A008475 -id:A008475 - cp's OEIS frontend

A092509 Möbius transform of sequence A008475.

0, 2, 3, 2, 5, 0, 7, 4, 6, 0, 11, 0, 13, 0, 0, 8, 17, 0, 19, 0, 0, 0, 23, 0, 20, 0, 18, 0, 29, 0, 31, 16, 0, 0, 0, 0, 37, 0, 0, 0, 41, 0, 43, 0, 0, 0, 47, 0, 42, 0, 0, 0, 53, 0, 0, 0, 0, 0, 59, 0, 61, 0, 0, 32, 0, 0, 67, 0, 0, 0, 71, 0, 73, 0, 0, 0, 0, 0, 79, 0, 54, 0, 83, 0, 0, 0, 0, 0, 89, 0, 0, 0, 0

Offset: 1

Leroy Quet, Dec 31 2004

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A008475, A008683.

f[n_] := Plus @@ Map[Power @@ #1 &, FactorInteger[n]]; mt[n_] := Block[{d = Divisors[n]}, Plus @@ (MoebiusMu /@ (n/d)*f /@ d)]; Table[ mt[n], {n, 93}] (* Robert G. Wilson v, Jan 12 2005 *)

A008475(n) = { my(f=factor(n)); vecsum(vector(#f~,i,f[i,1]^f[i,2])); };
A092509(n) = sumdiv(n,d,moebius(d)*A008475(n/d)); \\ Antti Karttunen, Nov 17 2017

a(n) = 0 if n is 1 or not a power of a prime;
a(n) = n if n is a prime;
a(n) = n*(1 -1/p) if n = p^k, k>= 2, p=prime.
a(n) = Sum_{d|n} A008683(d)*A008475(n/d). - Antti Karttunen, Nov 17 2017

More terms from Robert G. Wilson v, Jan 13 2005

A167515 The sum over the divisors of n, except the maximum-prime-power divisors collected in A008475.

Original entry on oeis.org

1, 1, 1, 3, 1, 7, 1, 7, 4, 11, 1, 21, 1, 15, 16, 15, 1, 28, 1, 33, 22, 23, 1, 49, 6, 27, 13, 45, 1, 62, 1, 31, 34, 35, 36, 78, 1, 39, 40, 77, 1, 84, 1, 69, 64, 47, 1, 105, 8, 66, 52, 81, 1, 91, 56, 105, 58, 59, 1, 156, 1, 63, 88, 63, 66, 128, 1, 105, 70

Offset: 1

Jaroslav Krizek, Dec 15 2010

nonn

If n = Product (p_j^k_j) is the standard prime power decomposition of n, there is a set of size A001221(n) which contains the divisors which are largest powers of primes, {p_1^k_1, p_2^k_2, ..., p_j^k_j}. a(n) sums all the divisors not in this set. If p, q, ..., z are distinct primes, k are natural numbers (A000027), p^k prime powers (A000961), the following formulas hold: a(p) = 1. a(pq) = pq+1. a(pq...z) = (p+1)* (q+1)* ... *(z+1) - (p+q+ ...+z). a(p^k) = (p^k-1)/(p-1).

			For n = 12, set of prime-power-factor divisors of 12: {3, 4}, set of non-(prime-power-factor) divisors on 12: {1, 2, 6, 12}. a(12) = 1+2+6+12=21.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A000203, A008475, A178636.

A008475 := proc(n) add( op(1,d)^op(2,d),d= ifactors(n)[2] ) ; end proc:
A167515 := proc(n) numtheory[sigma](n)-A008475(n) ; end proc:
seq(A167515(n),n=1..80) ; # R. J. Mathar, Dec 21 2010

a(n) = A000203(n) - A008475(n).

a(n) = A178636(n) + 1.

A333801 Numbers k such that A008475(k)+1 = A008475(k+1).

Original entry on oeis.org

2, 3, 4, 7, 8, 16, 20, 31, 35, 127, 143, 208, 256, 650, 1479, 2464, 2623, 4233, 4345, 5183, 8099, 8191, 9424, 11024, 11919, 12099, 14905, 16159, 20220, 20800, 21716, 22194, 24335, 26123, 27335, 27390, 30457, 34945, 38180, 40425, 52206, 56563, 65536, 67123, 68264

Offset: 1

Amiram Eldar, Apr 05 2020

nonn

A variation of A064111 and A228126 with unitary prime-power divisors instead of prime divisors.

			4 is a term since A008475(4) + 1 = 4 + 1 = 5 = A008475(5).

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

Cf. A006145, A008475, A039752, A064111, A228126, A331000, A333802.

s[1] = 0; s[n_] := Plus @@ (Power @@@ FactorInteger[n]); seq = {}; s1 = 0; Do[s2 = s[n]; If[s1 + 1 == s2, AppendTo[seq, n - 1]]; s1 = s2, {n, 2, 10^5}]; seq

A002497 Numbers N in A002809 such that there is rho > 0 such that for all A > 0, A008475(A)-A008475(N) >= rho*log(A/N).

