A355583 -id:A355583 - cp's OEIS frontend

A060753 Denominator of 1246...(prime(n-1)-1) / (2357...prime(n-1)).

1, 2, 3, 15, 35, 77, 1001, 17017, 323323, 676039, 2800733, 86822723, 3212440751, 131710070791, 5663533044013, 11573306655157, 47183480978717, 95993978542907, 5855632691117327, 392327390304860909

Offset: 1

Frank Ellermann, Apr 23 2001

Equivalently, numerator of Product_{k=1..n-1} prime(k)/(prime(k)-1) (cf. A038110). - N. J. A. Sloane, Apr 17 2015
a(n)/A038110(n) is the supremum of the abundancy index sigma(k)/k = A000203(k)/k of the prime(n-1)-smooth numbers, for n>1 (Laatsch, 1986). - Amiram Eldar, Oct 26 2021
From Amiram Eldar, Jul 10 2022: (Start)
a(n)/A038110(n) is the sum of the reciprocals of the prime(n-1)-smooth numbers, for n>1.
a(n)/A038110(n) is the asymptotic mean of the number of prime(n-1)-smooth divisors of the positive integers, for n>1 (cf. A001511, A072078, A355583). (End)

			A038110(50)/ a(50) = 0.1020..., exp(-gamma)/log(229) = 0.1033...
1*2*4/(2*3*5) = 4/15 has denominator a(4) = 15. - _Jonathan Sondow_, Jan 31 2014

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 429.

Michael De Vlieger, Table of n, a(n) for n = 1..423
Frank Ellermann, Illustration for A002110, A005867, A038110, A060753.
Richard Laatsch, Measuring the abundancy of integers, Mathematics Magazine, Vol. 59, No. 2 (1986), pp. 84-92.
Jonathan Sondow and Eric Weisstein, Euler Product, MathWorld.

Cf. A000203, A002110, A005867, A038110, A038111.
Cf. A236435, A236436.
Cf. A001511, A072078, A355583.

[1] cat [Denominator((&*[NthPrime(k-1)-1:k in [2..n]])/(&*[NthPrime(k-1):k in [2..n]])):n in [2..20]]; // Marius A. Burtea, Sep 19 2019

Table[Denominator@ Product[EulerPhi@ Prime[i]/Prime@ i, {i, n}], {n, 0, 19}] (* Michael De Vlieger, Jan 10 2015 *)
{1}~Join~Denominator@ FoldList[Times, Table[EulerPhi@ Prime[n]/Prime@ n, {n, 19}]] (* Michael De Vlieger, Jul 26 2016 *)
b[0] := 0; b[n_] := b[n - 1] + (1 - b[n - 1]) / Prime[n]
Denominator@ Table[b[n], {n, 0, 20}] (* Fred Daniel Kline, Jun 27 2017 *)
Join[{1},Denominator[With[{nn=20},FoldList[Times,Prime[Range[nn]]-1]/FoldList[ Times,Prime[Range[nn]]]]]] (* Harvey P. Dale, Apr 17 2022 *)

a(n) = A002110(n) / gcd( A005867(n), A002110(n) ).
A038110(n) / a(n) ~ exp( -gamma ) / log( prime(n) ), Mertens's theorem for x = prime(n) = A000040(n).
A038110(n) / a(n) = A005867(n) / A002110(n). - corrected by Simon Tatham, Jul 26 2016
a(n) = A038111(n) / prime(n). - Vladimir Shevelev, Jan 10 2014
a(n) = A038110(n) + A161527(n-1). - Jamie Morken, Jun 19 2019

Definition corrected by Jonathan Sondow, Jan 31 2014

A355582 a(n) is the largest 5-smooth divisor of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 1, 8, 9, 10, 1, 12, 1, 2, 15, 16, 1, 18, 1, 20, 3, 2, 1, 24, 25, 2, 27, 4, 1, 30, 1, 32, 3, 2, 5, 36, 1, 2, 3, 40, 1, 6, 1, 4, 45, 2, 1, 48, 1, 50, 3, 4, 1, 54, 5, 8, 3, 2, 1, 60, 1, 2, 9, 64, 5, 6, 1, 4, 3, 10, 1, 72, 1, 2, 75, 4, 1, 6, 1, 80

Offset: 1

Amiram Eldar, Jul 08 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)

Cf. A006519, A007814, A007949, A038500, A051037, A060904, A065331, A112765, A132741, A165725, A355583, A355584.

