A005579 -id:A005579 - cp's OEIS frontend

A005347 First differences of A005579.

1, 1, 1, 1, 2, 3, 5, 8, 13, 20, 34, 53, 88, 143, 236, 387, 641, 1061, 1763, 2937, 4903, 8202, 13750, 23095, 38850, 65461, 110465, 186665, 315827, 535011, 907341, 1540416, 2617782, 4452846, 7581016, 12917486, 22027745, 37591270, 64196610

Offset: 0

N. J. A. Sloane, R. K. Guy, Apr 12 1988

This is example 42 in Guy's paper. a(2)-a(8) are the same as the Fibonacci sequence A000045. Subsequent terms deviate from Fibonacci. - T. D. Noe, May 08 2006

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 0..43
Richard K. Guy, Letter to N. J. A. Sloane, 1988-04-12 (annotated scanned copy).
Richard K. Guy, Letters to N. J. A. Sloane, 1986-88.
Richard K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
Richard K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
Richard Laatsch, Measuring the abundancy of integers, Mathematics Magazine 59 (2) (1986) 84-92.

Cf. A000045, A005579.

prod = Interval[1]; k = k0 = 0; Join[{1, 1}, Table[While[Max[prod] <= n, k++; p = Prime[k]; prod = N[prod*p/(p - 1), 30]]; If[Min[prod] > n, If[k > 2, Print[k - k0] ]; k0 = k; k, "too few digits"], {n, 2, 39}] // Differences] (* Jean-François Alcover, Oct 07 2016, using T. D. Noe's code for A005579 *)

a(n) = A005579(n+1) - A005579(n) - T. D. Noe, May 08 2006

More terms from Harvey P. Dale, Aug 07 2013

Offset changed to 0, a(0) prepended, and a(1) inserted by Amiram Eldar, Apr 18 2025

A256969 Duplicate of A005579.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 9, 14, 22, 35, 55, 89, 142, 230, 373, 609, 996, 1637, 2698, 4461, 7398, 12301, 20503, 34253, 57348, 96198, 161659, 272124, 458789, 774616, 1309627, 2216968, 3757384, 6375166

Offset: 0

N. J. A. Sloane, Apr 17 2015

dead

More terms from Chai Wah Wu, Apr 17 2015
a(32)-a(33) from Chai Wah Wu, Apr 19 2015
a(0) corrected by Max Alekseyev, Jan 25 2025

A091456 Least number k such that n * phi(k) < k, where phi is Euler's totient function.

Original entry on oeis.org

2, 6, 30, 210, 30030, 223092870, 13082761331670030, 3217644767340672907899084554130, 1492182350939279320058875736615841068547583863326864530410

Offset: 1

Robert G. Wilson v, Jan 10 2004

nonn

By Mertens' theorem and the Prime Number Theorem log log a(n) ~ n / e^gamma. - Charles R Greathouse IV, Sep 07 2012

Amiram Eldar, Table of n, a(n) for n = 1..13

Subsequence of A002110.

Cf. A000010, A005579, A054741, A073087, A091439 (n * phi(k) <= k).

a(n) = {k = 1; while (n*eulerphi(k) >= k, k++); k;} \\ Michel Marcus, Sep 25 2013

a(n)=my(k=1);forprime(p=2,,if(n*eulerphi(k)Charles R Greathouse IV, Sep 25 2013

a(n) = A002110(A005579(n)). - Amiram Eldar, Nov 30 2024

A091440 Smallest number m such that m#/phi(m#) >= n, where m# indicates the primorial (A034386) of m and phi is Euler's totient function.

Original entry on oeis.org

1, 2, 3, 7, 13, 23, 43, 79, 149, 257, 461, 821, 1451, 2549, 4483, 7879, 13859, 24247, 42683, 75037, 131707, 230773, 405401, 710569, 1246379, 2185021, 3831913, 6720059, 11781551, 20657677, 36221753, 63503639, 111333529, 195199289, 342243479, 600036989

Offset: 1

T. D. Noe, Jan 09 2004

nonn

Does the ratio of adjacent terms converge?
It appears that lim_{n->infinity} a(n+1)/a(n) = 1.7532... - Jon E. Schoenfield, Feb 21 2019
For n > 1, a(n) is smallest prime p = prime(k) such that no fewer than (n-1)/n of any p# consecutive integers are divisible by a prime not greater than p. Cf. A053144(k)/A002110(k). - Peter Munn, Apr 29 2017
Also, the smallest prime p such that the sum of the reciprocals of the p-smooth numbers converges to at least n. - Keith F. Lynch, Apr 29 2023
Also, if m is a random integer much larger than the square of a(n), and m is not divisible by any prime less than or equal to a(n), the probability that m is prime is n/log(m). - Keith F. Lynch, Dec 17 2023

			7#/phi(7#) = (2*3*5*7)/(1*2*4*6) = 4.375 >= 4, 5#/phi(5#) = 3.75. Hence a(4) = 7.

