A065642 -id:A065642 - cp's OEIS frontend

A366825 Numbers of the form p^2 * m, squarefree m > 1, prime p < lpf(m), where lpf(m) = A020639(m).

12, 20, 28, 44, 45, 52, 60, 63, 68, 76, 84, 92, 99, 116, 117, 124, 132, 140, 148, 153, 156, 164, 171, 172, 175, 188, 204, 207, 212, 220, 228, 236, 244, 260, 261, 268, 275, 276, 279, 284, 292, 308, 315, 316, 325, 332, 333, 340, 348, 356, 364, 369, 372, 380, 387

Offset: 1

Michael De Vlieger, Dec 15 2023

Proper subset of A126706. Proper subset of A364996.
Prime signature of a(n) is 2 followed by at least one 1.
Numbers of the form A065642(A120944(k)) for some k.
The asymptotic density of this sequence is (6/Pi^2) * Sum_{p prime} (1/p^2) * (Product_{primes q <= p} (q/(q+1))) = 0.155068688392... . - Amiram Eldar, Dec 18 2023

			a(1) = 12 = 4*3 = p^2 * m, squarefree m > 1; sqrt(4) < lpf(3), i.e., 2 < 3.
a(5) = 45 = 9*5 = p^2 * m, squarefree m > 1; sqrt(9) < lpf(5), i.e., 3 < 5.
Prime powers p^k, k > 2, are not in the sequence since m = p^(k-2) is not squarefree and p = lpf(m).

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

Cf. A007947, A020639, A065642, A120944, A126706, A364996.

Select[Select[Range[500], PrimeOmega[#] > PrimeNu[#] > 1 &], First[#1] == 2 && Union[#2] == {1} & @@ TakeDrop[FactorInteger[#][[All, -1]], 1] &]

is(n) = {my(e = factor(n)[, 2]); #e > 1 && e[1] == 2 && vecmax(e[2..#e]) == 1;} \\ Amiram Eldar, Dec 18 2023

A285109 a(n) = n multiplied by its smallest prime factor; a(1) = 1.

Original entry on oeis.org

1, 4, 9, 8, 25, 12, 49, 16, 27, 20, 121, 24, 169, 28, 45, 32, 289, 36, 361, 40, 63, 44, 529, 48, 125, 52, 81, 56, 841, 60, 961, 64, 99, 68, 175, 72, 1369, 76, 117, 80, 1681, 84, 1849, 88, 135, 92, 2209, 96, 343, 100, 153, 104, 2809, 108, 275, 112, 171, 116, 3481, 120, 3721, 124, 189, 128, 325, 132, 4489, 136, 207

Offset: 1

Antti Karttunen, Apr 19 2017

nonn

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

Cf. A005117, A020639, A032742, A253560, A285100.

Differs from A065642 for the first time at n=12. See A284342 for all the differing points.

a[n_] := n * FactorInteger[n][[1, 1]]; Array[a, 100] (* Amiram Eldar, Jun 30 2022 *)

a(n)=if(n==1, 1, n*factor(n)[1,1]); \\ Joerg Arndt, Oct 27 2021

Scheme
```
(define (A285109 n) (* (A020639 n) n))
```

a(n) = A020639(n) * n.
Other identities. For all n >= 1:
a(A285100(n)) = A065642(A285100(n)). [Agrees with A065642 on all terms of A285100, but not on any other points.]

A285333 a(n) = A048675(A285332(n)).

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 4, 3, 6, 4, 9, 6, 5, 8, 8, 4, 15, 8, 12, 5, 14, 10, 27, 8, 10, 6, 25, 12, 7, 16, 16, 5, 210, 16, 45, 10, 35, 16, 18, 5, 105, 16, 28, 11, 462, 28, 81, 10, 21, 12, 20, 7, 154, 26, 125, 16, 30, 8, 49, 24, 11, 32, 32, 6, 10659, 212, 420, 17, 910, 46, 75, 10, 78, 36, 175, 20, 33, 20, 24, 6, 3094, 106, 315, 18, 385, 32, 56, 17, 780045

Offset: 0

Antti Karttunen, Apr 19 2017

Following A285332, also this sequence can be represented in a form of a binary tree:
                                    0
                                    |
                 ...................1...................
                2                                       2
      3......../ \........4                   4......../ \........3
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  6       4           9       6           5       8           8       4
15 8    12 5        14 10   27 8        10 6    25 12        7 16   16 5
etc.

Antti Karttunen, Table of n, a(n) for n = 0..1023

Cf. A001477, A048675, A109162, A285325, A285330, A285332 (even bisection).

