A385503 -id:A385503 - cp's OEIS frontend

A289661 a(n) is the smallest m such that p = n-th popular prime = A385503(n) is popular on the interval [2,m].

2, 3, 45, 70, 1456, 4845, 20332, 46345, 106812, 153032, 592760, 2484190, 2620033, 2623860, 41163150, 237321819, 1966462280, 13690728506, 64322151699, 79838726306, 220355977754, 232268764689, 618745965579, 1882062393429, 9607711921430, 19364051434020, 26393150922356, 37636607775855, 114514665136326, 173974642506024, 228013323182523, 259772852488365, 284600479332862, 644741545074402

Offset: 1

N. J. A. Sloane, Jul 25 2017

nonn

See A385503 for further links and information.

Nathan McNew, Popular values of the largest prime divisor function (corrected version), page 16, November 2015.

Cf. A006530, A385503, A289662-A289665.

Edited, with terms updated from revised version of McNew, by Peter Munn, Jul 04 2025

A289665 Number of integers in the interval [2,A289663(n)] whose greatest prime factor is equal to the n-th popular prime A385503(n).

Original entry on oeis.org

4, 14, 25, 77, 151, 428, 616, 1005, 1517, 2902, 7664, 7722, 8284, 48380, 161644, 698074, 2761234, 8357693, 9758410, 20285553, 21123128, 43031555, 96835113, 318539488, 534261087, 672081919, 873949289, 1997008416, 2720040151, 3321778833, 3663025425, 3924572413, 7241174991

Offset: 1

N. J. A. Sloane, Jul 25 2017

See A385503 for further links and information.

Nathan McNew, Popular values of the largest prime divisor function (corrected version), page 16, November 2015.

Cf. A006530, A385503, A289661-A289664.

Edited, with terms updated from revised version of McNew, by Peter Munn, Jul 04 2025

Name corrected by Pontus von Brömssen, Jul 05 2025

A289662 a(n) is the smallest m such that p = n-th popular prime = A385503(n) is uniquely popular on the interval [2,m] or -1 if p is never uniquely popular.

Original entry on oeis.org

2, 12, 80, 196, 1638, 4864, 22425, 46500, 109779, 158625, 603564, 2552416, 2620142, 2627250, 41163747, 237398795, 1966466950, 13690729828, 64322158656, 79838739611, 220355987735, 232268774850, 618745972214, 1882062476406, 9607713772982, 19364051829855, 26393150937218, 37636607806688, 114514665167797, 173974642809066, 228013323978930, 259772858868378, 284600479573629, 644741545246282

Offset: 1

N. J. A. Sloane, Jul 25 2017

nonn

See A385503 for further links and information.

Nathan McNew, Popular values of the largest prime divisor function (corrected version), page 16, November 2015.

Cf. A006530, A385503, A125624, A289661-A289665.

Deleted "first" from definition. - N. J. A. Sloane, Oct 03 2019

Edited, and terms updated from revised version of McNew, by Peter Munn, Jul 04 2025

A289663 a(n) is the largest m such that p = n-th popular prime = A385503(n) is popular on the interval [2,m].

Original entry on oeis.org

17, 119, 279, 1858, 5471, 29301, 53474, 117303, 220523, 611374, 2642391, 2672025, 2952463, 41192601, 237611044, 1967277194, 13692930957, 64358549949, 79880100420, 220369251374, 232880841877, 618765808209, 1882062587041, 9607847299025, 19364476224949, 26396066576762, 37636861534247, 114519204438979, 173975018331581, 228013888964263, 259777078505983, 284734190312531, 644744279642231

Offset: 1

N. J. A. Sloane, Jul 25 2017

nonn

There is a slight doubt about these numbers since McNew only carried a search up to a certain point.

See A385503 for further links and information.

Nathan McNew, Popular values of the largest prime divisor function (corrected version), page 16, November 2015.

Cf. A006530, A385503, A289661-A289665.