Cf. A379005 (rgs-transform), A379006 (ordinal transform).

a[n_] := Times @@ ({2, 3, 5}^IntegerExponent[n, {2, 3, 5}]); Array[a, 100]

a(n) = 3^valuation(n, 3) * 5^valuation(n, 5) << valuation(n, 2);

from sympy import multiplicity as v
def a(n): return 2**v(2, n) * 3**v(3, n) * 5**v(5, n)
print([a(n) for n in range(1, 81)]) # Michael S. Branicky, Jul 08 2022

Multiplicative with a(p^e) = p^e if p <= 5 and 1 otherwise.
a(n) = A006519(n) * A038500(n) * A060904(n).
a(n) = 2^A007814(n) * 3^A007949(n) * 5^A112765(n).
a(n) = n / A165725(n).
Dirichlet g.f.: zeta(s)*(2^s-1)*(3^s-1)*(5^s-1)/((2^s-2)*(3^s-3)*(5^s-5)). - Amiram Eldar, Dec 25 2022
Sum_{k=1..n} a(k) ~ 2*n*log(n)^3 / (45*log(2)*log(3)*log(5)) + O(n*log(n)^2). - Vaclav Kotesovec, Apr 20 2025

A355584 a(n) is the sum of the 5-smooth divisors of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 1, 15, 13, 18, 1, 28, 1, 3, 24, 31, 1, 39, 1, 42, 4, 3, 1, 60, 31, 3, 40, 7, 1, 72, 1, 63, 4, 3, 6, 91, 1, 3, 4, 90, 1, 12, 1, 7, 78, 3, 1, 124, 1, 93, 4, 7, 1, 120, 6, 15, 4, 3, 1, 168, 1, 3, 13, 127, 6, 12, 1, 7, 4, 18, 1, 195, 1, 3, 124, 7

Offset: 1

Amiram Eldar, Jul 08 2022

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

Sum of the p-smooth divisors of n: A038712 (2), A072079 (3), this sequence (5).

Cf. A000203, A007814, A007949, A051037, A112765, A355582, A355583.

a[n_] := (Times @@ ({2, 3, 5}^(IntegerExponent[n, {2, 3, 5}] + 1) - 1))/8; Array[a, 100]

a(n) = (2^(valuation(n, 2) + 1) - 1) * (3^(valuation(n, 3) + 1) - 1) * (5^(valuation(n, 5) + 1) - 1) / 8;

from sympy import multiplicity as v
def a(n): return (2**(v(2, n)+1)-1) * (3**(v(3, n)+1)-1) * (5**(v(5, n)+1)-1) // 8
print([a(n) for n in range(1, 77)]) # Michael S. Branicky, Jul 08 2022

Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if p <= 5, and 1 otherwise.
a(n) = (2^(A007814(n)+1)-1)*(3^(A007949(n)+1)-1)*(5^(A112765(n)+1)-1)/8.
a(n) = A000203(A355582(n)).
a(n) <= A000203(n), with equality if and only if n is in A051037.
Dirichlet g.f.: zeta(s)*(2^s/(2^s-2))*(3^s/(3^s-3))*(5^s/(5^s-5)). - Amiram Eldar, Dec 25 2022

A355710 Numbers k such that k and k+1 have the same number of 5-smooth divisors.

Original entry on oeis.org

2, 21, 33, 34, 38, 57, 85, 86, 93, 94, 104, 116, 122, 141, 145, 146, 154, 158, 171, 177, 182, 189, 201, 205, 213, 214, 218, 237, 265, 266, 273, 296, 302, 321, 326, 332, 334, 338, 344, 357, 362, 381, 385, 387, 393, 394, 398, 417, 445, 446, 453, 454, 475, 476, 482

Offset: 1

Amiram Eldar, Jul 15 2022

nonn

Numbers k such that A355583(k) = A355583(k+1).

			2 is a term since A355583(2) = A355583(3) = 2.