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ Michel Marcus, Oct 10 2016
A007947(n) = factorback(factorint(n)[, 1]); \\ From Andrew Lelechenko, May 09 2014
A065642(n) = { my(r=A007947(n)); if(1==n,n,n = n+r; while(A007947(n) <> r, n = n+r); n); };
A285332(n) = { if(n<=1,n+1,if(!(n%2),A019565(A285332(n/2)),A065642(A285332((n-1)/2)))); };
A285333(n) = if(!n,n,if(!(n%2),A285332(n/2),A048675(A285332(n))));

(define (A285333 n) (A048675 (A285332 n)))

a(n) = A048675(A285332(n)).
For all n >= 1, a(2n) = A285332(n).
a(2^n) = A109162(1+n). [The left edge of the tree.]
a(A000225(n)) = n. [The right edge of tree.]

A285325 Square array A(n,k) = A048675(A285321(n,k)), read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 4, 6, 4, 5, 8, 5, 8, 5, 6, 6, 12, 5, 10, 6, 7, 8, 7, 16, 6, 12, 7, 8, 8, 10, 9, 20, 6, 14, 8, 9, 16, 9, 10, 8, 24, 7, 16, 9, 10, 10, 24, 9, 12, 10, 28, 7, 18, 10, 11, 12, 11, 32, 11, 14, 9, 32, 7, 20, 11, 12, 12, 18, 17, 40, 10, 12, 11, 36, 8, 22, 12, 13, 16, 13, 14, 12, 48, 10, 14, 13, 40, 8, 24, 13

Offset: 1

Antti Karttunen, Apr 17 2017

			The top left 15x6 corner of the array:
  1,  2, 3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15
  2,  4, 4,  8,  6,  8,  8, 16, 10, 12, 12, 16, 14, 16, 16
  3,  6, 5, 12,  7, 10,  9, 24, 11, 18, 13, 20, 15, 18, 17
  4,  8, 5, 16,  9, 10,  9, 32, 17, 14, 13, 20, 17, 22, 17
  5, 10, 6, 20,  8, 12, 11, 40, 12, 20, 14, 24, 21, 18, 19
  6, 12, 6, 24, 10, 14, 10, 48, 18, 16, 19, 28, 16, 20, 18

Antti Karttunen, Table of n, a(n) for n = 1..120; the first 15 antidiagonals of array

Cf. A048675, A285321, A285333.
Row 1 & column 1: A000027.
Row 2: A285326, Row 3: A285327.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A007947(n) = factorback(factorint(n)[, 1]); \\ Andrew Lelechenko, May 09 2014
A065642(n) = { my(r=A007947(n)); if(1==n,n,n = n+r; while(A007947(n) <> r, n = n+r); n); };
A285321bi(row,col) = if(1==row,A019565(col),A065642(A285321bi(row-1,col)));
A002260(n)= { n-binomial((sqrtint(8*n)+1)\2, 2); }; \\ M. F. Hasler, Mar 10 2014
A004736(n)= { 1 + binomial(1 + floor(1/2 + sqrt(2*n)), 2) - n; };
A285321(n) = A285321bi(A002260(n), A004736(n));
A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ Michel Marcus, Oct 10 2016
A285325(n) = A048675(A285321(n));
for(n=1, 120, write("b285325.txt", n, " ", A285325(n)));

(define (A285325 n) (A285325bi (A002260 n) (A004736 n)))
(define (A285325bi row col) (A048675 (A285321bi row col)))

A(n,k) = A048675(A285321(n,k)).

A285326 a(0) = 0, for n > 0, a(n) = n + A006519(n).

Original entry on oeis.org

0, 2, 4, 4, 8, 6, 8, 8, 16, 10, 12, 12, 16, 14, 16, 16, 32, 18, 20, 20, 24, 22, 24, 24, 32, 26, 28, 28, 32, 30, 32, 32, 64, 34, 36, 36, 40, 38, 40, 40, 48, 42, 44, 44, 48, 46, 48, 48, 64, 50, 52, 52, 56, 54, 56, 56, 64, 58, 60, 60, 64, 62, 64, 64, 128, 66, 68, 68, 72, 70, 72, 72, 80, 74, 76, 76, 80, 78, 80, 80, 96, 82, 84, 84, 88, 86, 88, 88, 96, 90, 92, 92

Offset: 0

Antti Karttunen, Apr 19 2017

From M. F. Hasler, Oct 19 2019: (Start)
This sequence is equal to itself multiplied by 2 and interleaved with the positive even numbers: We have a(2n-1) = 2n (n >= 1) from the very definition, since A006519(m) = 1 for odd m. And a(2n) = 2n + A006519(2n) = 2*a(n), using A006519(2n) = 2*A006519(n).
The sequence repeats the pattern [A, B, C, C] where in the n-th occurrence C = 4n, B = C - 2, A = C if n is even, A = C + 4 if n = 3 (mod 4), and A = 16*a((n-1)/4) otherwise. (End)

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to binary expansion of n

Row 2 of A285325 (after the initial zero).
Cf. A006519, A019565, A048675, A065642, A129760, A285109.
Cf. A109168 (same terms divided by 2), also A140472.

