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Number of integer partitions of n whose product of parts is a power of a squarefree number ( A072774).
+20
7
1, 1, 2, 3, 5, 7, 11, 13, 18, 21, 31, 34, 45, 51, 63, 72, 88, 97, 120, 128, 158, 174, 201, 222, 264, 287, 333, 359, 416, 441, 518, 557, 631, 684, 770, 833, 954, 1017, 1141, 1222, 1378, 1475, 1643, 1755, 1939, 2097, 2327, 2471, 2758, 2928, 3233, 3470, 3813, 4085
EXAMPLE
The a(1) = 1 through a(8) = 18 integer partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (22) (32) (33) (52) (44)
(111) (31) (41) (42) (61) (53)
(211) (221) (51) (331) (71)
(1111) (311) (222) (421) (422)
(2111) (321) (511) (521)
(11111) (411) (2221) (611)
(2211) (3211) (2222)
(3111) (4111) (3311)
(21111) (22111) (4211)
(111111) (31111) (5111)
(211111) (22211)
(1111111) (32111)
(41111)
(221111)
(311111)
(2111111)
(11111111)
Missing from the list for n = 7 through 9:
(43) (62) (54)
(322) (332) (63)
(431) (432)
(3221) (522)
(621)
(3222)
(3321)
(4311)
(32211)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], SameQ@@Last/@FactorInteger[Times@@#]&]], {n, 30}]
CROSSREFS
Cf. A003963, A005117, A038041, A062503, A064573, A072774, A295193, A302505, A306021, A319169, A320322, A322526, A322528, A322530.
Numbers whose product of prime indices is a power of a squarefree number ( A072774).
+20
6
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 66, 67, 68, 72, 73, 76, 79, 80
COMMENTS
The complement is {35, 37, 39, 45, 61, 65, ...}.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}. This sequence lists all MM-numbers of regular multiset multisystems, where regularity means all vertex-degrees are equal.
EXAMPLE
Most small numbers belong to this sequence. However, the sequence of multiset multisystems whose MM-numbers do not belong to this sequence begins:
35: {{2},{1,1}}
37: {{1,1,2}}
39: {{1},{1,2}}
45: {{1},{1},{2}}
61: {{1,2,2}}
65: {{2},{1,2}}
69: {{1},{2,2}}
70: {{},{2},{1,1}}
71: {{1,1,3}}
74: {{},{1,1,2}}
75: {{1},{2},{2}}
77: {{1,1},{3}}
78: {{},{1},{1,2}}
87: {{1},{1,3}}
89: {{1,1,1,2}}
90: {{},{1},{1},{2}}
91: {{1,1},{1,2}}
95: {{2},{1,1,1}}
99: {{1},{1},{3}}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], SameQ@@Last/@FactorInteger[Times@@primeMS[#]]&]
Number of integer partitions of n whose parts all have the same number of prime factors (counted with multiplicity) and whose product of parts is a power of a squarefree number ( A072774).
+20
4
1, 1, 2, 2, 3, 3, 4, 3, 5, 4, 7, 2, 7, 4, 7, 7, 9, 3, 10, 5, 12, 9, 8, 6, 14, 10, 12, 10, 14, 11, 20, 13, 18, 13, 16, 16, 25, 16, 19, 20, 26, 18, 30, 19, 27, 26, 27, 22, 38, 30, 37, 28, 38, 32, 43, 37, 46, 40, 47, 40, 66, 49, 58, 56, 64, 56, 73, 58, 76, 70, 85
EXAMPLE
The a(1) = 1 through a(8) = 5 integer partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (32) (33) (52) (44)
(1111) (11111) (222) (1111111) (53)
(111111) (2222)
(11111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], And[SameQ@@PrimeOmega/@#, SameQ@@Last/@FactorInteger[Times@@#]]&]], {n, 30}]
CROSSREFS
Cf. A002865, A003963, A005117, A064573 , A072774, A295193, A302505, A306017, A319169, A320322, A322526, A322527, A322529, A322530, A322531.
