A051037 -id:A051037 - cp's OEIS frontend

A071520 Number of 5-smooth numbers (A051037) <= n.

1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 10, 10, 11, 12, 12, 13, 13, 14, 14, 14, 14, 15, 16, 16, 17, 17, 17, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 23, 23, 24, 24, 24, 24, 25, 25, 25, 25, 25, 25, 26, 26, 26, 26, 27, 27, 27, 27, 27, 27, 27, 27, 28, 28

Offset: 1

Benoit Cloitre, Jun 02 2002

A 5-smooth number is a number of the form 2^x*3^y*5^z (x,y,z) >= 0.

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

Cf. A008683, A051037, A106598, A112751.

Number of p-smooth numbers <= n: A070939 (p=2), A071521 (p=3), this sequence (p=5), A071604 (p=7), A071523 (p=11), A080684 (p=13), A080685 (p=17), A080686 (p=19).

Accumulate[Table[If[Max[FactorInteger[n][[;;,1]]]<6,1,0],{n,80}]] (* Harvey P. Dale, Aug 04 2024 *)

for(n=1,100,print1(sum(k=1,n,if(sum(i=4,n,if(k%prime(i),0,1)),0,1)),","))

a(n)=-sum(k=1,n,moebius(2*3*5*k)*floor(n/k)) \\ Benoit Cloitre, Jun 14 2007

from sympy import integer_log
def A071520(n):
    c = 0
    for i in range(integer_log(n,5)[0]+1):
        for j in range(integer_log(m:=n//5**i,3)[0]+1):
            c += (m//3**j).bit_length()
    return c # Chai Wah Wu, Sep 16 2024

a(n) = Card{ k | A051037(k) <= n }.
Asymptotically : let a = 1/(6*log(2)*log(3)*log(5)) and b = sqrt(30) then a(n) = a*log(b*n)^3 + O(log(n)).
a(n) = -Sum_{k=1,n} mu(30*k)*floor(n/k). - Benoit Cloitre, Jun 14 2007
a(n) = Sum_{i=0..floor(log_5(n))} Sum_{j=0..floor(log_3(n/5^i))} floor(log_2(2*n/(5^i*3^j))). - Ridouane Oudra, Jul 17 2020

Title corrected by Rainer Rosenthal, Aug 30 2020

A308456 Numbers that cannot be written as a difference of 5-smooth numbers (A051037).

Original entry on oeis.org

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

Offset: 1

Esteban Crespi de Valldaura, May 26 2019

nonn

Terms were found by generating in sequential order the 5-smooth numbers up to some limit and collecting the differences. The first 1000 candidates k were then proved to be correct by showing that each of the following congruences holds:
  {5} +- k !== {2,3} mod 205910575871,
  {3} +- k !== {2,5} mod 220411358713,
  {2} +- k !== {3,5} mod 3019333681,
where {a,b,...} represents the subgroup generated by a,b,... of the multiplicative subgroup modulo m. For a discussion iof this method of proof see A308247.

			281 = A308247(3) cannot be written as the difference of 5-smooth numbers. All smaller numbers can; for example, 277 = 3^4*5 - 2^7, 271 = 2^3*5^3 - 3^6.

Esteban Crespi de Valldaura, Table of n, a(n) for n = 1..1000

Cf. A051037 (5-smooth numbers).
Cf. numbers not the difference of p-smooth numbers for other values of p: A101082 (p=2), A290365 (p=3), A326318 (p=7), A326319 (p=11), A326320 (p=13).
Cf. A308247.