Table[If[n>0, n + GCD[2^n, n], 0], {n, 0, 100}] (* Indranil Ghosh, Apr 20 2017 *)

a(n) = if(n>0, n + gcd(2^n, n), 0); \\ Indranil Ghosh, Apr 20 2017

A285326(n)=n+bitand(n,-n) \\ Or: {a(n)=-bitand(-n,bitneg(n))}, not faster. - M. F. Hasler, Oct 19 2019

from sympy import gcd
def a(n): return n + gcd(2**n, n) if n>0 else 0 # Indranil Ghosh, Apr 20 2017

(define (A285326 n) (if (zero? n) n (+ n (A006519 n))))
(define (A285326 n) (A048675 (A065642 (A019565 n))))

a(0) = 0; for n > 0, a(n) = n + A006519(n).
a(n) = A048675(A065642(A019565(n))) = A048675(A285109(A019565(n))).
For n >= 1, a(n) = 2*A109168(n).
a(n) = 2*A140472(n) and a(2n) = 2*a(n) and a(2^n) = 2^(n+1) for all n >= 0, a(2n-1) = 2n for all n >= 1. - M. F. Hasler, Oct 19 2019

A286543 Restricted growth sequence of A286542.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 2, 5, 4, 6, 4, 5, 4, 3, 2, 7, 8, 6, 4, 6, 9, 6, 8, 7, 4, 6, 9, 5, 9, 3, 2, 10, 8, 11, 8, 6, 9, 6, 4, 12, 8, 13, 9, 12, 14, 11, 9, 10, 9, 6, 4, 12, 8, 13, 14, 7, 8, 13, 9, 5, 9, 3, 2, 15, 16, 11, 8, 11, 14, 11, 8, 12, 8, 13, 14, 6, 4, 6, 4, 17, 18, 11, 14, 13, 9, 13, 9, 6, 19, 20, 14, 11, 14, 13, 14, 15, 18, 13, 14, 6, 18, 6, 4, 6, 18, 11, 8

Offset: 0

Antti Karttunen, May 18 2017

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A046523, A285332, A286542, A286544.

allocatemem(2^30);
rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A000120(n) = hammingweight(n);
A002110(n) = prod(i=1,n,prime(i));
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A007947(n) = factorback(factorint(n)[, 1]); \\ From Andrew Lelechenko, May 09 2014
A065642(n) = { my(r=A007947(n)); if(1==n,n,n = n+r; while(A007947(n) <> r, n = n+r); n); };
A285332(n) = { if(n<=1,n+1,if(!(n%2),A019565(A285332(n/2)),A065642(A285332((n-1)/2)))); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A286542(n) = if(!n,1,if(!(n%2),A002110(A000120(A285332(n/2))),A046523(A285332(n))));
write_to_bfile(0,rgs_transform(vector(8192,n,A286542(n-1))),"b286543.txt");

A081381 Numbers n such that n and tau(n) = A000005(n) have the same prime factors (ignoring multiplicity).

Original entry on oeis.org

1, 2, 8, 9, 12, 18, 72, 80, 96, 108, 128, 288, 448, 486, 625, 720, 768, 864, 972, 1152, 1200, 1250, 1620, 1944, 2000, 2025, 2560, 4032, 4050, 5000, 5625, 6144, 6561, 6912, 7500, 7776, 8748, 9408, 10800, 11250, 11264, 12960, 13122, 16200, 18000, 18432, 19440

Offset: 1

Labos Elemer, Mar 26 2003

nonn

			n = 5000 = 2*2*2*5*5*5*5, tau(5000) = 20 = 2*2*5, common prime factors: {2,5}

Donovan Johnson and Giovanni Resta, Table of n, a(n) for n = 1..1096 (terms < 10^11, first 500 terms from Donovan Johnson)

Cf. A000005, A027598, A055744, A065642, A081377-A081380.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] Do[s=ba[DivisorSigma[0, n]]; If[Equal[s, ba[n]], Print[n]], {n, 1, 10000}]

is(n)=my(f=factor(n)); factor(numdiv(f))[,1]==f[,1] \\ Charles R Greathouse IV, Oct 19 2017

A081378 Numbers k for which the sums of prime factors (ignoring multiplicity) of sigma(k) and phi(k) are equal but the sets of prime factors of sigma and phi are different.

Original entry on oeis.org

412, 1142, 1236, 1328, 1339, 1703, 2855, 2875, 2884, 3406, 3426, 3668, 3708, 3984, 4017, 5109, 5356, 5710, 5750, 5924, 6003, 6281, 6399, 6413, 6640, 6812, 7994, 8054, 8318, 8515, 8565, 8611, 8625, 8652, 8843, 8858, 9373, 9707, 9991

Offset: 1

Labos Elemer, Mar 26 2003

nonn

			k = 412 = 2*2*103: sigma(412) = 728 = 2*2*2*7*13, phi(412) = 204 = 2*2*3*17, the sums of prime factors are identical (2 + 7 + 13 = 22 = 2 + 3 + 17) but the prime sets are different: {2,7,13} vs. {2,7,17}.