90, 126, 198, 234, 270, 300, 306, 342, 350, 378, 414, 522, 525, 550, 558, 588, 594, 650, 666, 702, 738, 774, 810, 825, 846, 850, 918, 950, 954, 975, 980, 1026, 1062, 1078, 1098, 1134, 1150, 1206, 1242, 1274, 1275, 1278, 1314, 1422, 1425, 1450, 1452, 1494
COMMENTS
All terms shown have exactly 3 distinct prime factors. a(101) = 2940 =2^2*3*5*7^2 is the first member with more than 3. a(107) = 3072 = 2^10*3 is the first member with less than 3.
EXAMPLE
90 = 2^1*3^2*5^1 is a member because the concatenation of the exponents is 121.
Number of factorizations of the n-th uniform number A072774(n) into uniform numbers > 1.
+20
2
1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 1, 2, 2, 5, 1, 1, 2, 2, 1, 2, 2, 3, 1, 5, 1, 7, 2, 2, 2, 7, 1, 2, 2, 1, 5, 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 1, 2, 11, 2, 5, 1, 2, 5, 1, 1, 2, 2, 5, 1, 5, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 7, 1, 5, 1, 5, 2, 1, 1, 5, 2, 1, 5, 2, 2, 2
COMMENTS
A number is uniform if its prime multiplicities are all equal, meaning it is a power of a squarefree number. Uniform numbers are listed in A072774.
EXAMPLE
The a(31) = 7 factorizations of 36 into uniform numbers together with the corresponding multiset partitions of {1,1,2,2}:
(2*2*3*3) {{1},{1},{2},{2}}
(2*2*9) {{1},{1},{2,2}}
(2*3*6) {{1},{2},{1,2}}
(3*3*4) {{2},{2},{1,1}}
(4*9) {{1,1},{2,2}}
(6*6) {{1,2},{1,2}}
(36) {{1,1,2,2}}
MATHEMATICA
nn=100;
facsusing[s_, n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facsusing[Select[s, Divisible[n/d, #]&], n/d], Min@@#>=d&]], {d, Select[s, Divisible[n, #]&]}]];
y=Select[Range[nn], SameQ@@Last/@FactorInteger[#]&];
Table[Length[facsusing[Rest[y], n]], {n, y}];
CROSSREFS
See link for additional cross-references.
Squarefree numbers: numbers that are not divisible by a square greater than 1.
(Formerly M0617)
+10
1751
1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113
COMMENTS
1 together with the numbers that are products of distinct primes.
Also smallest sequence with the property that a(m)*a(k) is never a square for k != m. - Ulrich Schimke (ulrschimke(AT)aol.com), Dec 12 2001
Numbers k such that there is only one Abelian group with k elements, the cyclic group of order k (the numbers such that A000688(k) = 1). - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 25 2001 a(n) is the smallest m with exactly n squarefree numbers <= m. - Amarnath Murthy, May 21 2002 k is squarefree <=> k divides prime(k)# where prime(k)# = product of first k prime numbers. - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 30 2004
The LCM of any finite subset is in this sequence. - Lekraj Beedassy, Jul 11 2006 This sequence and the Beatty Pi^2/6 sequence ( A059535) are "incestuous": the first 20000 terms are bounded within (-9, 14). - Ed Pegg Jr, Jul 22 2008 Let us introduce a function D(n) = sigma_0(n)/2^(alpha(1) + ... + alpha(r)), sigma_0(n) number of divisors of n ( A000005), prime factorization of n = p(1)^alpha(1) * ... * p(r)^alpha(r), alpha(1) + ... + alpha(r) is sequence ( A086436). Function D(n) splits the set of positive integers into subsets, according to the value of D(n). Squarefree numbers ( A005117) has D(n)=1, other numbers are "deviated" from the squarefree ideal and have 0 < D(n) < 1. For D(n)=1/2 we have A048109, for D(n)=3/4 we have A067295. - Ctibor O. Zizka, Sep 21 2008 Numbers k such that sqrt(k) cannot be simplified. - Sean Loughran, Sep 04 2011 Indices m where A057918(m)=0, i.e., positive integers m for which there are no integers k in {1,2,...,m-1} such that k*m is a square. - John W. Layman, Sep 08 2011 It appears that these are numbers j such that Product_{k=1..j} (prime(k) mod j) = 0 (see Maple code). - Gary Detlefs, Dec 07 2011. - This is the same claim as Mohammed Bouayoun's Mar 30 2004 comment above. To see why it holds: Primorial numbers, A002110, a subsequence of this sequence, are never divisible by any nonsquarefree number, A013929, and on the other hand, the index of the greatest prime dividing any n is less than n. Cf. A243291. - Antti Karttunen, Jun 03 2014 Conjecture: For each n=2,3,... there are infinitely many integers b > a(n) such that Sum_{k=1..n} a(k)*b^(k-1) is prime, and the smallest such an integer b does not exceed (n+3)*(n+4). - Zhi-Wei Sun, Mar 26 2013 The probability that a random natural number belongs to the sequence is 6/Pi^2, A059956 (see Cesà ro reference). - Giorgio Balzarotti, Nov 21 2013 Booker, Hiary, & Keating give a subexponential algorithm for testing membership in this sequence without factoring. - Charles R Greathouse IV, Jan 29 2014 Because in the factorizations into prime numbers these a(n) (n >= 2) have exponents which are either 0 or 1 one could call the a(n) 'numbers with a fermionic prime number decomposition'. The levels are the prime numbers prime(j), j >= 1, and the occupation numbers (exponents) e(j) are 0 or 1 (like in Pauli's exclusion principle). A 'fermionic state' is then denoted by a sequence with entries 0 or 1, where, except for the zero sequence, trailing zeros are omitted. The zero sequence stands for a(1) = 1. For example a(5) = 6 = 2^1*3^1 is denoted by the 'fermionic state' [1, 1], a(7) = 10 by [1, 0, 1]. Compare with 'fermionic partitions' counted in A000009. - Wolfdieter Lang, May 14 2014 The following is an Eratosthenes-type sieve for squarefree numbers. For integers > 1:
1) Remove even numbers, except for 2; the minimal non-removed number is 3.
2) Replace multiples of 3 removed in step 1, and remove multiples of 3 except for 3 itself; the minimal non-removed number is 5.
3) Replace multiples of 5 removed as a result of steps 1 and 2, and remove multiples of 5 except for 5 itself; the minimal non-removed number is 6.
4) Replace multiples of 6 removed as a result of steps 1, 2 and 3 and remove multiples of 6 except for 6 itself; the minimal non-removed number is 7.
5) Repeat using the last minimal non-removed number to sieve from the recovered multiples of previous steps.
Proof. We use induction. Suppose that as a result of the algorithm, we have found all squarefree numbers less than n and no other numbers. If n is squarefree, then the number of its proper divisors d > 1 is even (it is 2^k - 2, where k is the number of its prime divisors), and, by the algorithm, it remains in the sequence. Otherwise, n is removed, since the number of its squarefree divisors > 1 is odd (it is 2^k-1).
