A001747 -id:A001747 - cp's OEIS frontend

A077646 Solutions to lcm(n, tau(n)) = 2n which are neither prime nor twice a prime, where tau(k) = d(k) is the number of divisors of k.

45, 63, 75, 90, 99, 117, 120, 126, 147, 150, 153, 168, 171, 198, 207, 216, 234, 243, 261, 264, 279, 280, 294, 306, 312, 333, 342, 363, 369, 387, 405, 408, 414, 420, 423, 440, 456, 477, 486, 507, 520, 522, 531, 540, 549, 552, 558, 603, 616, 639, 657, 660

Offset: 1

Labos Elemer, Nov 18 2002

nonn

			n=63: d(n) = tau(n) = 6, lcm(63,6) = 126 = 2n.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A009530, A001747, A000040, A065091.

is(n)=!isprime(if(n%2,n,n/2)) && lcm(n,numdiv(n))==2*n \\ Charles R Greathouse IV, Oct 28 2013

Composite solutions to A009530(x)=2x which are not twice a prime.

A227534 Even numbers n such that the least e with n^e a totient is a new record.

Original entry on oeis.org

2, 22, 34, 62, 86, 202, 398, 2042, 6998, 12514, 12758, 33406, 48962, 101554, 154186, 197378, 298366

Offset: 1

Charles R Greathouse IV, Jul 14 2013

Essentially position of records in A227533.

Probably all terms beyond the first are even semiprimes: conjecturally this is a subsequence of A001747.

Cf. A227533, A227535.

r=0;forstep(n=2,1e5,2, t=1; while(!istotient(n^t++),); if(t>r,r=t;print1(n", ")))
\\ See also A227533 for a more efficient method of computing terms.

a(14) from Charles R Greathouse IV, Jul 16 2013
a(15) from Charles R Greathouse IV, Jul 17 2013
a(16)-a(17) from Charles R Greathouse IV, Jul 19 2013

A338356 Indices of records in A283312.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 13, 16, 20, 23, 27, 33, 36, 42, 46, 49, 53, 60, 66, 69, 75, 80, 83, 89, 93, 99, 107, 112, 115, 119, 122, 127, 140, 145, 151, 154, 163, 166, 173, 179, 183, 190, 196, 199, 210, 213, 217, 220, 231, 242, 246, 249, 254, 261, 264, 275, 281, 287, 294, 297, 303, 307, 310, 321

Offset: 1

N. J. A. Sloane, Nov 03 2020

nonn

The records themselves are essentially twice the primes (A001747).

N. J. A. Sloane, Table of n, a(n) for n = 1..7100

Cf. A001747, A283312.

A338360 Indices of records in A280985.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 14, 16, 21, 23, 28, 35, 37, 43, 47, 49, 54, 62, 67, 69, 76, 81, 83, 90, 95, 100, 109, 113, 115, 120, 122, 128, 142, 147, 152, 155, 166, 168, 174, 180, 185, 192, 199, 201, 212, 214, 219, 221, 234, 245, 247, 250, 257, 262, 264, 277, 284, 290, 295, 297, 304, 308, 310, 325

Offset: 1

N. J. A. Sloane, Nov 03 2020

nonn

The records themselves are essentially twice the primes (A001747).

N. J. A. Sloane, Table of n, a(n) for n = 1..7098

Cf. A001747, A280985.

A070309 Number of solutions 2<=x<=A060679(n) to the equation x^A060679(n)==1 (mod A060679(n)) where A060679(n) are the orders of non-cyclic groups.

Original entry on oeis.org

1, 1, 3, 2, 1, 3, 1, 7, 5, 7, 2, 1, 7, 4, 1, 8, 3, 3, 15, 1, 11, 1, 2, 15, 11, 3, 2, 1, 15, 6, 9, 7, 17, 4, 7, 2, 1, 15, 1, 8, 31, 3, 7, 3, 23, 1, 4, 3, 11, 31, 26, 1, 23, 1, 7, 11, 3, 2, 1, 31, 13, 2, 39, 3, 15, 2, 1, 35, 19, 2, 15, 11, 7, 8, 1, 31, 10, 1, 3, 24, 35, 63, 2, 3, 7, 1, 8, 31, 3

Offset: 1

Benoit Cloitre, May 10 2002

If there is only one solution 2<=x<=A060679(k) to x^A060679(k)==1 (mod A060679(k)) this solution is : x=A060679(k)-1 (also solution is A060679(k)+1). In this case A060679(k) is a term of A001747(n).