Jon E. Schoenfield, Table of n, a(n) for n = 1..44
Eric Weisstein's World of Mathematics, Primorial.
Eric Weisstein's World of Mathematics, Totient Function.

Cf. A091439, A000010, A002110, A005579, A034386, A053144, A038110, A060753, A164347.

Mathematica
```
prod=1; i=0; Table[While[prod
				
```

al(lim) = local(mm,n,m); mm=3; n=2; m=1; forprime(x=3,lim, n*=x; m*= (x-1); if (n\m >= mm, print1(x","); mm++)); /* This will generate all terms of this sequence from the 3rd onward, up to lim. The computation slows down for large values because of the size of the internal values. */ \\ Fred Schneider, Aug 13 2009, modified by Franklin T. Adams-Watters, Aug 29 2009

a(n) = prime(A005579(n)) for n >= 4. - Amiram Eldar, Apr 18 2025

More terms from David W. Wilson, Sep 28 2005

Sequence reference in name corrected by Peter Munn, Apr 29 2017

A134740 Number of distinct prime factors in the n-th multiply perfect number A007691(n).

Original entry on oeis.org

0, 2, 2, 3, 2, 3, 2, 4, 5, 4, 6, 5, 2, 5, 6, 6, 7, 5, 2, 7, 7, 6, 6, 6, 2, 7, 7, 7, 6, 8, 8, 8, 8, 6, 7, 8, 8, 9, 7, 8, 2, 9, 9, 9, 11, 10, 9, 10, 10, 8, 9, 11, 10, 10, 8, 11, 11, 10, 10, 11, 11, 9, 11, 9, 12, 11, 13, 13, 12, 11, 12, 12, 11, 13, 12, 12, 12, 11, 12, 13, 12, 11, 10, 13, 12, 11, 10

Offset: 1

T. D. Noe, Nov 07 2007

nonn

If a multiply perfect number k has abundancy m = sigma(k)/k, then k must have at least A005579(m) distinct prime factors.

T. D. Noe, Table of n, a(n) for n=1..5000 (using Flammenkamp's data)
Achim Flammenkamp, The Multiply Perfect Numbers Page

Cf. A007691.

A256968 Let b(n) = Product_{i=1..n} p_i/(p_i - 1), p_i = i-th prime; a(n) = minimum k such that b(k) >= n.

Original entry on oeis.org

0, 0, 1, 2, 4, 6, 9, 14, 22, 35, 55, 89, 142, 230, 373, 609, 996, 1637, 2698, 4461, 7398, 12301, 20503, 34253, 57348, 96198, 161659, 272124, 458789, 774616, 1309627, 2216968, 3757384, 6375166, 10828012, 18409028, 31326514, 53354259, 90945529, 155142139

Offset: 0

N. J. A. Sloane, Apr 17 2015

nonn

A001611 is similar but different.

Equal to A005579 except for n = 2 and n = 3. The following argument shows that they are equal for n > 3. First note that b(k+1) > b(k). Next, Product_{i=1..k} p_i is 2 times an odd number, i.e., it is not divisible by 4. Similarly since p_i - 1 is even for i > 1, Product_{i=1..k} (p_i - 1) is divisible by 2^(k-1), i.e., it is divisible by 4 for k >= 3. Thus b(k) is not an integer for k >= 3. Since b(3) = 15/4 > 3, this means that a(n) = A005579(n) for n > 3 - Chai Wah Wu, Apr 17 2015

			The sequence b(n) for n >= 0 begins 1, 2, 3, 15/4, 35/8, 77/16, 1001/192, 17017/3072, 323323/55296, 676039/110592, 2800733/442368, 86822723/13271040, 3212440751/477757440, 131710070791/19110297600, 5663533044013/802632499200, ... = A060753/A038110. So a(3) = 2.

