A007947 -id:A007947 - cp's OEIS frontend

A086486 Numbers k such that the sum of the distinct prime divisors divides rad(k)=A007947(k).

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 37, 41, 43, 47, 49, 53, 59, 60, 61, 64, 67, 70, 71, 73, 79, 81, 83, 89, 90, 97, 101, 103, 105, 107, 109, 113, 120, 121, 125, 127, 128, 131, 137, 139, 140, 149, 150, 151, 157, 163, 167

Offset: 1

Amarnath Murthy, Jul 28 2003

Every prime power is a member.
Numbers with exactly two distinct prime divisors are not members of the sequence. - Victoria A Sapko (vsapko(AT)canes.gsw.edu), Sep 23 2003
Numbers k such that A008472(k) divides A007947(k).

			30 is a member. The prime divisors of 30 are 2, 3 and 5 and 2+3+5 = 10, divides 30.
84, however, is not a member because the sum of its distinct prime divisors (2+3+7=12) does not divide the product of its distinct prime divisors (2*3*7=42), even though 12 does divide 84. - _Harvey P. Dale_, Nov 26 2011, based on a comment from _Ray Chandler_

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A086487, A066031. A proper subset of A089352.

sdpQ[n_]:=Module[{dpds=Transpose[FactorInteger[n]][[1]]}, Divisible[ Times@@dpds,Total[dpds]]]; Select[Range[2,200],sdpQ] (* Harvey P. Dale, Nov 26 2011 *)

More terms from Victoria A Sapko (vsapko(AT)canes.gsw.edu), Sep 23 2003

Edited by Franz Vrabec, Sep 03 2005

A143700 a(n) is the least odd number m minimizing A007947(m*(2^n-m)).

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 3, 13, 169, 25, 243, 375, 11, 49, 7, 3, 18225, 71875, 4913, 1701, 144027, 1825, 3483, 2197, 9156027, 131989, 1103, 5103, 38525, 458703, 1523, 3483891, 19283525

Offset: 1

Artur Jasinski, Nov 10 2008

Smallest odd number a(n) such that product of distinct prime divisors of (2^n)*a(n)*(2^n - a(n)) is the smallest available for a(n) <= 2^x - a(n) < 2^x.
Product of distinct prime divisors of (2^n)*a(n)*(2^n - a(n)) is also called radical: rad((2^n)*a(n)*(2^n - a(n))).
For numbers 2^n - a(n) see A143701.
For minimal values of rad((2^n)*a(n)*(2^n - a(n))) see A143702.
Related to the abc conjecture. - M. F. Hasler, Nov 13 2008

Cf. A007947, A085152, A085153, A147298-A147307, A147638-A147643.

a = {{1, 1}}; aa = {1}; bb = {}; rr = {}; Do[logmax = 0; k = 2^x; w = Floor[(k - 1)/2]; Do[m = FactorInteger[n (k - n)]; rad = 1; Do[rad = rad m[[s]][[1]], {s, 1, Length[m]}]; log = Log[k]/Log[rad]; If[log > logmax, bmin = k - n; amax = n; logmax = log; r = rad], {n, 1, w, 2}]; Print[{x, amax}]; AppendTo[aa, amax]; AppendTo[bb, bmin]; AppendTo[rr, r]; AppendTo[a, {x, logmax}], {x, 2, 15}]; aa (* Artur Jasinski with assistance of M. F. Hasler *)

A143700(n) = {my(b=1, m=2^n-b); forstep(a=3, 2^(n-1), 2, A007947(a)*A007947(2^n-a)A007947((2^n-a)*b=a)); b; } \\ M. F. Hasler, Nov 13 2008

a(28)-a(33) from M. F. Hasler, Nov 13 2008

A255334 Numbers n for which there exists k > n such that A000203(k) = A000203(n) and A007947(k) = A007947(n), where A000203 gives the sum of divisors, and A007947 gives the squarefree kernel of n.