Original entry on oeis.org

3, 12, 60, 420, 4620, 60060, 180180, 360360, 6126120, 116396280, 2677114440, 77636318760, 2406725881560, 89048857617720, 3651003162326520, 156993135980040360, 313986271960080720, 14757354782123793840, 14757354782123793840, 782139803452561073520, 46146248403701103337680

Offset: 1

N. J. A. Sloane

The numbers contain the starred entries on pp. 187-190 of Nicolas. It is a subsequence of A002809 by selecting only elements of a set/property "G" (page 150). G contains all N such that a real, strictly positive rho exists such that for all strictly positive integers A we have l(A)-l(N) >= rho*log(A/N). The function l()=A008475() is defined on page 139. - R. J. Mathar, Mar 23 2012

These numbers were named superior l-composite numbers (nombres l-composes superieurs, the function l(n) is A002809) by Massias, in analogy to Ramanujan's superior highly composite numbers (A002201). Deléglise and Nicolas named these numbers l-superchampion numbers. They are used by Deléglise et al. in calculating values of Landau's function g(n) (A000793). - Amiram Eldar, Aug 23 2019

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Marc Deléglise and Jean-Louis Nicolas, The Landau function and the Riemann hypothesis, arXiv preprint, arXiv:1907.07664 [math.NT] (2019).
Marc Deléglise, Jean-Louis Nicolas, and Paul Zimmermann, Landau's function for one million billions, Journal de théorie des nombres de Bordeaux, Vol. 20. No. 3 (2008), pp. 625-671.
Jean-Pierre Massias, Majoration explicite de l'ordre maximum d'un élément du groupe symétrique, Annales de la Faculté des Sciences de Toulouse: Mathématiques, Vol. 6, No. 3-4 (1984), pp. 269-281.
Jean-Louis Nicolas, Ordre maximum d'un élément du groupe S(n) des permutations et "highly composite numbers", Bull. Soc. Math. Française 97 (1969), 129-191.

Cf. A000793, A002201, A002498, A002809, A008475.

Edited by M. F. Hasler, Mar 29 2015

a(16)-a(21) from the paper by Massias added by Amiram Eldar, Aug 23 2019

A114521 a(n) = A008475(A114520(n)).

Original entry on oeis.org

5, 7, 7, 11, 13, 11, 11, 19, 13, 13, 19, 17, 29, 31, 17, 23, 43, 19, 29, 31, 23, 61, 73, 41, 37, 83, 19, 47, 31, 67, 53, 103, 29, 17, 109, 37, 127, 71, 23, 139, 41, 151, 83, 31, 43, 181, 193, 131, 23, 101, 23, 199, 29, 41, 107, 19, 61, 43, 37, 113, 71, 229, 23, 67, 241

Offset: 1

Leroy Quet, Dec 05 2005

nonn

			24 (which is composite) is the sixth term of sequence A114520. 24 = 2^3 * 3 and 2^3 + 3 = 11 (which is prime). So a(6) = 11.

Cf. A008475, A114518, A114519, A114520, A047820.

f[n_] := Plus @@ Power @@@ FactorInteger[n]; f /@ Select[Select[Range[500], PrimeQ[f[ # ]] &], ! PrimeQ[ # ] &] (* Ray Chandler, Dec 07 2005 *)

A008475(n)=local(t);if(n<1,0,t=factor(n);sum(k=1,matsize(t)[1],t[k,1]^t[k,2])); for(i=1,900,if(!isprime(i)&&isprime(A008475(i)),print1(A008475(i),","))) (Herrgesell)

Extended by Ray Chandler and Lambert Herrgesell (zero815(AT)googlemail.com), Dec 07 2005

A163408 Positive integers n such that A008475(n) is composite.

Original entry on oeis.org

4, 8, 9, 14, 15, 16, 20, 21, 25, 26, 27, 30, 32, 33, 35, 38, 39, 42, 44, 45, 46, 49, 50, 51, 55, 56, 57, 60, 62, 63, 64, 65, 66, 68, 69, 70, 74, 75, 77, 78, 80, 81, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 96, 98, 99, 102, 104, 105, 106, 110, 111, 114, 115, 116

Offset: 1

Leroy Quet, Jul 27 2009

nonn

Every integer >=2 is exclusively either in this sequence or in sequence A114518.

A008475, A114518, A163409

More terms from Franklin T. Adams-Watters, Aug 25 2011

A163409 Subsequence of composite terms of A008475.

Original entry on oeis.org

4, 8, 9, 9, 8, 16, 9, 10, 25, 15, 27, 10, 32, 14, 12, 21, 16, 12, 15, 14, 25, 49, 27, 20, 16, 15, 22, 12, 33, 16, 64, 18, 16, 21, 26, 14, 39, 28, 18, 18, 21, 81, 14, 22, 45, 32, 16, 20, 27, 34, 49, 24, 35, 51, 20, 22, 21, 15, 55, 18, 40, 24, 28, 33, 22, 24, 16, 121, 63, 44, 35, 125, 18, 128, 46, 20

Offset: 1

Leroy Quet, Jul 27 2009

nonn

A008475, A114519, A163408

a(n) = A008475(A163408(n)).