Popular Computing (Calabasas, CA), Problem 182 (Suggested by Victor Meally), Annotated and scanned copy of page 10 of Vol. 5 (No. 53, Aug 1977).

Cf. A005579, A060753, A038110, A001611.

from sympy import prime
A256968_list, count, bn, bd = [0,0], 2, 1, 1
for k in range(1,10**4):
    p = prime(k)
    bn *= p
    bd *= p-1
    while bn >= count*bd:
        A256968_list.append(k)
count += 1 # Chai Wah Wu, Apr 17 2015; corrected by Max Alekseyev, Jan 26 2025

More terms from Chai Wah Wu, Apr 17 2015
a(32)-a(33) from Chai Wah Wu, Apr 19 2015
a(0)-a(1) corrected and a(34)-a(39) copied over from A005579 by Max Alekseyev, Jan 26 2025

A005580 a(n) = smallest number k such that Product_{i=2..k+1} prime(i)/(prime(i)-1) > n.

Original entry on oeis.org

0, 1, 3, 8, 21, 54, 141, 372, 995, 2697, 7397, 20502, 57347, 161658, 458788, 1309626, 3757383, 10828011, 31326513, 90945528

Offset: 0

N. J. A. Sloane

Laatsch (1986) proved that a(n) gives the smallest number of distinct prime factors in odd numbers having an abundancy index > n.

The abundancy index of a number k is sigma(k)/k. - T. D. Noe, May 08 2006

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. K. Guy, Letter to N. J. A. Sloane, 1988-04-12 (annotated scanned copy)
Richard Laatsch, Measuring the abundancy of integers, Mathematics Magazine 59 (2) (1986) 84-92.

Cf. A005579.

prod=1; k=1; Table[While[prod<=n, k++; prod=prod*Prime[k]/(Prime[k]-1)]; k-1, {n,0,12}] (* T. D. Noe, May 08 2006 *)

For n > 0, a(n) = A005579(2n)-1. - T. D. Noe, May 08 2006

Edited by T. D. Noe, May 08 2006
a(14)-a(19) from the data at A005579 added by Amiram Eldar, Mar 21 2019
Edited and a(0)-a(1) prepended by Max Alekseyev, Jan 25 2025

A371701 a(n) is the least positive k such that Product_{i=1..k} 1 / (1 - 1/(2*i+1)) >= n.

Original entry on oeis.org

1, 1, 3, 7, 12, 19, 28, 38, 50, 63, 78, 95, 113, 132, 154, 176, 201, 227, 254, 283, 314, 346, 380, 415, 452, 491, 531, 572, 616, 660, 707, 755, 804, 855, 908, 962, 1018, 1075, 1134, 1194, 1256, 1320, 1385, 1452, 1520, 1590, 1662, 1735, 1809, 1885, 1963, 2043, 2123, 2206, 2290, 2376

Offset: 0

Ilya Gutkovskiy, Apr 03 2024

nonn

			a(3) = 7: (3/2) * (5/4) * (7/6) * (9/8) * (11/10) * (13/12) * (15/14) = 6435 / 2048 = 3.14208984375 > 3.

Cf. A000165, A001147, A001803, A005579, A046161, A256968, A371702.

a[n_] := For[k = 1, True, k++, If[(2 k + 1)!!/(2 k)!! >= n, Return[k]]]; Table[a[n], {n, 0, 55}]

a(n) ~ Pi * n^2 /4. - Vaclav Kotesovec, Apr 03 2024

cp's OEIS Frontend

A005347 First differences of A005579.

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A256969 Duplicate of A005579.

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A091456 Least number k such that n * phi(k) < k, where phi is Euler's totient function.

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A091440 Smallest number m such that m#/phi(m#) >= n, where m# indicates the primorial (A034386) of m and phi is Euler's totient function.

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A134740 Number of distinct prime factors in the n-th multiply perfect number A007691(n).

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A256968 Let b(n) = Product_{i=1..n} p_i/(p_i - 1), p_i = i-th prime; a(n) = minimum k such that b(k) >= n.

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A005580 a(n) = smallest number k such that Product_{i=2..k+1} prime(i)/(prime(i)-1) > n.

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A371701 a(n) is the least positive k such that Product_{i=1..k} 1 / (1 - 1/(2*i+1)) >= n.

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Mathematica