\\ Computes the first N elements in the sequence.
\\ At least the first 10000 are correct.
N=100;
\\computes the multiplicative subgroup generated
\\by the elements of the vector L modulo m.
SGR(L,m)={S=[1];for(l=1,length(L),z=znorder(Mod(L[l],m));T=[1];for(t=1,z,s=lift(Mod(L[l],m)^t);if(setsearch(S,s),break,T=concat(T,s);));for(t=1,length(T),S=Set(concat(S,lift(S*Mod(T[t],m))))));S}
m1=205910575871; L1= SGR([2,3],m1); M1 = SGR([5],m1);
m2=220411358713; L2= SGR([2,5],m2); M2 = SGR([3],m2);
m3=  3019333681; L3= SGR([3,5],m3); M3 = SGR([2],m3);
chkdif(k)={r=1;
   D=1;while(gcd(k/D,30)>1,D*=gcd(k/D,30));
   fordiv(D,d,
     if(vecmax(factor(k/d+1)[,1])<= 5 ,r=0);
     if(r,for(t=1,length(M1),
       if(setsearch(L1,(M1[t]+k/d)%m1),r=0;break)));
     if(r,for(t=1,length(M2),
       if(setsearch(L2,(M2[t]+k/d)%m2),r=0;break)));
     if(r,for(t=1,length(M3),
       if(setsearch(L3,(M3[t]+k/d)%m3),r=0;break)));
     if(r,for(t=1,length(M1),
       if(setsearch(L1,(M1[t]-k/d)%m1),r=0;break)));
     if(r,for(t=1,length(M2),
       if(setsearch(L2,(M2[t]-k/d)%m2),r=0;break)));
     if(r,for(t=1,length(M3),
       if(setsearch(L3,(M3[t]-k/d)%m3),r=0;break)));
     if(r==0, break)
   );
   r
}
for(k=1,m3,if(chkdif(k),print1(k,", ");if(N--==0, break))); print();

A188425 Position of 2^n in A051037 (5-smooth numbers).

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 27, 38, 52, 68, 87, 110, 137, 167, 201, 240, 284, 332, 386, 446, 511, 582, 660, 745, 836, 934, 1041, 1155, 1277, 1407, 1545, 1692, 1849, 2015, 2190, 2376, 2571, 2777, 2994, 3222, 3461, 3711, 3974, 4249, 4535, 4834, 5147, 5473, 5811, 6164, 6531, 6911, 7306, 7716, 8142, 8582, 9039, 9511, 9999, 10503, 11024, 11563, 12119, 12692, 13283, 13892, 14519

Offset: 0

Zak Seidov, Mar 30 2011

nonn

			2^0 = 1 = A051037(1), hence a(0) = 1.
a(25) = 934 because 2^25 = 33554432 = A051037(934).

Zak Seidov, Table of n, a(n) for n = 0..200

Cf. A051037, A188426, A188427 (positions of powers of 3 and 5 in A051037).

A051037(a(n)) = 2^n.

a(n) ~ c * n^3, where c = log(2)^2/(6*log(3)*log(5)) = 0.045287775775708... - Vaclav Kotesovec and Amiram Eldar, Sep 22 2024

A188426 Position of 3^n in A051037 (5-smooth numbers).

Original entry on oeis.org

1, 3, 8, 17, 31, 50, 77, 112, 157, 211, 276, 355, 448, 554, 676, 815, 973, 1149, 1344, 1561, 1801, 2064, 2351, 2663, 3003, 3371, 3766, 4190, 4647, 5135, 5657, 6212, 6802, 7430, 8094, 8797, 9539, 10322, 11147, 12014, 12925, 13881, 14883, 15933, 17031, 18178, 19375, 20624, 21925, 23281, 24690

Offset: 0

Zak Seidov, Mar 31 2011

nonn

Zak Seidov, Table of n, a(n) for n = 0..150

Cf. A051037, A188425, A188427 (powers of 2 and 5).

A051037(a(n)) = 3^n.

a(n) ~ c * n^3, where c = log(3)^2/(6*log(2)*log(5)) = 0.180317536836709... - Vaclav Kotesovec and Amiram Eldar, Sep 22 2024

A188427 Position of 5^n in A051037 (5-smooth numbers).

Original entry on oeis.org

1, 5, 16, 37, 72, 125, 198, 295, 419, 574, 764, 991, 1259, 1572, 1931, 2342, 2808, 3332, 3918, 4569, 5288, 6078, 6943, 7887, 8913, 10024, 11224, 12515, 13902, 15387, 16975, 18668, 20470, 22385, 24416, 26567, 28840, 31239, 33768, 36429, 39226, 42164, 45244, 48470, 51846, 55375, 59061, 62907, 66917, 71094, 75441

Offset: 0

Zak Seidov, Mar 31 2011

nonn

Zak Seidov, Table of n, a(n) for n = 0..102

Cf. A051037, A188425, A188426 (powers of 2 and 3).

