A002473 -id:A002473 - cp's OEIS frontend

A290365 Numbers that cannot be written as a difference of 3-smooth numbers (A003586).

41, 43, 59, 67, 82, 83, 85, 86, 89, 91, 97, 103, 109, 113, 118, 121, 123, 129, 131, 133, 134, 137, 145, 149, 151, 155, 157, 163, 164, 166, 167, 169, 170, 172, 173, 177, 178, 181, 182, 185, 187, 193, 194, 197, 199, 201, 203, 205, 206, 209, 218, 221, 223, 226

Offset: 1

Michel Marcus, Aug 03 2017

nonn

Called ndh-numbers in the da Silva et al. link.
From Jon E. Schoenfield, Aug 19 2017: (Start)
If (following da Silva et al.) we refer to these numbers as "ndh-numbers" (meaning that they cannot be expressed as the difference of two "harmonic numbers" [which, in this context, are 3-smooth numbers]), we could refer to the sequence of positive integers that are not in this sequence as "dh-numbers", and say that the set of positive integers <= 100 includes the 11 ndh-numbers listed at the link (i.e., a(1) = 41 through a(11) = 97) and 100 - 11 = 89 dh-numbers. Each of the 89 dh-numbers <= 100 can be written as the difference of two 3-smooth numbers using no 3-smooth number larger than 162 (which is required to obtain the difference 98 = 162 - 64). The table below shows results from checking every difference between two 3-smooth numbers < 10^50 (which seems very nearly certain to capture all differences in [1,10^10]):
.
    Number    Number
    of ndh-   of dh-
    numbers   numbers
      in        in     Largest 3-smooth number required
k  [1,10^k]  [1,10^k]   to obtain a dh-number in [1,10^k]
=  ========  ========  ==================================
1         0        10          12 =          3 +        9
2        11        89         162 =         64 +       98
3       522       478       13122 =      12288 +      834
4      8433      1567      531441 =     524288 +     7153
5     96065      3935     6377292 =    6291456 +    85836
6    991699      8301    68024448 =   67108864 +   915584
7   9984463     15537   688747536 =  679477248 +  9270288
8  99973546     26454  7346640384 = 7247757312 + 98883072
.
A101082 gives the numbers that cannot be written as a difference of 2-smooth numbers (i.e., the powers of 2: A000079).
Numbers that cannot be written as a difference of 5-smooth numbers (A051037) appear to be 281, 289, 353, 413, 421, 439, 443, 457, 469, 493, 541, 562, 563, 578, 581, 583, 641, 653, 661, 677, 683, 691, 701, 706, 707, 731, 733, 737, 751, 761, 769, 779, 787, 793, 803, 811, 817, 823, 826, 827, 829, 841, 842, 843, 853, 857, 867, 877, 878, 881, 883, 886, ...
Numbers that cannot be written as a difference of 7-smooth numbers (A002473) appear to be 1849, 2309, 2411, 2483, 2507, 2531, 2629, 2711, 2753, 2843, 2851, 2921, 2941, 3139, 3161, 3167, 3181, 3217, 3229, 3251, 3287, 3289, 3293, 3323, 3379, 3481, 3487, 3541, 3601, 3623, 3653, 3697, 3698, 3709, 3737, 3739, 3803, 3827, 3859, 3877, 3901, 3923, 3947, ...
Numbers that cannot be written as a difference of 11-smooth numbers (A051038) appear to be 9007, 10091, 10531, 10831, 11801, 12197, 12431, 12833, 12941, 13393, 13501, 13619, 13679, 13751, 13907, 13939, 14219, 14423, 14737, 14851, 14893, 15217, 15641, 15767, 15773, 15803, 15959, 16019, 16201, 16241, 16393, 16397, 16417, 16441, 16517, 16559, 16579, ...
(End)

Natalia da Silva, Serban Raianu, Hector Salgado, Differences of Harmonic Numbers and the abc-Conjecture, arXiv:1708.00620 [math.NT], 2017.

Cf. A000079, A002473, A003586, A051037, A051038, A101082.

terms = 54;
A3586 = Select[Range[3000], FactorInteger[#][[-1, 1]] <= 3&];
dd = Union[#[[2]] - #[[1]]& /@ Subsets[A3586, {2}]];
Complement[Range[u[[-1]]], dd][[1 ;; terms]] (* Jean-François Alcover, Sep 28 2018 *)

a(12)-a(54) from Jon E. Schoenfield, Aug 18 2017

A308247 a(n) is the least integer not the difference of two prime(n)-smooth numbers.