Original entry on oeis.org

1512, 7560, 16632, 19656, 25704, 28728, 34776, 37800, 43848, 44928, 46872, 55944, 61992, 65016, 71064, 80136, 83160, 89208, 92232, 98280, 101304, 107352, 110376, 119448, 125496, 128520, 134568, 143640, 146664, 152712, 155736, 161784, 164808, 170856, 173880, 182952, 189000, 192024, 198072, 207144, 210168, 216216

Offset: 1

Antti Karttunen, Mar 23 2015

nonn

None of the terms are squarefree, because if there were such x, then we would have rad(x) = x, and for any value k > x such that rad(k) = x we would have k = y*x, for some strictly positive integer y, and in that case sigma(k) > sigma(x). Thus all terms are members of sequence A013929.

None of the terms in range a(1) .. a(6589) occur in A255335. Are the sequences disjoint forever?

Antti Karttunen, Table of n, a(n) for n = 1..6589

Subsequence of A013929.
Cf. A000203, A007947.
Cf. also A255423 (gives the corresponding k), A255335 (same sequence sorted into ascending order, with duplicates removed), A255412 [gives sigma(a(n))], A255424 [gives rad(a(n))], A255425, A254035, A254791.

A007947(n) = factorback(factorint(n)[, 1]); \\ Andrew Lelechenko, May 09 2014
isA255334(n) = { my(r=A007947(n), s=sigma(n), k=n+r); while(kA007947(k) == r), return(1), k = k+r)); return(0); };
i=0; for(n=1, 2^25, if(isA255334(n), i++; write("b255334.txt", i, " ", n)))

;; With Antti Karttunen's IntSeq-library. Quite naive and slow implementation.
(define A255334 (MATCHING-POS 1 1 isA255334?))
(define (isA255334? n) (let ((sig_n (A000203 n)) (rad_n (A007947 n))) (let loop ((try (+ n rad_n))) (cond ((>= try sig_n) #f) ((and (= sig_n (A000203 try)) (= rad_n (A007947 try))) #t) (else (loop (+ try rad_n)))))))

a(n) = A255424(n) * A255425(n).

A336550 Numbers k such that A007947(k) divides sigma(k) and A003557(k)-1 either divides A326143(k) [= A001065(k) - A007947(k)], or both are zero.

Original entry on oeis.org

6, 24, 28, 96, 120, 234, 384, 496, 936, 1536, 1638, 6144, 8128, 24576, 42588, 98304, 393216, 1089270, 1572864, 6291456, 25165824, 33550336, 100663296, 115048440, 402653184, 1185125760, 1610612736

Offset: 1

Antti Karttunen, Jul 28 2020

Numbers k such that gcd(sigma(k)-A007947(k), A007947(k)) == A007947(k) are those in A175200. These are equal to k such that gcd(A326143(k), A007947(k)) = gcd(sigma(k)-A007947(k)-k, A007947(k)) are equal to A007947(k).
Sequence is infinite because all numbers of the form 6*4^n (A002023) are present.
Question: Are there any odd terms?

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Intersection of A175200 and A336552.
Cf. A007947, A326143, A336551.
Cf. A000396, A002023, A326145 (subsequences).
Cf. also A336641 for a similar construction.

A007947(n) = factorback(factorint(n)[, 1]);
isA336550(n) = { my(r=A007947(n), s=sigma(n), u=((n/r)-1)); (!(s%r) && (gcd(u,(s-r-n))==u)); };

A360765 Numbers k that are neither prime powers nor squarefree, such that A007947(k) * A053669(k) < k.

Original entry on oeis.org

36, 40, 45, 48, 50, 54, 56, 63, 72, 75, 80, 88, 96, 98, 99, 100, 104, 108, 112, 117, 135, 136, 144, 147, 152, 153, 160, 162, 171, 175, 176, 184, 189, 192, 196, 200, 207, 208, 216, 224, 225, 232, 240, 242, 245, 248, 250, 252, 261, 270, 272, 275, 279, 280, 288, 294, 296, 297, 300, 304, 315, 320, 324, 325

