0, 0, 0, 21714, 11355, 112864, 0, 0, 0, 10122540, 0, 0, 0, 1122234648, 1525, 112632, 0, 11242784, 0, 10112335560, 0, 11223648, 0, 131274, 13755, 0, 0, 123918, 0, 10012245480, 0, 11122234746, 0
Offset: 1

Views

Author

Amarnath Murthy, Nov 05 2002

Keywords

Comments

For a list of all values of n up to 10000 where a(n)=0, see A179197. - Jon E. Schoenfield, Jul 10 2010

Links

Crossrefs

Cf. A077352, A179197.

Programs

  • Maple
    A077351 := proc(n) local ndvs,ds,d,m,muldivs ; ndvs := [] ; ds := numtheory[divisors](n) ; for d from 1 to nops(ds) do ndvs := [op(ndvs), op(convert(op(d,ds),base,10))]; od ; ndvs := sort(ndvs) ; m := floor( 10^(nops(ndvs)-1)/n) ; while m*n < op(-1,ndvs)*10^(nops(ndvs)-1) do muldivs := sort(convert(m*n,base,10)) ; if muldivs = ndvs then RETURN(m*n) ; fi ; m := m+1 ; od ; RETURN(0) ; end: for n from 1 to 25 do print(n,A077351(n)) ; od ; # R. J. Mathar, Mar 20 2007

Extensions

Corrected and extended by R. J. Mathar, Mar 20 2007
a(24) from Don Reble, Nov 07 2007; a(25)-a(29) from R. J. Mathar, Mar 20 2007
More terms from Jon E. Schoenfield, Jul 02 2010

A077353 a(n) = (concatenation in ascending order of divisors of 5^n)/5^n.

Original entry on oeis.org

1, 3, 61, 12201, 2440201, 4880402001, 97608040020001, 1952160800400020001, 390432160080004000200001, 780864320160008000400002000001, 1561728640320016000800004000002000001, 31234572806400320016000080000040000020000001, 6246914561280064003200016000008000004000000200000001
Offset: 0

Views

Author

Amarnath Murthy, Nov 05 2002

Keywords

Examples

			a(5) = 15251256253125/3125 = 4880402001.

Links

Crossrefs

Cf. A000351, A055642, A077351, A069872, A077352.

Programs

  • Maple
    a:= n-> parse(cat(5^i$i=0..n))/5^n:
seq(a(n), n=0..12);  # Alois P. Heinz, May 16 2025
  • Mathematica
    A077353[n_] := FromDigits[Flatten[IntegerDigits[Divisors[#]]]]/# & [5^n];
Array[A077353, 16, 0] (* or *)
FoldList[10^IntegerLength[5^#2]/5*# + 1 &, 1, Range[15]] (* Paolo Xausa, May 19 2025 *)
  • PARI
    a(n) = eval(concat(apply(x->Str(x),divisors(5^n))))/5^n \\ Max Alekseyev, Dec 12 2011
  • PARI
    a(n) = if(n==0,1,(10^#Str(5^n)/5)*a(n-1)+1) \\ Jason Yuen, Aug 21 2024

Formula

a(0) = 1, a(n) = (10^A055642(5^n)/5)*a(n-1) + 1. - Jason Yuen, Aug 21 2024

Extensions

More terms from Max Alekseyev, Dec 12 2011
Showing 1-3 of 3 results.

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