(End)
The lexicographically least sequence of integers > 1 such that each entry has an even number of proper divisors occurring in the sequence (that's the sieve restated). - Glen Whitney, Aug 30 2015 0 is nonsquarefree because it is divisible by any square. - Jon Perry, Nov 22 2014, edited by M. F. Hasler, Aug 13 2015 The Heinz numbers of partitions with distinct parts. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product_{j=1..r} prime(j) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] the Heinz number is 2*2*3*7*29 = 2436. The number 30 (= 2*3*5) is in the sequence because it is the Heinz number of the partition [1,2,3]. - Emeric Deutsch, May 21 2015 It is possible for 2 consecutive terms to be even; for example a(258)=422 and a(259)=426. - Thomas Ordowski, Jul 21 2015. [These form a subsequence of A077395 since their product is divisible by 4. - M. F. Hasler, Aug 13 2015] There are never more than 3 consecutive terms. Runs of 3 terms start at 1, 5, 13, 21, 29, 33, ... ( A007675). - Ivan Neretin, Nov 07 2015 Numbers k such that b^(phi(k)+1) == b (mod k) for every integer b. - Thomas Ordowski, Oct 09 2016 Boreico shows that the set of square roots of the terms of this sequence is linearly independent over the rationals. - Jason Kimberley, Nov 25 2016 (reference found by Michael Coons). The prime zeta function P(s) "has singular points along the real axis for s=1/k where k runs through all positive integers without a square factor". See Wolfram link. - Maleval Francis, Jun 23 2018 The Schnirelmann density of the squarefree numbers is 53/88 (Rogers, 1964). - Amiram Eldar, Mar 12 2021 Numbers k such that all groups of order k have a trivial Frattini subgroup [Dummit and Foote].
Let the group G have order n. If n is squarefree and n > 1, then G is solvable, and thus by Hall's Theorem contains a subgroup H_p of index p for all p | n. Each H_p is maximal in G by order considerations, and the intersection of all the H_p's is trivial. Thus G's Frattini subgroup Phi(G), being the intersection of G's maximal subgroups, must be trivial. If n is not squarefree, the cyclic group of order n has a nontrivial Frattini subgroup. (End)
Numbers for which the squarefree divisors ( A206778) and the unitary divisors ( A077610) are the same; moreover they are also the set of divisors ( A027750). - Bernard Schott, Nov 04 2022 Numbers n such that mu(n) != 0, where mu(n) is the Möbius function ( A008683). Solutions to the equation Sum_{d|n} mu(d)*sigma(d) = mu(n)*n, where sigma(n) is the sum of divisors function ( A000203). (End) a(n) is the smallest root of x = 1 + Sum_{k=1..n-1} floor(sqrt(x/a(k))) greater than a(n-1). - Yifan Xie, Jul 10 2024
REFERENCES
Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 165, p. 53, Ellipses, Paris, 2008.
Dummit, David S., and Richard M. Foote. Abstract algebra. Vol. 1999. Englewood Cliffs, NJ: Prentice Hall, 1991.
Ivan M. Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 251.
Michael Pohst and Hans J. Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, page 432.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Srinivasa Ramanujan, Irregular numbers, J. Indian Math. Soc., Vol. 5 (1913), pp. 105-106. Eric Weisstein's World of Mathematics, Squarefree.
FORMULA
|a(n) - n*Pi^2/6| < 0.058377*sqrt(n) for n >= 268293; this result can be derived from Cohen, Dress, & El Marraki, see links. - Charles R Greathouse IV, Jan 18 2018 Sum_{n>=1} (-1)^(a(n)+1)/a(n)^2 = 9/Pi^2.
Sum_{k=1..n} 1/a(k) ~ (6/Pi^2) * log(n).
Sum_{k=1..n} (-1)^(a(k)+1)/a(k) ~ (2/Pi^2) * log(n).
