A065642 -id:A065642 - cp's OEIS frontend

A285322 Transpose of square array A285321.

2, 4, 3, 8, 9, 6, 16, 27, 12, 5, 32, 81, 18, 25, 10, 64, 243, 24, 125, 20, 15, 128, 729, 36, 625, 40, 45, 30, 256, 2187, 48, 3125, 50, 75, 60, 7, 512, 6561, 54, 15625, 80, 135, 90, 49, 14, 1024, 19683, 72, 78125, 100, 225, 120, 343, 28, 21

Offset: 1

Antti Karttunen, Apr 17 2017

See A285321.

Transpose: A285321.

from functools import reduce
from operator import mul
from sympy import prime, primefactors
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 = n + r
    while a007947(n)!=r:
        n+=r
    return n
def A(n, k): return a019565(k) if n==1 else a065642(A(n - 1, k))
for n in range(1, 11): print([A(n - k + 1, k) for k in range(1, n + 1)]) # Indranil Ghosh, Apr 19 2017

(define (A285322 n) (A285321bi (A004736 n) (A002260 n))) ;; For A285321bi, see A285321.

A366460 Odd terms in A366825.

Original entry on oeis.org

45, 63, 99, 117, 153, 171, 175, 207, 261, 275, 279, 315, 325, 333, 369, 387, 423, 425, 475, 477, 495, 531, 539, 549, 575, 585, 603, 637, 639, 657, 693, 711, 725, 747, 765, 775, 801, 819, 833, 855, 873, 909, 925, 927, 931, 963, 981, 1017, 1025, 1035, 1071, 1075

Offset: 1

Michael De Vlieger, Jan 05 2024

Proper subset of A364997, in turn a proper subset of A364996, which is a proper subset of A126706.
Prime signature of a(n) is 2 followed by at least one 1.
Numbers of the form A065642(k) where k is an odd term in A120944.
Numbers of the form p^2 * m, squarefree m > 1, odd prime p < lpf(m), where lpf(m) = A020639(m).
The asymptotic density of this sequence is (2/(3*Pi^2)) * Sum_{p odd prime} ((1/p^2) * (Product_{odd primes q <= p} (q/(q+1)))) = 0.0537475047... . - Amiram Eldar, Jan 08 2024

			a(1) = 45 = 9*5 = p^2 * m, squarefree m > 1; sqrt(9) < lpf(5), i.e., 3 < 5.
a(2) = 63 = 9*7 = p^2 * m, squarefree m > 1; sqrt(9) < lpf(7), i.e., 3 < 7.
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, A364997, A364999.

Select[Select[Range[1, 1100, 2], PrimeOmega[#] > PrimeNu[#] > 1 &], And[OddQ[#1], #1/(Times @@ #2) == #2[[1]]] & @@ {#, FactorInteger[#][[All, 1]]} &]

is(n) = {my(e); n%2 && e = factor(n)[, 2]; #e > 1 && e[1] == 2 && vecmax(e[2..#e]) == 1; } \\ Amiram Eldar, Jan 08 2024

{a(n)} = {A366825 \ A364999}.

A079228 Least number > n with greater squarefree kernel than that of n.

Original entry on oeis.org

2, 3, 5, 5, 6, 7, 10, 9, 10, 11, 13, 13, 14, 15, 17, 17, 19, 19, 21, 21, 22, 23, 26, 26, 26, 29, 28, 29, 30, 31, 33, 33, 34, 35, 37, 37, 38, 39, 41, 41, 42, 43, 46, 46, 46, 47, 51, 49, 50, 51, 53, 53, 55, 55, 57, 57, 58, 59, 61, 61, 62, 65, 65, 65, 66, 67, 69, 69, 70, 71, 73, 73

Offset: 1

Reinhard Zumkeller, Jan 02 2003

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

a(n) = n + A079229(n). Cf. A007947, A065642.

a079228 n = head [k | k <- [n+1..], a007947 k > a007947 n]
-- Reinhard Zumkeller, Oct 11 2011

rad[n_] := Times @@ FactorInteger[n][[All, 1]]; a[1] = 2; a[n_] := For[k = 1, True, k++, If[rad[n + k] > rad[n], Return[n + k]]]; Table[a[n], {n, 1, 72}] (* Jean-François Alcover, Oct 05 2012 *)

rad(n)=my(f=factor(n)[,1]);prod(i=1,#f,f[i])
a(n)=my(r=rad(n));while(rad(n++)<=r,); n \\ Charles R Greathouse IV, Aug 21 2013

A286544 Restricted growth sequence of A285333.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 18 2017

nonn

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

Cf. A285332, A285333, A286543.