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

Cf. A355583, A355709 (3-smooth analog).
Subsequences: A355711, A355712.
Similar sequences: A005237, A006049, A332386, A343819, A344312, A344313, A344314, A348345.

s[n_] := Times @@ (1 + IntegerExponent[n, {2, 3, 5}]); Select[Range[500], s[#] == s[#+1] &]

s(n) = (valuation(n, 2) + 1) * (valuation(n, 3) + 1) * (valuation(n, 5) + 1);
s1 = s(1); for(k = 2, 500, s2 = s(k); if(s1 == s2, print1(k-1,", ")); s1 = s2);

A355711 Starts of runs of 3 consecutive numbers with the same number of 5-smooth divisors.

Original entry on oeis.org

33, 85, 93, 145, 213, 265, 393, 445, 453, 475, 505, 633, 685, 753, 805, 813, 865, 933, 985, 993, 1045, 1113, 1165, 1293, 1345, 1353, 1405, 1430, 1533, 1585, 1624, 1653, 1705, 1713, 1765, 1833, 1885, 1893, 1945, 2013, 2065, 2193, 2245, 2253, 2275, 2305, 2433, 2485

Offset: 1

Amiram Eldar, Jul 15 2022

nonn

Numbers k such that A355583(k) = A355583(k+1) = A355583(k+2).

			33 is a term since A355583(33) = A355583(34) = A355583(35) = 2.

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

Cf. A355583.
Subsequence of A355710.
A355712 is a subsequence.
Similar sequences: A005238, A006073, A045939, A332313, A332387.

f[n_] := Times @@ (1 + IntegerExponent[n, {2, 3, 5}]); s = {}; m = 3; fs = f /@ Range[m]; Do[If[Equal @@ fs, AppendTo[s, n - m]]; fs = Rest @ AppendTo[fs, f[n]], {n, m + 1, 2500}]; s

s(n) = (valuation(n, 2) + 1) * (valuation(n, 3) + 1) * (valuation(n, 5) + 1);
s1 = s(1); s2 = s(2); for(k = 3, 2500, s3 = s(k); if(s1 == s2 && s2 == s3, print1(k-2,", ")); s1 = s2; s2 = s3);

A355712 Starts of runs of 4 consecutive numbers with the same number of 5-smooth divisors.

Original entry on oeis.org

28374, 133623, 136374, 187623, 190374, 298374, 349623, 352374, 457623, 460374, 511623, 619623, 622374, 673623, 676374, 781623, 838374, 943623, 946374, 997623, 1000374, 1108374, 1159623, 1162374, 1267623, 1270374, 1321623, 1429623, 1432374, 1483623, 1486374, 1591623

Offset: 1

Amiram Eldar, Jul 15 2022

nonn

Numbers k such that A355583(k) = A355583(k+1) = A355583(k+2) = A355583(k+3).

Are there runs of 5 consecutive numbers with the same number of 5-smooth divisors? There are no such runs below 10^10.

			28374 is a term since A355583(28374) = A355583(28375) = A355583(28376) = A355583(28377) = 4.

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

Cf. A355583.
Subsequence of A355710 and A355711.
Similar sequences: A006601, A332314, A332388.

f[n_] := Times @@ (1 + IntegerExponent[n, {2, 3, 5}]); s = {}; m = 4; fs = f /@ Range[m]; Do[If[Equal @@ fs, AppendTo[s, n - m]]; fs = Rest @ AppendTo[fs, f[n]], {n, m + 1, 10^6}]; s

s(n) = (valuation(n, 2) + 1) * (valuation(n, 3) + 1) * (valuation(n, 5) + 1);
s1 = s(1); s2 = s(2); s3 = s(3); for(k = 4, 1.6e6, s4 = s(k); if(s1 == s2 && s2 == s3 && s3 == s4, print1(k-3,", ")); s1 = s2; s2 = s3; s3 = s4);

A356006 The number of prime divisors of n that are not greater than 5, counted with multiplicity.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jul 23 2022

Equivalently, the number of prime divisors, counted with multiplicity, of the largest 5-smooth divisor of n.

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

Cf. A007814, A007949, A112765.

Cf. A001222, A112759, A169611, A355582, A355583, A355584.

a[n_] := Plus @@ IntegerExponent[n, {2, 3, 5}]; Array[a, 100]

a(n) = valuation(n, 2) + valuation(n, 3) + valuation(n, 5);

from sympy import multiplicity as v
def a(n): return v(2, n) + v(3, n) + v(5, n)
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Jul 25 2022

Totally additive with a(p) = 1 if p <= 5, and 0 otherwise.
a(n) = A007814(n) + A007949(n) + A112765(n).
a(n) = A001222(A355582(n)).
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 7/4.