Cf. A060679, A001747, A003277.

for(n=1,200,if(prod(i=2,n-1,(i^n-1)%n)==0,print1(sum(i=2,n-1,if((i^n-1)%n,0,1)),",")))

A075813 Palindromic even numbers with exactly 2 prime factors (counted with multiplicity). Equivalently, palindromic numbers of the form 2*p with p prime.

Original entry on oeis.org

4, 6, 22, 202, 262, 454, 626, 818, 838, 878, 898, 20302, 20602, 22322, 22522, 22622, 22822, 24142, 24842, 26662, 26762, 28682, 41014, 41414, 41614, 41714, 43034, 43234, 43534, 43634, 45454, 45554, 45754, 47074, 47374, 47774, 49094, 49394

Offset: 1

Jani Melik, Oct 13 2002

			4=2^2, 6=2*3 and 22=2*11 are palindromic, even and have exactly 2 prime factors.

Cf. A001747.

Even subsequence of A046328.

test := proc(n) local d; d := convert(n,base,10); return ListTools[Reverse](d)=d and numtheory[bigomega](n)=2; end; a := []; for n from 2 to 50000 by 2 do if test(n) then a := [op(a),n]; end; od; a;

Select[Range[50000],EvenQ[#]&&PalindromeQ[#]&&PrimeOmega[#]==2&] (* Harvey P. Dale, Oct 16 2024 *)

Edited by Dean Hickerson, Oct 21 2002

A282671 Twice composite numbers.

Original entry on oeis.org

8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 48, 50, 52, 54, 56, 60, 64, 66, 68, 70, 72, 76, 78, 80, 84, 88, 90, 92, 96, 98, 100, 102, 104, 108, 110, 112, 114, 116, 120, 124, 126, 128, 130, 132, 136, 138, 140, 144, 148, 150, 152, 154, 156, 160, 162, 164, 168, 170, 172, 174

Offset: 1

Alessandro Polcini, Feb 20 2017

Even numbers greater than 2 that do not appear in A001747.

Cf. A001747, A139270, A176100.

is(n)=n%2==0 && n>6 && !isprime(n/2) \\ Charles R Greathouse IV, Feb 21 2017

a(n) = 2*A002808(n). - R. J. Mathar, Feb 23 2017

a(n) = A139270(n+1). - R. J. Mathar, Feb 25 2017

A347047 Smallest squarefree semiprime whose prime indices sum to n.

Original entry on oeis.org

6, 10, 14, 21, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514, 526

Offset: 3

Gus Wiseman, Aug 22 2021

nonn

Compared to A001747, we have 21 instead of 22 and lack 2 and 4.
Compared to A100484 (shifted) we have 21 instead of 22 and lack 4.
Compared to A161344, we have 21 instead of 22 and lack 4 and 8.
Compared to A339114, we have 11 instead of 9 and lack 4.
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.
A squarefree semiprime (A006881) is a product of any two distinct prime numbers.

			The initial terms and their prime indices:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   21: {2,4}
   26: {1,6}
   34: {1,7}
   38: {1,8}
   46: {1,9}

The opposite version (greatest instead of smallest) is A332765.
These are the minima of rows of A338905.
The nonsquarefree version is A339114 (opposite: A339115).
A001358 lists semiprimes (squarefree: A006881).
A024697 adds up semiprimes by weight (squarefree: A025129).
A056239 adds up prime indices, row sums of A112798.
A246868 gives the greatest squarefree number whose prime indices sum to n.
A320655 counts factorizations into semiprimes (squarefree: A320656).
A338898, A338912, A338913 give the prime indices of semiprimes.
A338899, A270650, A270652 give the prime indices of squarefree semiprimes.
A339116 groups squarefree semiprimes by greater factor, sums A339194.
A339362 adds up prime indices of squarefree semiprimes.
Cf. A001221, A087112, A089994, A098350, A176504, A338900, A338901, A338904, A338907/A338908, A339005, A339191.