A051037(a(n)) = 5^n.

a(n) ~ c * n^3, where c = log(5)^2/(6*log(2)*log(3)) = 0.566927196003501... - Vaclav Kotesovec and Amiram Eldar, Sep 22 2024

A372400 Position of 30^n among 5-smooth numbers A051037.

Original entry on oeis.org

1, 18, 83, 228, 486, 888, 1466, 2255, 3283, 4583, 6189, 8134, 10445, 13158, 16305, 19916, 24027, 28667, 33870, 39665, 46086, 53166, 60937, 69429, 78675, 88709, 99561, 111263, 123849, 137347, 151793, 167219, 183658, 201139, 219695, 239359, 260165, 282141, 305320

Offset: 0

Michael De Vlieger, Jun 03 2024

nonn

Also position of 30^(n+1) in A143207.

Charles R Greathouse IV, Table of n, a(n) for n = 0..1000

Cf. A002110, A051037, A143207, A202821, A372401, A372402.

Table[Sum[Floor@ Log[5, 30^n/(2^i*3^j)] + 1, {i, 0, Log[2, 30^n]}, {j, 0, Log[3, 30^n/2^i]}], {n, 0, 38}]

a(n)=my(t=30^n,u=5*t); sum(a=0,logint(t,5), u\=5; sum(b=0,logint(u,3), logint(u\3^b,2)+1)) \\ Charles R Greathouse IV, Sep 18 2024

# uses imports/function in A372401
print(list(islice(A372401gen(p=5), 40))) # Michael S. Branicky, Jun 05 2024

from sympy import integer_log
def A372400(n):
    c, x = 0, 30**n
    for i in range(integer_log(x,5)[0]+1):
        for j in range(integer_log(y:=x//5**i,3)[0]+1):
            c += (y//3**j).bit_length()
    return c # Chai Wah Wu, Sep 16 2024

a(n) = k*n^3 + (3k/2)*n^2 + O(n) where k = (log 30)^3/(6 log 2 log 3 log 5) = 5.35057081984.... - Charles R Greathouse IV, Sep 19 2024

A250089 5-smooth numbers (A051037) written in base 60, concatenating the decimal values of the sexagesimal digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 100, 104, 112, 115, 120, 121, 130, 136, 140, 148, 200, 205, 208, 215, 224, 230, 240, 242, 300, 312, 320, 336, 345, 400, 403, 410, 416, 430, 448, 500, 520, 524, 600

Offset: 1

Michael De Vlieger, Nov 11 2014

Each pair of digits constitutes the decimal value of a single sexagesimal digit, as on a digital clock, eliminating the colon (:). Any leading zeros are truncated. Thus decimal 64 appears as "104" and not "0104".

			a(28) = 112 since A051037(28) = 72. 72 = 1 * 60 + 12, thus sexagesimal 1,12. Concatenating the decimal values of the sexagesimal places gives "112".

D. E. Knuth, Ancient Babylonian Algorithms, Communications of the ACM 15 (1972): 671-677.

Michael De Vlieger, Table of n, a(n) for n = 1..10540 (Hamming numbers <= 60^10)

Cf. A051037, A250073, A254334, A254335, A254336.

a250089[n_Integer] := FromDigits /@ Map[StringJoin, If[# < 10, StringJoin["0", ToString[#]], ToString[#]] & /@ IntegerDigits[#, 60] & /@ Select[Range[n], Last@Map[First, FactorInteger@#] < 7 &], 2]; a250089[360] (* Michael De Vlieger, Nov 11 2014, after Robert G. Wilson v at A051037 *)
With[{n = 360}, Map[FromDigits@ IntegerDigits[#, MixedRadix[ Flatten@ ConstantArray[{6, 10}, {2 Ceiling@ Log[60, n]}]]] &, Union@ Flatten@ Table[2^p1*3^p2*5^p3, {p1, 0, Log[2, n/(1)]}, {p2, 0, Log[3, n/(2^p1)]}, {p3, 0, Log[5, n/(2^p1*3^p2)]}]]] (* Version 10.2, or *)
With[{n = 360}, FromDigits@ StringJoin@ Map[If[# < 10, StringJoin["0", ToString@ #], ToString@ #] &, IntegerDigits[#, 60]] & /@ Union@ Flatten@ Table[2^p1*3^p2*5^p3, {p1, 0, Log[2, n/(1)]}, {p2, 0, Log[3, n/(2^p1)]}, {p3, 0, Log[5, n/(2^p1*3^p2)]}]] (* Michael De Vlieger, Feb 20 2017 *)