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])); }
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))));
write_to_bfile(0,rgs_transform(vector(8192,n,A285333(n-1))),"b286544.txt");

A340305 Numbers k such that k and the least number that is larger than k and has the same set of distinct prime divisors as k also has the same prime signature as k.

Original entry on oeis.org

12, 45, 60, 63, 84, 132, 156, 175, 204, 228, 275, 276, 315, 325, 348, 350, 372, 420, 425, 444, 475, 492, 495, 516, 525, 539, 540, 564, 575, 585, 636, 637, 660, 675, 693, 700, 708, 732, 765, 780, 804, 819, 833, 852, 855, 876, 924, 931, 948, 996, 1020, 1035, 1068

Offset: 1

Amiram Eldar, Jan 03 2021

nonn

Numbers k such that k and A065642(k) have the same prime signature (A118914).

			12 is a term since the least number that is larger than 12 and has the same set of distinct prime divisors as 12, {2, 3}, is 18 = 2 * 3^2 which also has the same prime signature as 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to prime signature

Cf. A065642, A118914.

Subsequence: A251720.

rad[n_] := Times @@ FactorInteger[n][[;; , 1]]; next[n_] := Module[{r = rad[n]}, SelectFirst[Range[n + 1, n^2], rad[#] == r &]]; sig[n_] := Sort @ FactorInteger[n][[;; , 2]]; Select[Range[2, 300], sig[#] == sig[next[#]] &]

A360529 a(n) is the smallest k > A024619(n) such that rad(k) = rad(A024619(n)), where rad(n) = A007947(n).

Original entry on oeis.org

12, 20, 18, 28, 45, 24, 40, 63, 44, 36, 52, 56, 60, 99, 68, 175, 48, 76, 117, 50, 84, 88, 75, 92, 54, 80, 153, 104, 72, 275, 98, 171, 116, 90, 124, 147, 325, 132, 136, 207, 140, 96, 148, 135, 152, 539, 156, 100, 164, 126, 425, 172, 261, 176, 120, 637, 184, 279, 188, 475, 108, 112, 297, 160, 204, 208

Offset: 1

Michael De Vlieger, May 01 2023

nonn

Permutation of A126706.

Let m = A024619(n) and let R_m be the sequence of numbers k such that rad(k) = rad(m). a(n) gives the successor to m in R_m.

			A024619(1) = 6; the smallest k > 6 such that rad(k) = 6 is a(1) = 12.
A024619(2) = 10; the smallest k > 10 such that rad(k) = 10 is a(2) = 20.
A024619(3) = 12; the smallest k > 12 such that rad(k) = rad(12) = 6 is a(3) = 18.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16.

Cf. A001597, A007947, A020639, A024619, A065642, A360719.

rad[x_] := rad[x] = Times@ FactorInteger[x][[All, 1]]; Table[k = m + 1; Function[r, If[SquareFreeQ[m], m*FactorInteger[m][[1, 1]],  While[rad[k] != r, k++]; k]][rad[m]], {m, Select[Range[2, 104], ! PrimePowerQ[#] &]}]

a(n) = A065642 \ A001597.

Squarefree m implies a(n) = lpf(m)*m = A020639(m)*m.

A366807 a(n) = A020639(A120944(n))*A120944(n).

Original entry on oeis.org

12, 20, 28, 45, 63, 44, 52, 60, 99, 68, 175, 76, 117, 84, 92, 153, 275, 171, 116, 124, 325, 132, 207, 140, 148, 539, 156, 164, 425, 172, 261, 637, 279, 188, 475, 204, 315, 212, 220, 333, 228, 575, 236, 833, 244, 369, 387, 260, 931, 268, 276, 423, 284, 1573, 725

Offset: 1

Michael De Vlieger, Dec 16 2023

Define f(x) to be lpf(k)*k, where lpf(k) = A020639(k). This sequence contains the mappings of f(x) across composite squarefree numbers A120944.
Define sequence R_k = { m : rad(m) | k }, where rad(n) = A007947(n) and k is squarefree. Then the sequence k*R_k contains all numbers divisible by squarefree k that are also not divisible by any prime q coprime to k. It is plain to see that k is the first term in the sequence k*R_k. This sequence gives the second term in k*R_k since lpf(k) is the second term in R_k.
Permutation of A366825. Contains numbers whose prime signature has at least 2 terms, of which is 2, the rest of which are 1s.
Proper subset of A364996, which itself is contained in A126706.