Offset: 1

Michael De Vlieger, Mar 05 2023

nonn

Let rad(k) = A007947(k), and let q = A053669(k).
Let j = A007947(k)*A053669(k) = rad(k)*q.
Composite prime powers p^e such that e > 1 and p^e > 4 have the property j < k. With rad(p^e) = p, in the case of p = 2, pq = 6, 6 < 2^e for e > 2. In the case of odd p, we have 2p < p^e for e > 1.
Squarefree k do not have this property, since rad(k) = k, thus, kq > k by definition of prime q.
For k in this sequence, omega(j) > omega(k), but Omega(j) <= Omega(k), where omega(n) = A001221(n), and Omega(n) = A001222(n).
Subset of A126706.

			k = 12 is not in the sequence since rad(k)*q(k) = 6*5 = 30, and 30 exceeds k. 18 and 24 are also not in the sequence for the same reason.
k = 36 is in the sequence since rad(36)*q(36) = 6*5 = 30, and 30 < 36.
k = 45 is in the sequence since rad(45)*q(45) = 15*2 = 30, and 30 < 45.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, 1016 X 1016 pixel bitmap of n, n = 1..1032256, showing n in black if A126706(n) is in this sequence, else white.

Cf. A001221, A001222, A007947, A053669, A126706, A246547.

rad[n_] := rad[n] = Times @@ FactorInteger[n][[All, 1]];
q[n_] := If[OddQ[n], 2, p = 2; While[Divisible[n, p], p = NextPrime[p]]; p];
Select[Select[Range[325], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], rad[#]*q[#] < # &] (* Michael De Vlieger, Mar 05 2023 *)

A379552 Number of pairs (d, k/d), d < k/d, such that d|k, rad(d) = rad(k/d) = rad(k), but d|k/d, for k = A376936(n), where rad = A007947.

Original entry on oeis.org

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

Offset: 1

Michael De Vlieger, Dec 25 2024

In other words, one half the number of coreful complementary divisor pairs (d, k/d), d|k, that do not divide one another, for k in A376936, the sequence of numbers k that have at least 1 such pair.

Divisors d and k/d are both composite, further, are neither squarefree nor prime powers, hence in A126706.

			Let b(n) = A376936(n) and define property Q pertaining to (d, k/d), d|k, to be rad(d) = rad(k/d) = rad(k) but neither d | k/d nor k/d | d. Examples below show only (d, k/d) that have property Q:
a(1) = 1 since b(1) = 216 = 12*18.
a(2) = 1 since b(2) = 432 = 18*24.
a(3) = 1 since b(3) = 648 = 12*54.
a(4) = 2 since b(4) = 864 = 18*48 = 24*36.
a(14) = 3 since b(14) = 3456 = 18*192 = 36*96 = 48*72.
a(22) = 4 since b(22) = 7776 = 24*324 = 48*162 = 54*144 = 72*108, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007947, A126706, A376936, A379553, A379554.

nn = 2^16;
rad[x_] := Times @@ FactorInteger[x][[All, 1]];
s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
  Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &];
Table[k = s[[n]];
  Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2] ]] &@ Divisors[k],
    _?(And[1 < GCD @@ {##},
       rad[#1] == rad[#2],
       Mod[#1, #2] != 0,
       Mod[#2, #1] != 0] & @@ # &)], {n, Length[s]}]

A062760 a(n) is n divided by the largest power of the squarefree kernel of n (A007947) which divides it.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 8, 1, 5, 1, 2, 1, 9, 1, 4, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 5, 2, 1, 1, 1, 8, 1, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 16, 1, 7, 3, 1, 1, 1, 1, 4

Offset: 1

Labos Elemer, Jul 16 2001

nonn

a(n) divides A003557 but is not equal to it.

a(n) is least d such that the prime power exponents of n/d are all equal; see also A066636. - David James Sycamore, Jun 13 2024

			n=1800: the squarefree kernel is 2*3*5 = 30 and 900 = 30^2 divides n, a(1800) = 2, the quotient of 1800/900.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A001694, A003557, A007947, A051904, A003557, A052485, A062759.
Cf. A059404 (n such that a(n)>1), A072774 (n such that a(n)=1).
Cf. A066636.