A352219 a(n) is the least k such that A051037(n) | 60^k.

Original entry on oeis.org

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

Offset: 1

Michael De Vlieger, Mar 08 2022

Also, number of digits in the sexagesimal expansion of terminating unit fractions 1/A051037.

			a(1) = 0 since A051037(1) = 1 | 60^0.
a(2) = 1 since A051037(2) = 2 | 60^1; 1/2 in base 60 is represented by digit 30 after the radix point ":", i.e., :30.
a(7) = 2 since A051037(7) = 8 | 60^2; 1/8 in base 60 is :7,30, etc.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Chapter IX: The Representation of Numbers by Decimals, Theorem 136. 8th ed., Oxford Univ. Press, 2008, 144-145.

Michael De Vlieger, Table of n, a(n) for n = 1..10540 (A051037(10540) = 60^10)
Eric Weisstein's World of Mathematics, Sexagesimal
Wikipedia, Regular number.

Cf. A051037, A086415, A117920.

With[{nn = 1024}, Sort[Flatten[Table[{2^a * 3^b * 5^c, Max[Ceiling[a/2], b, c]}, {a, 0, Log2[nn]}, {b, 0, Log[3, nn/(2^a)]}, {c, 0, Log[5, nn/(2^a*3^b)]}], 2]][[All, -1]] ]

a(n) ≍ n^(1/3), with lim sup a(n)/n^(1/3) being (6*log(2)*log(3)*log(5))^(1/3)/log(3) = 1.770... where A051037(n) is a power of 3 and the lim inf being (6*log(2)*log(3)*log(5))^(1/3)/log(60) = 0.4749... where A051037(n) is a power of 60. - Charles R Greathouse IV, Mar 08 2022

A219794 First differences of 5-smooth numbers (A051037).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 2, 4, 1, 2, 3, 2, 4, 4, 5, 3, 2, 4, 6, 4, 8, 3, 5, 1, 9, 6, 4, 8, 12, 5, 3, 7, 9, 6, 10, 2, 18, 12, 8, 16, 9, 15, 3, 7, 6, 14, 18, 12, 20, 4, 36, 15, 9, 16, 5, 27, 18, 30, 6, 14, 12, 28, 36, 24, 25, 15, 8, 27, 45, 9, 21, 18

Offset: 1

Zak Seidov, Nov 28 2012

nonn

lim inf a(n) >= 2 by Størmer's theorem. Is lim a(n) = infinity? It would be very surprising if this were false, since then there is some k such that n and n+k are both 5-smooth for infinitely many n. - Charles R Greathouse IV, Nov 28 2012

A085152 gives all n's for which a(n) = 1. Thue-Siegel-Roth theorem gives lim a(n) = infinity. With the aid of lower bounds for linear forms in logarithms, Tijdeman showed that a(n+1)-a(n) > a(n)/(log a(n))^C for some effectively computable constant C. - Tomohiro Yamada, Apr 15 2017

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (First 1000 terms from Zak Seidov)
Carl Størmer, Quelques théorèmes sur l'équation de Pell x^2 - Dy^2 = +-1 et leurs applications, Skrifter Videnskabs-selskabet (Christiania), Mat.-Naturv. Kl. I (2), 48 pp.
Robert Tijdeman, On integers with many small prime factors, Compos. Math. 26 (1973), 319-330.

Cf. A051037, A085152.