			Let b(n) = A120944(n).
a(1) = 12 = 2^2*3^1 = b(1)*lpf(b(1)) = 6*lpf(6) = 6*2. In {6*A003586}, 12 is the second term.
a(2) = 20 = 2^2*5^1 = b(2)*lpf(b(2)) = 10*lpf(10) = 10*2. In {10*A003592}, 20 is the second term.
a(4) = 45 = 3^2*5^1 = b(4)*lpf(b(4)) = 15*lpf(15) = 15*3. In {15*A003593}, 45 is the second term, etc.

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

Cf. A000040, A007947, A020639, A065642, A120944, A126706, A285109, A364996, A366825.

nn = 150; s = Select[Range[nn], And[SquareFreeQ[#], CompositeQ[#]] &];
Array[#*FactorInteger[#][[1, 1]] &[s[[#]]] &, Length[s]]

from math import isqrt
from sympy import primepi, mobius, primefactors
def A366807(n):
    def f(x): return n+1+primepi(x)+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n+1, f(n+1)
    while m != k:
        m, k = k, f(k)
    return m*min(primefactors(m)) # Chai Wah Wu, Aug 02 2024

a(n) = A065642(A120944(n)), n > 1.

a(n) = A285109(A120944(n)).

A357126 a(n) is the smallest positive integer k such that k > n and A071364(k) = A071364(n).

Original entry on oeis.org

3, 5, 9, 7, 10, 11, 27, 25, 14, 13, 20, 17, 15, 21, 81, 19, 50, 23, 28, 22, 26, 29, 40, 49, 33, 125, 44, 31, 42, 37, 243, 34, 35, 38, 100, 41, 39, 46, 56, 43, 66, 47, 45, 52, 51, 53, 80, 121, 75, 55, 63, 59, 250, 57, 88, 58, 62, 61, 84, 67, 65, 68, 729, 69, 70, 71, 76, 74, 78, 73, 200, 79, 77, 98

Offset: 2

Gleb Ivanov, Oct 26 2022

nonn

			a(12) = 20 as 12 has (2, 1) sequence of exponents in canonical prime factorization via 12 = 2^2 * 3^1 and the smallest positive integer > 12 with the same sequence of exponents in canonical prime factorization being (2, 1) is 20 as 20 = 2^2 * 5^1. - _David A. Corneth_, Oct 26 2022

Cf. A000961, A003961, A071364, A065642, A081382, A081761.

f4(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = prime(i)); factorback(f); \\ A071364
a(n) = my(k=n+1, f=f4(n)); while (f4(k) != f, k++); k; \\ Michel Marcus, Oct 26 2022

first(n) = { my(res = vector(n + 1), todo = n, m = Map(), u = precprime(n)); for(e = 2, logint(n, 2), u = max(u, nextprime(sqrtnint(n, e) + 2)^e) ); forfactored(i = 2, u, cs = i[2][,2]; if(mapisdefined(m, cs), ci = mapget(m, cs); if(ci <= n + 1, res[ci] = i[1]; mapput(m, cs, i[1]); todo--; if(todo <= 0, res = res[^1]; return(res) ) ) , if(i[1] <= n + 1, mapput(m, cs, i[1]) ) ) ) } \\ David A. Corneth, Oct 26 2022

from sympy import factorint
to_s_exp = lambda n: tuple(i[1] for i in sorted(factorint(n).items()))
terms = []
for i in range(2, 100):
    k = i+1;t = to_s_exp(i)
    while t != to_s_exp(k):k+=1
    terms.append(k)
print(terms)

a(A000961(k)) = a(A003961(A000961(k))) for k > 1. - David A. Corneth, Oct 26 2022

a(n) >= A081761(n). - Rémy Sigrist, Feb 16 2023

A362432 a(n) is the smallest k > A126706(n) such that rad(k) = rad(A126706(n)) and k mod n != 0, where rad(n) = A007947(n).

Original entry on oeis.org

18, 24, 50, 36, 98, 48, 50, 242, 75, 54, 80, 338, 72, 98, 90, 147, 578, 96, 135, 722, 100, 126, 242, 120, 1058, 108, 112, 363, 160, 338, 144, 196, 1682, 507, 150, 1922, 168, 198, 225, 578, 350, 162, 189, 2738, 180, 722, 867, 234, 200, 192, 3362, 252, 1083, 3698, 245, 242, 240, 1058, 4418, 441, 216, 224

Offset: 1

Michael De Vlieger, May 19 2023

nonn

Let m = A126706(n) and r = rad(m).
Smallest number k greater than m that shares the same squarefree kernel as m, yet does not divide m.
a(n) is in A126706, not a permutation of A126706.
k/r and m/r are coprime.
a(n) < m^2, since k/m < r.