f:= proc(n) local F,m,t;
  F:= ifactors(n)[2];
  m:= min(seq(t[2],t=F));
  mul(t[1]^(t[2]-m),t=F)
end proc:
map(f, [$1..200]); # Robert Israel, Nov 03 2017

{1}~Join~Table[n/#^IntegerExponent[n, #] &@ Last@ Select[Divisors@ n, SquareFreeQ], {n, 2, 104}] (* Michael De Vlieger, Nov 02 2017 *)
a[n_] := Module[{f = FactorInteger[n], e}, e = Min[f[[;; , 2]]]; f[[;; , 2]] -= e; Times @@ Power @@@ f]; Array[a, 100] (* Amiram Eldar, Feb 12 2023 *)

A007947(n) = factorback(factorint(n)[, 1]); \\ Andrew Lelechenko, May 09 2014
A051904(n) = if(1==n,0,vecmin(factor(n)[, 2])); \\ After Charles R Greathouse IV's code
A062760(n) = n/(A007947(n)^A051904(n)); \\ Antti Karttunen, Sep 23 2017

a(n) = n/(A007947(n)^A051904(n)).

a(n) = n/A062759(n). - Amiram Eldar, Feb 12 2023

A078325 Squarefree numbers of the form m*rad(m)+1, where rad = A007947 (squarefree kernel).

Original entry on oeis.org

2, 5, 10, 17, 26, 33, 37, 65, 73, 82, 101, 109, 122, 129, 145, 170, 197, 201, 217, 226, 257, 290, 362, 393, 401, 433, 442, 485, 501, 530, 577, 626, 649, 677, 730, 785, 842, 865, 901, 962, 969, 973, 1001, 1090, 1126, 1153, 1157, 1226, 1297, 1353, 1370, 1373

Offset: 1

Reinhard Zumkeller, Nov 23 2002

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Intersection of A005117 and A224866.

Cf. A008966, A078310, A078324.

a078325 n = a078325_list !! (n-1)
a078325_list = filter ((== 1) . a008966) a224866_list
-- Reinhard Zumkeller, Jul 23 2013

powQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;; , 2]], # > 1 &]; Select[Range[1400], SquareFreeQ[#] && powQ[# - 1] &] (* Amiram Eldar, Jul 31 2022 *)

is(n) = n>1 && issquarefree(n) && ispowerful(n-1); \\ Amiram Eldar, Jul 31 2022

A285329 a(n) = A013928(A007947(n)).

Original entry on oeis.org

0, 1, 2, 1, 3, 4, 5, 1, 2, 6, 7, 4, 8, 9, 10, 1, 11, 4, 12, 6, 13, 14, 15, 4, 3, 16, 2, 9, 17, 18, 19, 1, 20, 21, 22, 4, 23, 24, 25, 6, 26, 27, 28, 14, 10, 29, 30, 4, 5, 6, 31, 16, 32, 4, 33, 9, 34, 35, 36, 18, 37, 38, 13, 1, 39, 40, 41, 21, 42, 43, 44, 4, 45, 46, 10, 24, 47, 48, 49, 6, 2, 50, 51, 27, 52, 53, 54, 14, 55, 18, 56, 29, 57, 58, 59, 4, 60, 9, 20, 6

Offset: 1

Antti Karttunen, Apr 17 2017

nonn

For n > 1, a(n) gives the (one-based) index of the column where n is located in array A284311, or respectively, index of the row where n is in A284457. A008479 gives the other index.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A008479 (the other index).
Cf. A005117, A007947, A008683, A013928, A019565, A064273, A087207, A285328.
Cf. array A284311 (A284457).

from operator import mul
from sympy import primefactors
from sympy.ntheory.factor_ import core
from functools import reduce
def a007947(n): return 1 if n<2 else reduce(mul, primefactors(n))
def a013928(n): return sum(1 for i in range(1, n) if core(i) == i)
print([a013928(a007947(n)) for n in range(1, 101)]) # Indranil Ghosh, Apr 18 2017

from math import prod, isqrt
from sympy import primefactors, mobius
def A285329(n):
    m=prod(primefactors(n))-1
    return sum(mobius(k)*(m//k**2) for k in range(1,isqrt(m)+1)) # Chai Wah Wu, May 12 2024

a(n) = A013928(A007947(n)).
Other identities. For all n >= 0:
If A008683(n) <> 0 [when n is squarefree, A005117], a(n) = A013928(n), otherwise a(n) = a(A285328(n)).
a(A019565(n)) = A064273(n).