More terms from Max Alekseyev, Oct 14 2012

A038701 Prime powers q for which f(g(m(q))) = m(q), where f = A051703, g = A008475 and m = A003418.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 29, 31, 32, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 103, 107, 109, 113

Offset: 1

Vladeta Jovovic, May 01 2000

These functions are defined for all natural numbers > 1 by: g(x) = Sum (p_j^k_j) where x = Product (p_j^k_j) is prime factorization of x (A008475); f(n) = max{x:g(x)=n} (A051703); m(n) = lcm(1,2,3,...,n) (A003418).
There are no more prime powers in the list <= 199. Conjecture: The sequence is finite, i.e., f(g(m(q))) > m(q) for sufficiently great prime powers q.
No other terms below 409. - Max Alekseyev, Sep 05 2023

			27 is not in the list because m(27) = 2^4*3^3*5^2*7*11*13*17*19*23, g(m(27))=158, f(158) = 3*5*7*11*13*17*19*23*29*31 > m(27).

D. W. Wilson, Answers to sci.math questions

Cf. A000961, A003418, A008475, A051703.

Offset changed to 1 by Jinyuan Wang, Mar 16 2020

A082085 Fixed points when A008475 is iterated started at factorials of prime numbers.

Original entry on oeis.org

2, 5, 16, 37, 11, 23, 31, 67, 25, 193, 7, 7, 19, 19, 5939, 27, 13, 11, 11, 503, 15889, 37, 11, 4651, 52960025378359863409578953, 8, 13, 11, 25, 79, 19, 25, 56707367, 7, 103, 23, 9, 61

Offset: 1

Labos Elemer, Apr 08 2003

nonn

			n=25: p(25)=97, start with 97!, end at a large prime: 52960025378359863409578953=a(25); it seems difficult to predict magnitude of fixed point.

Cf. A008475, A082081, A000142, A000040, A000961.

a(n) = A082081(A000142(A000040(n))).

A354424 Numbers k for which the ratio A008475(k)/k reaches a record low.

Original entry on oeis.org

2, 6, 10, 12, 15, 20, 28, 30, 40, 42, 56, 60, 84, 105, 120, 140, 168, 180, 210, 252, 280, 315, 330, 360, 385, 390, 420, 616, 630, 660, 770, 780, 840, 924, 1092, 1155, 1260, 1540, 1820, 1848, 1980, 2184, 2310, 2520, 2730, 3080, 3465, 3640, 3960, 4095, 4290, 4620, 5460, 6552, 6930

Offset: 1

Chris Grossack, Jul 11 2022

nonn

Sequence gives the numbers k for which m/k reaches a record low, where m is minimal so that the symmetric group S_m has an element of order k.

			First, an element of order 2 shows up in S_2, so the smallest ratio we've seen so far is 1. This is the smallest ratio we see until we reach 6, since there's an element of order 6 in S_5. Next is 10, since there's an element of order 10 in S_7, and 7/10 is the next ratio smaller than 5/6. Then comes 12, since S_7 also has an element of order 12, and 7/12 is the next ratio less than 7/10, etc.

Cf. A008475.

s = {}; fm = 2; Do[If[(f = Plus @@ Power @@@ FactorInteger[n]/n) < fm, fm = f; AppendTo[s, n]], {n, 2, 7000}]; s (* Amiram Eldar, Jul 12 2022 *)

b(n) = my(f=factor(n)); vecsum(vector(#f~, i, f[i, 1]^f[i, 2])); \\ A008475
lista(nn) = my(m=oo, list=List(), x); for (n=2, nn, if ((x=b(n)/n) < m, m = x; listput(list, n););); Vec(list); \\ Michel Marcus, Jul 12 2022

memo = {1: (2,1)}
def a(n):
    if n in memo.keys(): return memo[n]
    _ = a(n-1)
    prev, prevRatio = memo[n-1]
    ratio = 1
    N = prev
    while ratio >= prevRatio:
        N += 1
        # compute m so that S_m has an element of order N
        principalDivisors = list(factor(N))
        m = sum([a^b for (a,b) in principalDivisors])
        ratio = m/N
    memo[n] = (N, ratio)
    return N

cp's OEIS Frontend

A092509 Möbius transform of sequence A008475.

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A167515 The sum over the divisors of n, except the maximum-prime-power divisors collected in A008475.

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A333801 Numbers k such that A008475(k)+1 = A008475(k+1).

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A002497 Numbers N in A002809 such that there is rho > 0 such that for all A > 0, A008475(A)-A008475(N) >= rho*log(A/N).

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A114521 a(n) = A008475(A114520(n)).

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A163408 Positive integers n such that A008475(n) is composite.

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A163409 Subsequence of composite terms of A008475.

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A038701 Prime powers q for which f(g(m(q))) = m(q), where f = A051703, g = A008475 and m = A003418.

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A082085 Fixed points when A008475 is iterated started at factorials of prime numbers.

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