			A126706(1) = 12; the smallest k > 12 such that both rad(k) = rad(12) = 6 and 12 does not divide k is a(1) = 18.
A126706(2) = 18; the smallest k > 18 such that both rad(k) = rad(18) = 6 and 18 does not divide k is a(2) = 24.
A126706(3) = 20; the smallest k > 20 such that rad(k) = rad(20) = 10, indivisible by 20, is a(3) = 50.
A126706(7) = 40; the smallest k > 40 such that rad(k) = rad(40) = 10, indivisible by 40, is a(7) = 50.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^12, showing records in red.

Cf. A007947, A065642, A126706.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[k = m + 1; Function[r, While[Nand[rad[k] == r, ! Divisible[k, m]], k++]][rad[m]]; k, {m, Select[Range[196], Nor[PrimePowerQ[#], SquareFreeQ[#]] &]}]

A365656 Array T(n,k) read by antidiagonals (downward): T(n,1) = A005117(n) (squarefree numbers > 1); for k > 1, columns are nonsquarefree numbers (in descending order) with exactly the same prime factors as T(n,1).

Original entry on oeis.org

1, 2, 4, 3, 8, 9, 5, 16, 27, 25, 6, 32, 81, 125, 12, 7, 64, 243, 625, 18, 49, 10, 128, 729, 3125, 24, 343, 20, 11, 256, 2187, 15625, 36, 2401, 40, 121, 13, 512, 6561, 78125, 48, 16807, 50, 1331, 169, 14, 1024, 19683, 390625, 54, 117649, 80, 14641, 2197, 28, 15

Offset: 0

Michael De Vlieger, Nov 17 2023

Permutation of natural numbers.
Transpose of A284311 with a(0) = 1 prepended.
Essentially the same as A284457. - R. J. Mathar, Jan 23 2024

			Table T(n,k) for n = 1..12 and k = 1..6 shown below:
  n\k |  1    2     3       4        5         6 ...
  ----------------------------------------------
   1  |  1
   2  |  2    4     8      16       32        64
   3  |  3    9    27      81      243       729
   4  |  5   25   125     625     3125     15625
   5  |  6   12    18      24       36        48
   6  |  7   49   343    2401    16807    117649
   7  | 10   20    40      50       80       100
   8  | 11  121  1331   14641   161051   1771561
   9  | 13  169  2197   28561   371293   4826809
  10  | 14   28    56      98      112       196
  11  | 15   45    75     135      225       375
  12  | 17  289  4913   83521  1419857  24137569
  ...
Triangle begins:
   1;
   2;
   4,   3;
   8,   9,   5;
  16,  27,  25,  6;
  32,  81, 125, 12,  7;
  64, 243, 625, 18, 49, 10;
 ...

Michael De Vlieger, Table of n, a(n) for n = 0..11326 (rows 0..150, flattened)
Michael De Vlieger, Log log scatterplot of log_10(a(n)), n = 1..64981 (i.e., 360 rows, flattened).
Michael De Vlieger, Log log scatterplot of log_10(a(n)), n = 1..4096, showing primes in red, squarefree composites in green, composite prime powers in gold, and numbers neither squarefree nor prime powers in blue and purple; we show squareful numbers that are not prime powers in purple.

Cf. A002260, A005117, A007947, A065642, A284311, A284457.

f[n_, k_ : 1] := Block[{c = 0, s = Sign[k], m}, m = n + s;
    While[c < Abs[k], While[! SquareFreeQ@ m, If[s < 0, m--, m++]];
     If[s < 0, m--, m++]; c++];
    m + If[s < 0, 1, -1] ] (* after Robert G.Wilson v at A005117 *);
  T[n_, k_] := T[n, k] =
    Which[And[n == 1, k == 1], 2, k == 1, f@T[n - 1, k],
     PrimeQ@ T[n, 1], T[n, 1]^k, True,
     Module[{j = T[n, k - 1]/T[n, 1] + 1},
      While[PowerMod[T[n, 1], j, j] != 0, j++]; j T[n, 1]]]; {1}~Join~
   Table[T[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // TableForm

For prime n = p, T(p,k) = p^k.

cp's OEIS Frontend

A285322 Transpose of square array A285321.

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A366460 Odd terms in A366825.

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A079228 Least number > n with greater squarefree kernel than that of n.

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A286544 Restricted growth sequence of A285333.

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A340305 Numbers k such that k and the least number that is larger than k and has the same set of distinct prime divisors as k also has the same prime signature as k.

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A360529 a(n) is the smallest k > A024619(n) such that rad(k) = rad(A024619(n)), where rad(n) = A007947(n).

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A366807 a(n) = A020639(A120944(n))*A120944(n).

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A357126 a(n) is the smallest positive integer k such that k > n and A071364(k) = A071364(n).

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