Original entry on oeis.org

5, 41, 281, 1849, 9007, 35803

Offset: 1

Esteban Crespi de Valldaura, May 16 2019

The known terms have been found by exhaustive search and then proved not to be the difference of prime(n)-smooth numbers using assertions such as +- a(n) !== (modulo m) meaning that no element of the subgroup of Z/m generated by a,b,... added to a(n) is congruent modulo m with an element of the subgroup generated by . For example: <2> +- 41 !== <3> (mod 91) and the fact that 41+1 is not 3-smooth is enough to prove that 41 is not the difference of 3-smooth numbers; <2> + 281 !== <3,5> (mod 13981), <2> - 281 !== <3,5> (mod 76627) and <3> +- 281 !== <2,5> along with the fact that 281+1 is not 5-smooth is enough to show that 281 is not the difference of 5-smooth numbers. The proofs get exponentially harder as n increases. For example, <2, 11> + 9007 !== <3, 5, 7> (mod 308859288230831), or <2,5,7> + 35803 !== <3,11,13> (mod 2219897250633559197203).

The next few terms are conjectured to be 158857, 681179, 2516509, 10772123, 51292187, 186323681; if they were not, they would provide examples of ABC-triples with quality greater than 2.

			We see that 1 = 2-1, 2 = 4-2, 3 = 4-1, and 4 = 8-4. It is easy to see that 5 is not the difference of two powers of 2, so a(1) = 5. In the same way we can see that all the integers up to 40 are the difference of 3-smooth numbers, but as shown above 41 is not, so a(2)=41.

Esteban Crespi de Valldaura, Proof that a(n) is not prime(n)-smooth for n=2,3,4,5,6
Wikipedia, abc conjecture

P-smooth_numbers: A000079, A003586, A051037, A002473, A051038, ...

a(i) is the first term in each of A101082, A290365, A308456, A326318, A326319, A326320.

A333953 Recursively superabundant numbers: numbers m such that A330575(m)/m > A330575(k)/k for all k < m.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 24, 36, 48, 72, 96, 120, 144, 240, 288, 360, 480, 576, 720, 1152, 1440, 2160, 2880, 4320, 5760, 8640, 11520, 17280, 25920, 30240, 34560, 51840, 60480, 69120, 103680, 120960, 172800, 181440, 207360, 241920, 345600, 362880, 414720, 483840, 725760

Offset: 1

Amiram Eldar, Apr 11 2020

nonn

Fink (2019) defined this sequence. He asked whether 720 is the largest term that is also superabundant number (A004394).
He noted that up to 10^6 all the recursively superabundant numbers are also recursively highly composite numbers (A333952), except for 181440 (the next term which is not recursively highly composite is 2177280). He asked whether there are a finite number of numbers that are both recursively highly composite and recursively superabundant (in analogy to A166981).
From David A. Corneth, Apr 13 2020: (Start)
The 76 terms in the b-file are products of primorials (Cf. A025487) and 7-smooth numbers  (Cf. A002473). All terms are in A025487.
Proof: As A330575(n) = Sum_{d|n} A074206(d) * n/d we have A330575(n) / n = Sum_{d|n} A074206(d)/d which is maximal for some prime signature when n is a product of primorials.
Assuming terms below 10^17 are 13-smooth gives the 213 11-smooth numbers in the Corneth a-file. (End)

David A. Corneth, Table of n, a(n) for n = 1..213 (first 76 terms from Amiram Eldar)
Thomas Fink, Recursively divisible numbers, arXiv:1912.07979 [math.NT], 2019. See section 5.

Cf. A002473, A004394, A166981, A025487, A330575, A333952.

s[1] = 1; s[n_] := s[n] = n + DivisorSum[n, s[#] &, # < n &]; seq={}; rm = 0; Do[r1 = s[n]/n; If[r1 > rm, rm = r1; AppendTo[seq, n]], {n, 1, 10^4}]; seq

A063072 Sum of divisors of Ramanujan's highly composite numbers, or sigma(A002182(n)).

Original entry on oeis.org

1, 3, 7, 12, 28, 60, 91, 124, 168, 360, 546, 744, 1170, 2418, 2880, 4368, 5952, 9360, 19344, 28800, 39312, 59520, 79248, 99944, 112320, 180048, 203112, 232128, 345600, 471744, 714240, 950976, 1199328, 1451520, 2160576, 2437344, 2926080

Offset: 1

Jason Earls, Aug 02 2001

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..50 from Harry J. Smith)

Cf. A000005, A004394, A002182, A002183, A002473, A000203.