A325126 a(1) = 1; a(n) = -Sum_{d|n, dA007947.

Original entry on oeis.org

1, -2, -3, 2, -5, 6, -7, -2, 6, 10, -11, -6, -13, 14, 15, 2, -17, -12, -19, -10, 21, 22, -23, 6, 20, 26, -12, -14, -29, -30, -31, -2, 33, 34, 35, 12, -37, 38, 39, 10, -41, -42, -43, -22, -30, 46, -47, -6, 42, -40, 51, -26, -53, 24, 55, 14, 57, 58, -59, 30

Offset: 1

Ilya Gutkovskiy, Sep 04 2019

Dirichlet inverse of A007947.

Moebius transform of A125131.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A007947, A008836, A125131, A326297.

Cf. also A327564, A332732, A349340, A349612, A349618.

a[n_] := If[n == 1, n, -Sum[If[d < n, Last[Select[Divisors[n/d], SquareFreeQ]] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 60}]
f[p_, e_] := -p*(1 - p)^(e - 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 14 2020 *)

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
lista(nn) = {my(va=vector(nn)); va[1] = 1; for (n=2, nn, va[n] = -sumdiv(n, d, if (dMichel Marcus, Jun 01 2020

G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} rad(k) * A(x^k).
From Isaac Saffold, May 30 2020: (Start)
a(n) = A008836(n)*A326297(n)*A007947(n).
Proof:
Define lambda(n) := A008836(n); h(n) := A326297(n); rad(n) := A007947(n).
As lambda(n), h(n), and rad(n) are multiplicative, the identity needs only to be proved for prime power n.
It is clear that the identity holds for n = 1 = p^0. For a given nonnegative integer k, assume the identity holds for all v such that 0 <= v <= k. Then, by the recursive formula for Dirichlet inverses,
a(p^(k+1)) = -Sum_{v=0..k} lambda(p^v)*h(p^v)*rad(p^v)*rad(p^(k+1-v))
= -p * (1 + p*Sum_{v=1..k}((-1)^v * (p-1)^(v-1)))
= -p * (1 - p*Sum_{v=0..(k-1)}((1 - p)^v))
= -p * (1 - p*(((1-p)^k - 1) / -p))
= -p * (1-p)^k
= (-1)^(k+1) * (p-1)^k * p
= lambda(p^(k+1)) * h(p^(k+1)) * rad(p^(k+1))
Thus the identity holds for p^(k+1), k >= 0.
As k is arbitrary and the identity holds for p^0, it holds for the prime powers, and thus for all positive integers. Q.E.D. (End)

cp's OEIS Frontend

A086486 Numbers k such that the sum of the distinct prime divisors divides rad(k)=A007947(k).

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A143700 a(n) is the least odd number m minimizing A007947(m*(2^n-m)).

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A255334 Numbers n for which there exists k > n such that A000203(k) = A000203(n) and A007947(k) = A007947(n), where A000203 gives the sum of divisors, and A007947 gives the squarefree kernel of n.

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A336550 Numbers k such that A007947(k) divides sigma(k) and A003557(k)-1 either divides A326143(k) [= A001065(k) - A007947(k)], or both are zero.

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A360765 Numbers k that are neither prime powers nor squarefree, such that A007947(k) * A053669(k) < k.

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A379552 Number of pairs (d, k/d), d < k/d, such that d|k, rad(d) = rad(k/d) = rad(k), but d|k/d, for k = A376936(n), where rad = A007947.

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A062760 a(n) is n divided by the largest power of the squarefree kernel of n (A007947) which divides it.

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A078325 Squarefree numbers of the form m*rad(m)+1, where rad = A007947 (squarefree kernel).

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