(all from Scott, 2006) (End)
MAPLE
with(numtheory); a := [ ]; for n from 1 to 200 do if issqrfree(n) then a := [ op(a), n ]; fi; od:
t:= n-> product(ithprime(k), k=1..n): for n from 1 to 113 do if(t(n) mod n = 0) then print(n) fi od; # Gary Detlefs, Dec 07 2011 A005117 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do if numtheory[issqrfree](a) then return a; end if; end do: end if; end proc: # R. J. Mathar, Jan 09 2013
MATHEMATICA
Select[Range[150], Max[Last /@ FactorInteger[ # ]] < 2 &] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 26 2006 *)
NextSquareFree[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sf = n + sgn; While[c < Abs[k], While[ ! SquareFreeQ@ sf, If[sgn < 0, sf--, sf++]]; If[ sgn < 0, sf--, sf++]; c++]; sf + If[ sgn < 0, 1, -1]]; NestList[ NextSquareFree, 1, 70] (* Robert G. Wilson v, Apr 18 2014 *)
PROG
(Magma) [ n : n in [1..1000] | IsSquarefree(n) ];
(PARI) bnd = 1000; L = vector(bnd); j = 1; for (i=1, bnd, if(issquarefree(i), L[j]=i; j=j+1)); L
(PARI) {a(n)= local(m, c); if(n<=1, n==1, c=1; m=1; while( c<n, m++; if(issquarefree(m), c++)); m)} /* Michael Somos, Apr 29 2005 */ (PARI) list(n)=my(v=vectorsmall(n, i, 1), u, j); forprime(p=2, sqrtint(n), forstep(i=p^2, n, p^2, v[i]=0)); u=vector(sum(i=1, n, v[i])); for(i=1, n, if(v[i], u[j++]=i)); u \\ Charles R Greathouse IV, Jun 08 2012 (PARI)
S(n) = my(s); forsquarefree(k=1, sqrtint(n), s+=n\k[1]^2*moebius(k)); s;
a(n) = my(min=1, max=231, k=0, sc=0); if(n >= 144, min=floor(zeta(2)*n - 5*sqrt(n)); max=ceil(zeta(2)*n + 5*sqrt(n))); while(min <= max, k=(min+max)\2; sc=S(k); if(abs(sc-n) <= sqrtint(n), break); if(sc > n, max=k-1, if(sc < n, min=k+1, break))); while(!issquarefree(k), k-=1); while(sc != n, my(j=1); if(sc > n, j = -1); k += j; sc += j; while(!issquarefree(k), k += j)); k; \\ Daniel Suteu, Jul 07 2022 (PARI) first(n)=my(v=vector(n), i); forsquarefree(k=1, if(n<268293, (33*n+30)\20, (n*Pi^2/6+0.058377*sqrt(n))\1), if(i++>n, return(v)); v[i]=k[1]); v \\ Charles R Greathouse IV, Jan 10 2023 (Haskell)
a005117 n = a005117_list !! (n-1)
a005117_list = filter ((== 1) . a008966) [1..]
(Python)
from sympy.ntheory.factor_ import core
def ok(n): return core(n, 2) == n
(Python)
from itertools import count, islice
from sympy import factorint
def A005117_gen(startvalue=1): # generator of terms >= startvalue return filter(lambda n:all(x == 1 for x in factorint(n).values()), count(max(startvalue, 1)))
(Python)
from math import isqrt
from sympy import mobius
def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
m, k = n, f(n)
while m != k:
m, k = k, f(k)
CROSSREFS
Cf. A076259 (first differences), A173143 (partial sums), A000688, A003277, A013928, A020753, A020754, A020755, A030059, A030229, A033197, A034444, A039956, A048672, A053797, A057918, A059956, A071403, A072284, A120992, A133466, A136742, A136743, A160764, A243289, A243347, A243348, A243351, A215366, A046660, A265668, A265675.
Tek's flip: Write n as the product of distinct factors of the form prime(i)^(2^(j-1)) with i and j integers, and replace each such factor with prime(j)^(2^(i-1)).
+10
94
1, 2, 4, 3, 16, 8, 256, 6, 9, 32, 65536, 12, 4294967296, 512, 64, 5, 18446744073709551616, 18, 340282366920938463463374607431768211456, 48, 1024, 131072, 115792089237316195423570985008687907853269984665640564039457584007913129639936, 24, 81, 8589934592, 36, 768
COMMENTS
This is a multiplicative self-inverse permutation of the integers.
This sequence operates on the Fermi-Dirac factors of a number. As arranged in array form, in A329050, this sequence reflects these factors about the main diagonal of the array, substituting A329050[j,i] for A329050[i,j], and this results in many relationships including significant homomorphisms. This sequence provides a relationship between the operations of squaring and prime shift ( A003961) because each successive column of the A329050 array is the square of the previous column, and each successive row is the prime shift of the previous row. A329050 gives examples of how significant sets of numbers can be formed by choosing their factors in relation to rows and/or columns. This sequence therefore maps equivalent derived sets by exchanging rows and columns. Thus odd numbers are exchanged for squares, squarefree numbers for powers of 2 etc.