s={}; dm=0; Do[d = DivisorSigma[0,n]; If[d > dm, dm=d; AppendTo[s, DivisorSigma[1,n]]], {n, 1, 10^5}]; s (* Amiram Eldar, Jun 28 2019 *)

a=0; j=[]; for(n=1,200000,b=numdiv(n); if(b>a,a=b; j=concat(j, sigma(n)))); j

{ n=a=0; for (m=1, 10^9, b=numdiv(m); if(b>a, a=b; write("b063072.txt", n++, " ", sigma(m)); if (n==50, break)) ) } \\ Harry J. Smith, Aug 16 2009

a(n) = A000203(A002182(n)). - Michel Marcus, Jun 28 2018

More terms from Reiner Martin, Dec 22 2001

A085908 Smallest 7-smooth number beginning with n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 112, 12, 135, 14, 15, 16, 175, 18, 192, 20, 21, 224, 2304, 24, 25, 2625, 27, 28, 294, 30, 315, 32, 336, 343, 35, 36, 375, 384, 392, 40, 4116, 42, 432, 441, 45, 4608, 4704, 48, 49, 50, 512, 525, 5376, 54, 55125, 56, 576, 588, 59049, 60

Offset: 1

Amarnath Murthy, Jul 09 2003

			a(23) = 2304 = 2^8*3^2 is the smallest 7-smooth number beginning with 23. (23, 230, 231, 232, ..., 239, 2301, 2302, 2303 etc. have a divisor > 10.)

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A002473.

N:= 300: # for a(1) .. a(N)
V:= Vector(N): count:= 0:with(priqueue):
initialize(pq);
insert([-1,0,0,0,0],pq);
while count < N do
  t:= extract(pq);
  x:= -t[1];
  for d from ilog10(x) to 0 by -1 do
    xd:= floor(x/10^d);
    if xd > N then break fi;
    if V[xd] = 0 then V[xd]:= x; count:= count+1; fi;
  od;
  insert([-7*x,t[2],t[3],t[4],t[5]+1],pq);
  if t[5]=0 then
    insert([-5*x,t[2],t[3],t[4]+1,0],pq);
    if t[4] = 0 then
      insert([-3*x,t[2],t[3]+1,0,0],pq);
      if t[3] = 0 then
        insert([-2*x,t[2]+1,0,0,0],pq);
  fi fi fi;
od:
convert(V,list); # Robert Israel, Sep 18 2024

a[n_] := Module[{d = IntegerDigits[n], k = 1}, While[Max[FactorInteger[k][[;; , 1]]] > 7 || Length[IntegerDigits[k]] < Length[d] || IntegerDigits[k][[1 ;; Length[d]]] != d, k++]; k]; Array[a, 60] (* Amiram Eldar, Apr 30 2022 *)

hc(n) = local(f); f = factor(n); f[matsize(f)[1], 1] < 10;
a(n) = local(d, x); if (hc(n), return(n)); d = 1; while (d, for (i = 1, 10^d - 1, x = n*10^d + i; if (hc(x), return(x))); d++); \\ David Wasserman, Feb 11 2005

from itertools import count
from sympy import integer_log
def A085908(n):
    if n<2: return n
    def f(x):
        c = 0
        for i in range(integer_log(x,7)[0]+1):
            for j in range(integer_log(m:=x//7**i,5)[0]+1):
                for k in range(integer_log(r:=m//5**j,3)[0]+1):
                    c += (r//3**k).bit_length()
        return c
    for l in count(0):
        kmin, kmax = n*10**l-1, (n+1)*10**l-1
        mmin, mmax = f(kmin), f(kmax)
        if mmax>mmin:
            while kmax-kmin > 1:
                kmid = kmax+kmin>>1
                mmid = f(kmid)
                if mmid > mmin:
                    kmax, mmax = kmid, mmid
                else:
                    kmin, mmin = kmid, mmid
            return kmax # Chai Wah Wu, Sep 17 2024

Corrected and extended by David Wasserman, Feb 11 2005

Name corrected by J. Lowell, Apr 30 2022

A086290 Minimal exponent in prime factorization of 7-smooth numbers.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 2, 3, 1, 1, 5, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 6, 1, 2, 1, 1, 4, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 7, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 3, 1, 2, 1, 5, 1, 1, 1, 8, 1, 1, 2, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 2, 1, 1

Offset: 1

Reinhard Zumkeller, Jul 15 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002473, A051904, A086289, A086291.

minExp[1] = 0; minExp[n_] := Min @@ Last /@ FactorInteger[n]; minExp/@Select[Range[500], Max[Transpose[FactorInteger[#]][[1]]] <= 7 &] (* Amiram Eldar, Jan 06 2020 *)

a(n) = A051904(A002473(n)).

a(n) <= A086291(n) <= A086289(n).