A382489 The number of unitary 5-smooth divisors of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 2, 4, 2, 1, 4, 1, 4, 2, 2, 1, 4, 2, 2, 2, 2, 1, 8, 1, 2, 2, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 2, 4, 2, 1, 4, 1, 4, 2, 2, 1, 4, 2, 2, 2, 2, 1, 8, 1, 2, 2, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 2, 4, 2, 1, 4, 1, 4, 2, 2, 1, 4, 2, 2, 2

Offset: 1

Amiram Eldar, Mar 29 2025

Period 30: repeat [1, 2, 2, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 2, 4, 2, 1, 4, 1, 4, 2, 2, 1, 4, 2, 2, 2, 2, 1, 8].
In general, the sequence of the number of unitary prime(k)-smooth divisors of n, for k >= 1, is periodic with period A002110(k).
Decimal expansion of 135804580460138015713571358020/111111111111111111111111111111.
Continued fraction expansion of 808690/(525316 + sqrt(382161348866)) (with offset 0).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A002110, A005117, A034444, A051037, A054640, A236435, A236436, A355582, A355583.

The number of unitary prime(k)-smooth divisors of n: A134451 (k = 1), A382488 (k = 2), this sequence (k = 3).

a[n_] := Product[If[Divisible[n, p], 2, 1], {p, {2, 3, 5}}]; Array[a, 100]

a(n) = vecprod(apply(x -> !((n % 30) % x) + 1, [2, 3, 5]))

Multiplicative with a(p^e) = 2 if p <= 5, and 1 otherwise.
a(n) = A034444(A355582(n)).
a(n) = A034444(n) if and only if n is 5-smooth (A051037).
a(n) = A355583(n) if and only if n is squarefree (A005117).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 12/5.
In general, the asymptotic mean of the number of unitary prime(k)-smooth divisors of n is A054640(k)/A002110(k) = A236435(k)/A236436(k).
Dirichlet g.f.: (1 + 1/2^s) * (1 + 1/3^s) * (1 + 1/5^s) * zeta(s).
In general, Dirichlet g.f. of the number of unitary prime(k)-smooth divisors of n is zeta(s) * Product_{p prime <= prime(k)} (1 + 1/p^s).

A076302 Triangle T(n,k) = number of k-smooth divisors of n, read by rows.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 3, 3, 3, 1, 1, 1, 1, 2, 1, 2, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 2, 1, 4, 4, 4, 4, 4, 4, 4, 1, 1, 3, 3, 3, 3, 3, 3, 3, 1, 2, 2, 2, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 1

Reinhard Zumkeller, Mar 14 2003

			Triangle begins:
                   1
                 1   2
               1   1   2
             1   3   3   3
           1   1   1   1   2
         1   2   4   4   4   4
       1   1   1   1   1   1   2
     1   4   4   4   4   4   4   4
   1   1   3   3   3   3   3   3   3

Eric Weisstein's World of Mathematics, Divisor Function.
Eric Weisstein's World of Mathematics, Smooth Number.

Cf. A000005, A001511, A072078, A355583.

T[n_, k_] := Times@@(IntegerExponent[n, #]+1& /@ Select[Range[2, k], PrimeQ]);
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 15 2021 *)

T(n,n) = A000005(n);
T(n,2) = A001511(n) for n>1.
T(n,3) = A072078(n) for n>2.
T(n,5) = A355583(n) for n>4.
Limit_{m->oo} (1/m) * Sum_{n=k..m} T(n,k) = 1/Product_{p prime <= k} (1 - 1/p). - Amiram Eldar, Apr 17 2025

cp's OEIS Frontend

A060753 Denominator of 1*2*4*6*...*(prime(n-1)-1) / (2*3*5*7*...*prime(n-1)).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A355582 a(n) is the largest 5-smooth divisor of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A355584 a(n) is the sum of the 5-smooth divisors of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A355710 Numbers k such that k and k+1 have the same number of 5-smooth divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A355711 Starts of runs of 3 consecutive numbers with the same number of 5-smooth divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A355712 Starts of runs of 4 consecutive numbers with the same number of 5-smooth divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A356006 The number of prime divisors of n that are not greater than 5, counted with multiplicity.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A060753 Denominator of 1246...(prime(n-1)-1) / (2357...prime(n-1)).