Edited, with terms updated from revised version of McNew, by Peter Munn, Jul 04 2025

A289664 Number of integers in the interval [2,A289661(n)] whose greatest prime factor is equal to the n-th popular prime A385503(n).

Original entry on oeis.org

1, 1, 8, 10, 67, 140, 344, 563, 947, 1197, 2846, 7357, 7621, 7629, 48357, 161507, 697875, 2760913, 8354317, 9754751, 20284680, 21082412, 43030537, 96835105, 318536223, 534252383, 672026918, 873944930, 1996949860, 2720035791, 3321772681, 3662980704, 3923186891, 7241151976

Offset: 1

N. J. A. Sloane, Jul 25 2017

See A385503 for further links and information.

Nathan McNew, Popular values of the largest prime divisor function (corrected version), page 16, November 2015.

Cf. A006530, A385503, A289661-A289665.

Edited, with terms updated from revised version of McNew, by Peter Munn, Jul 04 2025

Name corrected by Pontus von Brömssen, Jul 05 2025

A386007 Least k such that there are exactly n primes that are popular on the interval [2,k] (see A385503); i.e., exactly n primes share the lead as the most common greatest prime factor of the numbers 2..k.

Original entry on oeis.org

2, 3, 70, 2626355

Offset: 1

Pontus von Brömssen, Jul 14 2025

			a(3) = 70, because the 3 primes 3, 5, and 7 all occur A385652(70) = 10 times (the maximum) as the greatest prime factor of the numbers 2..70, and for earlier intervals there is never a tie between 3 numbers.
a(4) = 2626355, because the 4 primes 73, 83, 109, and 113 all occur A385652(2626355) = 7634 times (the maximum) as the greatest prime factor of the numbers 2..2626355, and for earlier intervals there is never a tie between 4 numbers.

Cf. A006530, A385503, A385652.

gpf[n_]:=FactorInteger[n][[-1,1]];a[n_]:=Module[{k=1},Until[Length[Commonest[gpf/@Range[2,k]]]==n,k++];k] (* James C. McMahon, Jul 20 2025 *)

A006530 Gpf(n): greatest prime dividing n, for n >= 2; a(1)=1.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5, 2, 17, 3, 19, 5, 7, 11, 23, 3, 5, 13, 3, 7, 29, 5, 31, 2, 11, 17, 7, 3, 37, 19, 13, 5, 41, 7, 43, 11, 5, 23, 47, 3, 7, 5, 17, 13, 53, 3, 11, 7, 19, 29, 59, 5, 61, 31, 7, 2, 13, 11, 67, 17, 23, 7, 71, 3, 73, 37, 5, 19, 11, 13, 79, 5, 3, 41, 83, 7, 17, 43

Offset: 1

N. J. A. Sloane

The initial term a(1)=1 is purely conventional: The unit 1 is not a prime number, although it has been considered so in the past. 1 is the empty product of prime numbers, thus 1 has no largest prime factor. - Daniel Forgues, Jul 05 2011
Greatest noncomposite number dividing n, (cf. A008578). - Omar E. Pol, Aug 31 2013
Conjecture: Let a, b be nonzero integers and f(n) denote the maximum prime factor of a*n + b if a*n + b <> 0 and f(n)=0 if a*n + b=0 for any integer n. Then the set {n, f(n), f(f(n)), ...} is finite of bounded size. - M. Farrokhi D. G., Jan 10 2021