A086293 Sum of divisors of 7-smooth numbers.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 28, 24, 24, 31, 39, 42, 32, 60, 31, 40, 56, 72, 63, 48, 91, 90, 96, 78, 124, 57, 93, 120, 120, 168, 104, 127, 144, 195, 124, 186, 121, 224, 234, 252, 171, 217, 192, 280, 248, 360, 156, 312, 255, 240, 336, 403, 228, 372, 378

Offset: 1

Reinhard Zumkeller, Jul 15 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A002473, A086292.

DivisorSigma[1, Select[Range[500], Max[Transpose[FactorInteger[#]][[1]]] <= 7 &]] (* Amiram Eldar, Jan 06 2020 after G. C. Greubel at A086288 *)

a(n) = A000203(A002473(n)).

A086299 a(n) = if n is 7-smooth then 1 else 0: characteristic function of 7-smooth numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0

Offset: 1

Reinhard Zumkeller, Jul 15 2003

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Smooth Number.
Index entries for characteristic functions.

Cf. A002473, A006530, A068191.

a086299 = fromEnum . (<= 7) . a006530  -- Reinhard Zumkeller, Apr 01 2012

Table[If[Max[Transpose[FactorInteger[n]][[1]]]<11,1,0],{n,110}] (* Harvey P. Dale, Oct 08 2013 *)
smooth7Q[n_] := n == Times@@({2, 3, 5, 7}^IntegerExponent[n, {2, 3, 5, 7}]);
a[n_] := Boole[smooth7Q[n]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 17 2021 *)

Multiplicative with: a(p) = if p<=7 then 1 else 0, p prime.
a(A002473(n)) = 1; a(A068191(n)) = 0. - Reinhard Zumkeller, Apr 01 2012
Dirichlet g.f.: 1/((1-2^(-s))*(1-3^(-s))*(1-5^(-s))*(1-7^(-s))). - Amiram Eldar, Dec 27 2022

A262401 In prime factorization of n: replace each factor with its largest decimal digit.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 12, 3, 14, 15, 16, 7, 18, 9, 20, 21, 2, 3, 24, 25, 6, 27, 28, 9, 30, 3, 32, 3, 14, 35, 36, 7, 18, 9, 40, 4, 42, 4, 4, 45, 6, 7, 48, 49, 50, 21, 12, 5, 54, 5, 56, 27, 18, 9, 60, 6, 6, 63, 64, 15, 6, 7, 28, 9, 70, 7, 72, 7, 14

Offset: 1

Reinhard Zumkeller, Sep 25 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A054055, A027746, A001222, A002473, A006530, A060418, A068191.

a262401 = product . map a054055 . a027746_row'

Array[Times @@ (Power[Max@ IntegerDigits[#1], #2] & @@@ FactorInteger[#]) &, 74] (* Michael De Vlieger, Jan 23 2022 *)

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = vecmax(digits(f[k,1]))); factorback(f); \\ Michel Marcus, Jan 22 2022

Multiplicative with p -> A054055(p), p prime.
a(n) = Product_{k=1..A001222(n)} A054055(A027746(n,k)).
a(n) <= n.
a(m) = m iff m is 7-smooth:
  a(A002473(n)) = A002473(n) and a(A068191(n)) < A068191(n).
A006530(a(n)) <= 7.
a(a(n)) = a(n).

A068184 Smallest number whose product of digits equals n!.

Original entry on oeis.org

1, 1, 2, 6, 38, 358, 2589, 25789, 257889, 2578879, 45578899

Offset: 0

Labos Elemer, Feb 18 2002

The sequence is finite because n! for n>10 has 2-digit prime factors.

			For n=4 the solutions having digit products equal 24 excluding those with digit 1 are: {38, 46, 64, 83, 226, 234, 243, 262, 324, 342, 423, 432, 622, 2223, 2232, 2322, 3222} of which the smallest is 38. For n>1, numbers with a digit 1 are too big.

Cf. A000142, A001222, A002473, A067734, A068183-A068187, A068189-A068191.

Edited by Henry Bottomley, Feb 26 2002

cp's OEIS Frontend

A290365 Numbers that cannot be written as a difference of 3-smooth numbers (A003586).

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A308247 a(n) is the least integer not the difference of two prime(n)-smooth numbers.

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A333953 Recursively superabundant numbers: numbers m such that A330575(m)/m > A330575(k)/k for all k < m.

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A063072 Sum of divisors of Ramanujan's highly composite numbers, or sigma(A002182(n)).

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A085908 Smallest 7-smooth number beginning with n.

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A086290 Minimal exponent in prime factorization of 7-smooth numbers.

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A086293 Sum of divisors of 7-smooth numbers.

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A086299 a(n) = if n is 7-smooth then 1 else 0: characteristic function of 7-smooth numbers.

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