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

Cf. A000010, A000203, A065642, A076533, A081377.

ffi[x_] := Flatten[FactorInteger[x]]; lf[x_] := Length[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; spf[x_] := Apply[Plus, ba[x]]; k=0; Do[s=ba[DivisorSigma[1, n]]; s1=ba[EulerPhi[n]]; s3=spf[DivisorSigma[1, n]]; s4=spf[EulerPhi[n]]; If[ !Equal[s, s1]&&Equal[s3, s4], k=k+1; Print[{n, s, s1, ba[n], s3}]], {n, 1, 10000}]

is(n) = {my(f = factor(n), p1 = factor(sigma(f))[, 1], p2 = factor(eulerphi(f))[, 1]); p1 != p2 && vecsum(p1) == vecsum(p2) ;} \\ Amiram Eldar, Mar 25 2024

A081382 a(1) = 1, for n > 1 a(n) = Min{x > n, A008472(x) = A008472(n)}.

Original entry on oeis.org

1, 4, 9, 8, 6, 12, 10, 16, 27, 20, 121, 18, 22, 28, 45, 32, 210, 24, 34, 40, 30, 44, 273, 25, 36, 52, 81, 56, 399, 60, 58, 64, 70, 68, 42, 48, 435, 76, 55, 49, 651, 84, 82, 88, 75, 92, 777, 54, 50, 80, 91, 104, 903, 72, 66, 98, 85, 116, 1645, 63, 118, 124, 90, 128, 77, 117

Offset: 1

Labos Elemer, Mar 26 2003

nonn

Differs from A065642, a(n) <= A065642(n).

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

Cf. A065642, A008472.

a081382 1 = 1
a081382 n = head [x | let sopf = a008472 n, x <- [n+1..], a008472 x == sopf]
-- Reinhard Zumkeller, Jun 12 2015

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] spf[x_] := Apply[Plus, ba[x]] Table[Min[Flatten[Position[Table[spf[w], {w, n+1, n^2}]-spf[n], 0]]+n], {n, 1, 100}]

A285110 a(n) = A001222(A285323(n)).

Original entry on oeis.org

0, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1

Offset: 0

Antti Karttunen, Apr 19 2017

nonn

The sequence is completely determined by the positions of two least significant 1-bits of n: After initial zero, if n is a power of two (only one 1-bit present) or if prime(1+A285099(n)) > prime(1+A007814(n))^2, a(n) = 2, otherwise a(n) = 1.

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A000040, A000290, A001222, A007814, A285099, A285323, A285324.

from operator import mul
from sympy import prime, primefactors
from functools import reduce
def a001222(n): return 0 if n<2 else a001222(n//min(primefactors(n))) + 1
def a019565(n): return reduce(mul, (prime(i+1) for i, v in enumerate(bin(n)[:1:-1]) if v == '1')) if n > 0 else 1 # This function from Chai Wah Wu
def a007947(n): return 1 if n<2 else reduce(mul, primefactors(n))
def a065642(n):
    if n==1: return 1
    r=a007947(n)
    n += r
    while a007947(n)!=r:
        n+=r
    return n
def a285323(n): return a065642(a065642(a019565(n)))//a019565(n)
def a(n): return a001222(a285323(n))
print([a(n) for n in range(121)]) # Indranil Ghosh, Apr 20 2017

(define (A285110 n) (A001222 (A285323 n)))
(define (A285110 n) (cond ((zero? n) n) ((or (= 1 (A000120 n)) (> (A000040 (+ 1 (A285099 n))) (A000290 (A000040 (+ 1 (A007814 n)))))) 2) (else 1)))

a(n) = A001222(A285323(n)).

cp's OEIS Frontend

A366825 Numbers of the form p^2 * m, squarefree m > 1, prime p < lpf(m), where lpf(m) = A020639(m).

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A285109 a(n) = n multiplied by its smallest prime factor; a(1) = 1.

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A285333 a(n) = A048675(A285332(n)).

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A285325 Square array A(n,k) = A048675(A285321(n,k)), read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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A285326 a(0) = 0, for n > 0, a(n) = n + A006519(n).

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A286543 Restricted growth sequence of A286542.

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A081381 Numbers n such that n and tau(n) = A000005(n) have the same prime factors (ignoring multiplicity).

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A081378 Numbers k for which the sums of prime factors (ignoring multiplicity) of sigma(k) and phi(k) are equal but the sets of prime factors of sigma and phi are different.

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