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section IV.1.
H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 210.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
K. Alladi and P. Erdős, On an additive arithmetic function, Pacific J. Math., Volume 71, Number 2 (1977), 275-294. MR 0447086 (56 #5401).
G. Back and M. Caragiu, The greatest prime factor and recurrent sequences, Fib. Q., 48 (2010), 358-362.
A. E. Brouwer, Two number theoretic sums, Stichting Mathematisch Centrum. Zuivere Wiskunde, Report ZW 19/74 (1974): 3 pages. [Cached copy, included with the permission of the author]
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
Mats Granvik, Mathematica program to check the conjectured formula, Feb 28 2021.
Nathan McNew, The Most Frequent Values of the Largest Prime Divisor Function, Exper. Math., 2017, Vol. 26, No. 2, 210-224.
OEIS Wiki, Greatest prime factor of n
H. P. Robinson, Letter to N. J. A. Sloane, Oct 1981
David Singmaster, Letter to N. J. A. Sloane, Oct 03 1982.
Eric Weisstein's World of Mathematics, Greatest Prime Factor
Index entries for "core" sequences

Cf. A000040, A020639 (smallest prime divisor), A034684, A028233, A034699, A053585.
See also A032742, A052126, A070087, A070089, A061395, A175723.
Cf. A046670 (partial sums), A104350 (partial products).
See A385503 for "popular" primes.

[ #f eq 0 select 1 else f[ #f][1] where f is Factorization(n): n in [1..86] ]; // Klaus Brockhaus, Oct 23 2008

with(numtheory,divisors); A006530 := proc(n) local i,t1,t2,t3,t4,t5; t1 := divisors(n); t2 := convert(t1,list); t3 := sort(t2); t4 := nops(t3); t5 := 1; for i from 1 to t4 do if isprime(t3[t4+1-i]) then return t3[t4+1-i]; fi; od; 1; end;
# alternative
A006530 := n->max(1,op(numtheory[factorset](n))); # Peter Luschny, Nov 02 2010

Table[ FactorInteger[n][[ -1, 1]], {n, 100}] (* Ray Chandler, Nov 12 2005 and modified by Robert G. Wilson v, Jul 16 2014 *)

A006530(n)=if(n>1,vecmax(factor(n)[,1]),1) \\ Edited to cover n=1. - M. F. Hasler, Jul 30 2015

from sympy import factorint
def a(n): return 1 if n == 1 else max(factorint(n))
print([a(n) for n in range(1, 87)]) # Michael S. Branicky, Aug 08 2022

def A006530(n): return list(factor(n))[-1][0] if n > 1 else 1
print([A006530(n) for n in range(1, 87)])  # Peter Luschny, Jan 07 2024

;; The following uses macro definec for the memoization (caching) of the results. A naive implementation of A020639 can be found under that entry. It could be also defined with definec to make it faster on the later calls. See http://oeis.org/wiki/Memoization#Scheme
(definec (A006530 n) (let ((spf (A020639 n))) (if (= spf n) spf (A006530 (/ n spf)))))
;; Antti Karttunen, Mar 12 2017

a(n) = A027748(n, A001221(n)) = A027746(n, A001222(n)); a(n)^A071178(n) = A053585(n). - Reinhard Zumkeller, Aug 27 2011
a(n) = A000040(A061395(n)). - M. F. Hasler, Jan 16 2015
a(n) = n + 1 - Sum_{k=1..n} (floor((k!^n)/n) - floor(((k!^n)-1)/n)). - Anthony Browne, May 11 2016
n/a(n) = A052126(n). - R. J. Mathar, Oct 03 2016
If A020639(n) = n [when n is 1 or a prime] then a(n) = n, otherwise a(n) = a(A032742(n)). - Antti Karttunen, Mar 12 2017
a(n) has average order Pi^2*n/(12 log n) [Brouwer]. See also A046670. - N. J. A. Sloane, Jun 26 2017

Edited by M. F. Hasler, Jan 16 2015

A124661 Primes prime(n) such that prime(n-k)+prime(n+k) >= 2*prime(n) for k = 1..n-2.