Alternative construction: For n > 1, form a vector v of length A299090(n), where each element v[i] for i=1.. A299090(n) is a product of those distinct prime factors p(i) of n whose exponent e(i) has the bit (i-1) "on", or 1 (as an empty product) if no such exponents are present. a(n) is then Product_{i=1.. A299090(n)} A000040(i)^ A048675(v[i]). Note that because each element of vector v is squarefree, it means that each exponent A048675(v[i]) present in the product is a "submask" (not all necessarily proper) of the binary string A087207(n). This permutation effects the following mappings:
(End)
Moreover, we see also that this sequence maps between A016825 (Numbers of the form 4k+2) and A001105 (2*squares) as well as between A008586 (Multiples of 4) and A028983 (Numbers with even sum of the divisors). (End)
FORMULA
a(prime(i)) = 2^(2^(i-1)).
The previous formula implies a(n*k) = a(n) * a(k) if A059895(n,k) = 1. (End)
For all n >= 1, a(2n) = A334747(a(n)). In particular, for n = A003159(m), m >= 1, a(2n) = 2*a(n). [Note that A003159 includes all odd numbers] (End)
EXAMPLE
7744 = prime(1)^2^(2-1)*prime(1)^2^(3-1)*prime(5)^2^(2-1).
a(7744) = prime(2)^2^(1-1)*prime(3)^2^(1-1)*prime(2)^2^(5-1) = 645700815.
MATHEMATICA
Array[If[# == 1, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]] &, 28] (* Michael De Vlieger, Jan 21 2020 *)
PROG
(PARI)
A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));
a(n) = {my(f=factor(n)); for (i=1, #f~, my(p=f[i, 1]); f[i, 1] = A019565(f[i, 2]); f[i, 2] = 2^(primepi(p)-1); ); factorback(f); } \\ Michel Marcus, Nov 29 2019 (PARI)
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A225546(n) = if(1==n, 1, my(f=factor(n), u=#binary(vecmax(f[, 2])), prods=vector(u, x, 1), m=1, e); for(i=1, u, for(k=1, #f~, if(bitand(f[k, 2], m), prods[i] *= f[k, 1])); m<<=1); prod(i=1, u, prime(i)^ A048675(prods[i]))); \\ Antti Karttunen, Feb 02 2020
(Python)
from math import prod
from sympy import prime, primepi, factorint
def A225546(n): return prod(prod(prime(i) for i, v in enumerate(bin(e)[:1:-1], 1) if v == '1')**(1<<primepi(p)-1) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 17 2023
CROSSREFS
Cf. A225547 (fixed points) and the subsequences listed there. Pairs of sequences (f,g) that satisfy a(f(n)) = g(a(n)): ( A000265, A008833), ( A000290, A003961), ( A005843, A334747), ( A006519, A007913), ( A008586, A334748). Pairs of sequences (f,g) that satisfy f(a(n)) = g(n), possibly with offset change: ( A000035, A010052), ( A008966, A209229), ( A007814, A248663), ( A061395, A299090), ( A087207, A267116), ( A225569, A227291). Cf. A331740 [= A001222(a(n)), number of prime factors with multiplicity]. A self-inverse isomorphism between pairs of A059897 subgroups: ( A000079, A005117), ( A000244, A062503), ( A000290\{0}, A005408), ( A000302, A056911), ( A000351, A113849 U {1}), ( A000400, A062838), ( A001651, A252895), ( A003586, A046100), ( A007310, A000583), ( A011557, A113850 U {1}), ( A028982, A042968), ( A053165, A065331), ( A262675, A268390).
Squarefree numbers squared.