Table[Min@@Select[Table[Times@@Prime/@y,{y,IntegerPartitions[n,{2}]}],SquareFreeQ],{n,3,50}]

from sympy import prime, sieve
def a(n):
    p = [0] + list(sieve.primerange(1, prime(n)+1))
    return min(p[i]*p[n-i] for i in range(1, (n+1)//2))
print([a(n) for n in range(3, 58)]) # Michael S. Branicky, Sep 05 2021

A363473 Triangle read by rows: T(n, k) = k * prime(n - k + A061395(k)) for 1 < k <= n, and T(n, 1) = A008578(n).

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 5, 10, 15, 8, 7, 14, 21, 12, 25, 11, 22, 33, 20, 35, 18, 13, 26, 39, 28, 55, 30, 49, 17, 34, 51, 44, 65, 42, 77, 16, 19, 38, 57, 52, 85, 66, 91, 24, 27, 23, 46, 69, 68, 95, 78, 119, 40, 45, 50, 29, 58, 87, 76, 115, 102, 133, 56, 63, 70, 121, 31, 62, 93, 92, 145, 114, 161, 88, 99, 110, 143, 36

Offset: 1

Werner Schulte, Jan 05 2024

Conjecture: this is a permutation of the natural numbers.

Generalized conjecture: Let T(n, k) = b(k) * prime(n - k + A061395(b(k))) for 1 < k <= n, and T(n, 1) = A008578(n), where b(n), n > 0, is a permutation of the natural numbers with b(1) = 1, then T(n, k), read by rows, is a permutation of the natural numbers.

			Triangle begins:
n\k :   1    2    3    4    5    6    7    8    9   10   11   12   13
=====================================================================
 1  :   1
 2  :   2    4
 3  :   3    6    9
 4  :   5   10   15    8
 5  :   7   14   21   12   25
 6  :  11   22   33   20   35   18
 7  :  13   26   39   28   55   30   49
 8  :  17   34   51   44   65   42   77   16
 9  :  19   38   57   52   85   66   91   24   27
10  :  23   46   69   68   95   78  119   40   45   50
11  :  29   58   87   76  115  102  133   56   63   70  121
12  :  31   62   93   92  145  114  161   88   99  110  143   36
13  :  37   74  111  116  155  138  203  104  117  130  187   60  169
etc.

Cf. A000040, A001747, A001749, A001750, A008578, A112773, A138636, A253560, A272470, A061395.

T(n, k) = { if(k==1, if(n==1, 1, prime(n-1)), i=floor((k+1)/2);
            while(k % prime(i) != 0, i=i-1); k*prime(n-k+i)) }

def prime(n): return sloane.A000040(n)
def A061395(n): return prime_pi(factor(n)[-1][0]) if n > 1 else 0
def T(n, k):
     if k == 1: return prime(n - 1) if n > 1 else 1
     return k * prime(n - k + A061395(k))
for n in range(1, 11): print([T(n,k) for k in range(1, n+1)])
# Peter Luschny, Jan 07 2024

T(n, n) = A253560(n) for n > 0.
T(n, 1) = A008578(n) for n > 0.
T(n, 2) = A001747(n) for n > 1.
T(n, 3) = A112773(n) for n > 2.
T(n, 4) = A001749(n-3) for n > 3.
T(n, 5) = A001750(n-2) for n > 4.
T(n, 6) = A138636(n-4) for n > 5.
T(n, 7) = A272470(n-3) for n > 6.

cp's OEIS Frontend

A077646 Solutions to lcm(n, tau(n)) = 2n which are neither prime nor twice a prime, where tau(k) = d(k) is the number of divisors of k.

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A227534 Even numbers n such that the least e with n^e a totient is a new record.

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PARI

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A338356 Indices of records in A283312.

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A338360 Indices of records in A280985.

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A070309 Number of solutions 2<=x<=A060679(n) to the equation x^A060679(n)==1 (mod A060679(n)) where A060679(n) are the orders of non-cyclic groups.

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A075813 Palindromic even numbers with exactly 2 prime factors (counted with multiplicity). Equivalently, palindromic numbers of the form 2*p with p prime.

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Maple

Mathematica

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A282671 Twice composite numbers.

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A347047 Smallest squarefree semiprime whose prime indices sum to n.

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Mathematica

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A363473 Triangle read by rows: T(n, k) = k * prime(n - k + A061395(k)) for 1 < k <= n, and T(n, 1) = A008578(n).

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