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 23, 31, 43, 47, 73, 83, 109, 113, 181, 199, 283, 293, 313, 317, 463, 467, 503, 509, 523, 619, 661, 683, 691, 887, 1063, 1069, 1103, 1109, 1123, 1129, 1303, 1307, 1321, 1327, 1613, 1621, 1627, 1637, 1669, 1789

Offset: 1

Artur Jasinski, Dec 23 2006

The first two primes, 2 and 3, are tested against an empty set of k, and we include them, defining such a test to have a positive outcome.
McNew's "popular primes" sequence (A385503) has the same first 14 terms, differing first by excluding 181. McNew says that a prime p is "popular" on an interval [2, k] if no prime occurs more frequently than p as the greatest prime factor (gpf, A006530) of the integers in that interval. - N. J. A. Sloane, Jul 25 2017 and Peter Munn, Jul 01 2025
See the Pomerance link for a proof that the sequence is infinite. - Peter Munn, Jul 01 2025

			prime(11)=31 is in the sequence because prime(10)+prime(12) = 66, prime(9)+prime(13) = 64,..., prime(2)+prime(20) = 74 are all >= 62 = 2*31.
prime(10) = 29 is not in the sequence because prime(9)+prime(11) = 54 for example is smaller than 58 = 2*29.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Nathan McNew, Popular values of the largest prime divisor function, arXiv:1504.05985 [math.NT], 2015.
Nathan McNew, The Most Frequent Values of the Largest Prime Divisor Function, Exper. Math., 2017, Vol. 26, No. 2, 210-224.
C. Pomerance, The prime number graph, Math. Comp. 33 (1979) 399--408. - _Nathan McNew_, Apr 04 2014

Cf. A006530, A051635, A385503.

Select[Prime@ Range@ 300, Function[{p, n}, NoneTrue[Range[n - 2], Prime[n - #] + Prime[n + #] < 2 p &]] @@ {#, PrimePi@ #} &] (* Michael De Vlieger, Jul 25 2017 *)

isok(p) = {n = primepi(p); for (k=1, n-2, if (prime(n-k) + prime(n+k) < 2*p, return (0));); return (1);}
lista(nn) = {for(n=1, nn, if (isok(prime(n)), print1(prime(n), ", ");););} \\ Michel Marcus, Nov 03 2013

from sympy import prime
A124661_list = []
for n in range(1,10**6):
    p = prime(n)
    for k in range(1,n-1):
        if prime(n-k)+prime(n+k) < 2*p:
            break
    else:
        A124661_list.append(p) # Chai Wah Wu, Jul 25 2017

Sequence extended by R. J. Mathar, Mar 28 2010

Edited, restoring previous name, by Peter Munn, Jul 01 2025

A385652 Maximum frequency of gpf(k) for 2 <= k <= n, where gpf(k) = A006530(k) is the greatest prime factor of k.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12

Offset: 2

Pontus von Brömssen, Jul 06 2025

nonn

The prime p is popular on the interval [2,n] if it is the greatest prime factor of a(n) numbers in that interval; see A385503.

			     |     | cumulative frequencies for gpf's |
   n | gpf |    2  3  5  7 11 13 17 19 23     | a(n)
  ---+-----+----------------------------------+-----
   2 |   2 |    1  0  0  0  0  0  0  0  0     |  1
   3 |   3 |    1  1  0  0  0  0  0  0  0     |  1
   4 |   2 |    2  1  0  0  0  0  0  0  0     |  2
   5 |   5 |    2  1  1  0  0  0  0  0  0     |  2
   6 |   3 |    2  2  1  0  0  0  0  0  0     |  2
   7 |   7 |    2  2  1  1  0  0  0  0  0     |  2
   8 |   2 |    3  2  1  1  0  0  0  0  0     |  3
   9 |   3 |    3  3  1  1  0  0  0  0  0     |  3
  10 |   5 |    3  3  2  1  0  0  0  0  0     |  3
  11 |  11 |    3  3  2  1  1  0  0  0  0     |  3
  12 |   3 |    3  4  2  1  1  0  0  0  0     |  4
  13 |  13 |    3  4  2  1  1  1  0  0  0     |  4
  14 |   7 |    3  4  2  2  1  1  0  0  0     |  4
  15 |   5 |    3  4  3  2  1  1  0  0  0     |  4
  16 |   2 |    4  4  3  2  1  1  0  0  0     |  4
  17 |  17 |    4  4  3  2  1  1  1  0  0     |  4
  18 |   3 |    4  5  3  2  1  1  1  0  0     |  5
  19 |  19 |    4  5  3  2  1  1  1  1  0     |  5
  20 |   5 |    4  5  4  2  1  1  1  1  0     |  5
  21 |   7 |    4  5  4  3  1  1  1  1  0     |  5
  22 |  11 |    4  5  4  3  2  1  1  1  0     |  5
  23 |  23 |    4  5  4  3  2  1  1  1  1     |  5
  24 |   3 |    4  6  4  3  2  1  1  1  1     |  6