Differences@ Union@ Flatten@ Table[2^i * 3^j * 5^k, {i, 0, Log2[#]}, {j, 0, Log[3, #/(2^i)]}, {k, 0, Log[5, #/(2^i*3^j)] } ] &[1000] (* Michael De Vlieger, Mar 16 2024 *)

a(n) = A051037(n+1) - A051037(n).

A353385 Irregular triangle T(n,k) with row n listing A051037(j) not divisible by 60 such that A352219(j) = n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 8, 9, 16, 18, 24, 25, 36, 40, 45, 48, 50, 72, 75, 80, 90, 100, 144, 150, 200, 225, 400, 450, 27, 32, 54, 64, 96, 108, 125, 135, 160, 192, 216, 250, 270, 288, 320, 375, 432, 500, 576, 675, 750, 800, 864, 1000, 1125, 1350, 1600

Offset: 0

Michael De Vlieger, Apr 15 2022

All terms in A051037 are products T(n,k)*60^j, j >= 0.
When expressed in base 60, T(n,k) does not end in zero, yet 1/T(n,k) is a terminating fraction, regular to 60.
The first 11 terms are the proper divisors of 60.
For these reasons, the terms may be called sexagesimal "proper regular" numbers.

			For row w, plot terms m = 2^x * 3^y * 5^z at (x,y,z). Rows are labeled below the figures parenthetically for clarity. The x axis points toward the bottom right, the y axis to the bottom left, and the z axis upward. In the plot, we mark terms from previous rows by ".", and use "*" to show the origin, that is, the empty product 1:
                                                   125
                                                375   250
                                             1125   750   500
                                         3375  2250        1000
                                            6750              2000
                         25                          .           4000
                      75    50                    .     .           8000
                  225   150   100              .     .     .
                     450         200      675     .           .
                                    400     1350                 .
         5                .                          .            800
     15    10          .     .                    .     .           1600
         30    20   45     .     .              .     .     .
                      90          40      135     .           .
                                      80      270                 .
1         *                *                          *            160
       3     2          .     .                    .     .            320
          6     4    9     .     .              .     .     .
            12         18     .     8       27     .     .     .
                          36    24    16       54     .     .     .
                             72    48            108     .     .    32
                               144                  216     .    96    64
                                                       432   288   192
                                                          864   432
                                                            1728
(0)      (1)              (2)                        (3)
The terms in row w are sorted, hence row 1 has {2, 3, 4, 5, 6, 10, 12, 15, 20, 30}.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Chapter IX: The Representation of Numbers by Decimals, Theorem 136. 8th ed., Oxford Univ. Press, 2008, 144-145.

Michael De Vlieger, Table of n, a(n) for n = 0..10250 (rows n = 1..40, flattened)
Eric Weisstein's World of Mathematics, Sexagesimal
Wikipedia, Regular number.

Cf. A017641, A051037, A352219.

Block[{t, s = DeleteCases[Sort[Flatten[Table[{2^a* 3^b * 5^c, Max[Ceiling[a/2], b, c]}, {a, 0, Log2[#]}, {b, 0, Log[3, #/(2^a)]}, {c, 0, Log[5, #/(2^a*3^b)]}], 2]] &[60^3], _?(Mod[First[#], 60] == 0 &)]}, #[[1 ;; 2 + LengthWhile[Rest@ Differences[Length /@ #], # == 12 &]]] &@ Map[s[[#, 1]] &, Values@ PositionIndex[s[[All, -1]]]]] // Flatten

Row 0 contains the empty product, thus row length = 1.

For n > 0, length of row n = 12(n-1) + 10 = A017641(n-1).

cp's OEIS Frontend

A071520 Number of 5-smooth numbers (A051037) <= n.

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A308456 Numbers that cannot be written as a difference of 5-smooth numbers (A051037).

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A188425 Position of 2^n in A051037 (5-smooth numbers).

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A188426 Position of 3^n in A051037 (5-smooth numbers).

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A188427 Position of 5^n in A051037 (5-smooth numbers).

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A372400 Position of 30^n among 5-smooth numbers A051037.

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A250089 5-smooth numbers (A051037) written in base 60, concatenating the decimal values of the sexagesimal digits.

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A352219 a(n) is the least k such that A051037(n) | 60^k.

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