+10
61
1, 4, 9, 25, 36, 49, 100, 121, 169, 196, 225, 289, 361, 441, 484, 529, 676, 841, 900, 961, 1089, 1156, 1225, 1369, 1444, 1521, 1681, 1764, 1849, 2116, 2209, 2601, 2809, 3025, 3249, 3364, 3481, 3721, 3844, 4225, 4356, 4489, 4761, 4900, 5041, 5329, 5476
COMMENTS
Also, except for the initial term, numbers whose prime factors are squared. - Cino Hilliard, Jan 25 2006 Also cubefree numbers that are squares. - Gionata Neri, May 08 2016 All positive integers have a unique factorization into powers of squarefree numbers with distinct exponents that are powers of two. So every positive number is a product of at most one squarefree number ( A005117), at most one square of a squarefree number (term of this sequence), at most one 4th power of a squarefree number ( A113849), at most one 8th power of a squarefree number, and so on. - Peter Munn, Mar 12 2020
FORMULA
Numbers k such that Sum_{d|k} mu(d)*mu(k/d) = 1. - Benoit Cloitre, Mar 03 2004 For all k in the sequence, Omega(k) = 2*omega(k). - Wesley Ivan Hurt, Apr 30 2020
MATHEMATICA
Select[Range[100], SquareFreeQ]^2
PROG
(PARI) je=[]; for(n=1, 200, if(issquarefree(n), je=concat(je, n^2), )); je
(PARI) n=0; for (m=1, 10^5, if(issquarefree(m), write("b062503.txt", n++, " ", m^2); if (n==1000, break))) \\ Harry J. Smith, Aug 08 2009 (Haskell)
(Python)
from math import isqrt
from sympy import mobius
def f(x): return n-1+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
kmin, kmax = 1, 2
while f(kmax) >= kmax:
kmax <<= 1
while True:
kmid = kmax+kmin>>1
if f(kmid) < kmid:
kmax = kmid
else:
kmin = kmid
if kmax-kmin <= 1:
break
CROSSREFS
Characteristic function is A227291. A329332 column 2 in ascending order.
Numbers whose multiset of prime indices has all equal run-sums.
+10
33
1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 40, 41, 43, 47, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 112, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 169, 173, 179
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. For example, the runs of (2,2,1,1,1,3,2,2) are (2,2), (1,1,1), (3), (2,2), with sums (4,3,3,4).
EXAMPLE
The prime indices of 12 are {1,1,2}, with run-sums (2,2), so 12 is in the sequence.
MATHEMATICA
Select[Range[100], SameQ@@Cases[FactorInteger[#], {p_, k_}:>PrimePi[p]*k]&]
CROSSREFS
For run-lengths instead of run-sums we have A072774, counted by A047966. These partitions are counted by A304442. These are the positions of powers of primes in A353832. The restriction to nonprimes is A353834. For distinct instead of equal run-sums we have A353838, counted by A353837. A005811 counts runs in binary expansion, distinct run-lengths A165413.
Numbers with different exponents in their prime factorizations.
+10
29
12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189
COMMENTS
Former name: Numbers k such that k/(largest power of squarefree kernel of k) is larger than 1.
Complement of A072774 (powers of squarefree numbers). Also numbers k = p(1)^alpha(1)* ... * p(r)^alpha(r) such that RootMeanSquare(alpha(1), ..., alpha(r)) is not an integer. - Ctibor O. Zizka, Sep 19 2008
EXAMPLE
440 is in the sequence because 440 = 2^3*5*11 and it has two distinct exponents, 3 and 1.
PROG
(Python)
from sympy import factorint
def ok(n): return len(set(factorint(n).values())) > 1
(Python)
from math import isqrt
from sympy import mobius, integer_nthroot
def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
def f(x): return n+1-(y:=x.bit_length())+sum(g(integer_nthroot(x, k)[0]) for k in range(1, y))
kmin, kmax = 1, 2
while f(kmax) >= kmax:
kmax <<= 1
while True:
kmid = kmax+kmin>>1
if f(kmid) < kmid:
kmax = kmid
else:
kmin = kmid
if kmax-kmin <= 1:
break
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