Pontus von Brömssen, Table of n, a(n) for n = 2..10000

Cf. A006530, A078899, A385503, A385653, A385654.

gpf(n) = if (n==1,1, vecmax(factor(n)[,1])); \\ A006530
a(n) = my(v=vector(n, k, gpf(k)), s=Set(v)); vecmax(apply(x->#x, vector(#s, i, select(x->(x==s[i]), v)))); \\ Michel Marcus, Jul 06 2025

from collections import Counter
from itertools import count
from sympy import factorint
def A385652_generator():
    c = Counter()
    M = 0
    for n in count(2):
        gpf = max(factorint(n))
        c[gpf] += 1
        if c[gpf] > M: M += 1
        yield M

a(n) = max_{k=2..n} A078899(k).

A385653 Least k such that A385652(k) = n.

Original entry on oeis.org

2, 4, 8, 12, 18, 24, 27, 36, 48, 54, 72, 80, 90, 100, 120, 125, 135, 150, 160, 180, 196, 210, 224, 245, 252, 280, 294, 315, 336, 343, 350, 378, 392, 420, 441, 448, 490, 504, 525, 560, 567, 588, 630, 672, 686, 700, 735, 756, 784, 840, 875, 882, 896, 945, 980

Offset: 1

Pontus von Brömssen, Jul 06 2025

nonn

A385654(n) is uniquely popular on the interval [2,a(n)]; see A289662.

Equivalently, a(n) is the least k >= 2 such that A078899(k) = n.

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A006530, A078899, A289662, A385503, A385652, A385654.

gpf(n) = if (n==1,1, vecmax(factor(n)[,1])); \\ A006530
f(n) = my(v=vector(n, k, gpf(k)), s=Set(v)); vecmax(apply(x->#x, vector(#s, i, select(x->(x==s[i]), v)))); \\ A385652
a(n) = my(k=2); while (f(k) !=n, k++); k; \\ Michel Marcus, Jul 06 2025

cp's OEIS Frontend

A289661 a(n) is the smallest m such that p = n-th popular prime = A385503(n) is popular on the interval [2,m].

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A289665 Number of integers in the interval [2,A289663(n)] whose greatest prime factor is equal to the n-th popular prime A385503(n).

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A289662 a(n) is the smallest m such that p = n-th popular prime = A385503(n) is uniquely popular on the interval [2,m] or -1 if p is never uniquely popular.

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A289663 a(n) is the largest m such that p = n-th popular prime = A385503(n) is popular on the interval [2,m].

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A289664 Number of integers in the interval [2,A289661(n)] whose greatest prime factor is equal to the n-th popular prime A385503(n).

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A386007 Least k such that there are exactly n primes that are popular on the interval [2,k] (see A385503); i.e., exactly n primes share the lead as the most common greatest prime factor of the numbers 2..k.

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A006530 Gpf(n): greatest prime dividing n, for n >= 2; a(1)=1.

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A124661 Primes prime(n) such that prime(n-k)+prime(n+k) >= 2*prime(n) for k = 1..n-2.

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