A000040 -id:A000040 - cp's OEIS frontend

A246372 Numbers n such that 2n-1 = product_{k >= 1} (p_k)^(c_k), then n <= product_{k >= 1} (p_{k-1})^(c_k), where p_k indicates the k-th prime, A000040(k).

1, 2, 3, 4, 6, 7, 9, 10, 12, 15, 16, 19, 20, 21, 22, 24, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 40, 42, 44, 45, 46, 47, 48, 49, 51, 52, 54, 55, 56, 57, 60, 62, 64, 65, 66, 67, 69, 70, 71, 72, 75, 76, 78, 79, 80, 81, 82, 84, 85, 87, 89, 90, 91, 92, 93, 96, 97, 99, 100, 101, 102, 103, 105, 106, 107, 108, 109, 110

Offset: 1

Antti Karttunen, Aug 24 2014

nonn

Numbers n such that A064216(n) >= n.

Numbers n such that A064989(2n-1) >= n.

			1 is present, as 2*1 - 1 = empty product = 1.
2 is present, as 2*2 - 1 = 3 = p_2, and p_{2-1} = p_1 = 2 >= 2.
3 is present, as 2*3 - 1 = 5 = p_3, and p_{3-1} = p_2 = 3 >= 3.
5 is not present, as 2*5 - 1 = 9 = p_2 * p_2, and p_1 * p_1 = 4, with 4 < 5.
6 is present, as 2*6 - 1 = 11 = p_5, and p_{5-1} = p_4 = 7 >= 6.
25 is present, as 2*25 - 1 = 49 = p_4^2, and p_3^2 = 5*5 = 25 >= 25.
35 is present, as 2*35 - 1 = 69 = 3*23 = p_2 * p_9, and p_1 * p_8 = 2*19 = 38 >= 35.

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

Complement: A246371
Union of A246362 and A048674.
Subsequences: A006254 (A111333), A246373 (the primes present in this sequence).
Cf. A000040, A064216, A064989, A246352.

default(primelimit, 2^30);
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A064216(n) = A064989((2*n)-1);
isA246372(n) = (A064216(n) >= n);
n = 0; i = 0; while(i < 10000, n++; if(isA246372(n), i++; write("b246372.txt", i, " ", n)));
(Scheme, with Antti Karttunen's IntSeq-library)
(define A246372 (MATCHING-POS 1 1 (lambda (n) (>= (A064216 n) n))))

A056221 Image of primes (A000040) under "little Hankel" transform that sends [c_0, c_1, ...] to [d_0, d_1, ...] where d_n = c_n^2 - c_{n+1}*c_{n-1}.

Original entry on oeis.org

-1, 4, -6, 30, -18, 42, -30, -22, 128, -112, 98, 90, -78, -70, 36, 248, -232, 158, 150, -280, 182, -142, -130, 420, 210, -198, 222, -210, -1074, 1326, -238, 560, -1092, 1212, -592, 36, 350, -310, 36, 728, -1428, 1548, -378, 402, -1966, 144, 1832, 462, -450, -442

Offset: 1

N. J. A. Sloane, Aug 06 2000

sign

a(n) > 0 if and only if n+1 is in A046868. a(n) < 0 if and only if n+1 is in A233671. - Chai Wah Wu, Sep 10 2019

Stefano Spezia, Table of n, a(n) for n = 1..10000
L. Panaitopol, On the sequence p(n)^2=p(n-1)*p(n+1), J. Inequal. Pure Appl. Math., Volume 3, Issue 4, Article 53, 2002.

Cf. A000040, A046868, A056222, A233671, A342567.

A056221 := proc(n)
        ithprime(n+1)^2-ithprime(n)*ithprime(n+2) ;
end proc:
seq(A056221(n),n=1..10) ; # R. J. Mathar, Dec 10 2011

a[n_]:=Prime[n+1]^2-Prime[n]Prime[n+2]; Array[a,50] (* Stefano Spezia, Jul 15 2024 *)

a(n) = determinant of matrix
| prime(n+1) prime(n)|
| prime(n+2) prime(n+1)|. - Zak Seidov, Jul 23 2008, indices corrected by Gary Detlefs, Dec 09 2011
a(n) = 2*A342567(n+1) for n >= 2. - Hugo Pfoertner, Jun 20 2021

A124270 a(n) = prime(A014612(n)) - A014612(prime(n)). Commutator [A000040,A014612] at n.

Original entry on oeis.org

7, 19, 34, 41, 53, 44, 38, 103, 91, 73, 99, 75, 135, 142, 147, 118, 133, 125, 118, 193, 229, 191, 212, 202, 197, 201, 216, 213, 248, 239, 209, 248, 279, 279, 277, 277, 333, 325, 350, 327, 299, 308, 264, 309, 314, 322, 297, 281, 363, 374, 461, 488, 484, 482

Offset: 1

Jonathan Vos Post, Oct 23 2006

			a(1) = prime(3almostprime(1)) - 3almostprime(prime(1)) = prime(8) - 3almostprime(2) = 19 - 12 = 7.
a(2) = prime(3almostprime(2)) - 3almostprime(prime(2)) = prime(12) - 3almostprime(3) = 37 - 18 = 19.
a(3) = prime(3almostprime(3)) - 3almostprime(prime(3)) = prime(18) - 3almostprime(5) = 61 - 27 = 34.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000040 (primes), A014612 (3-almost primes).
Cf. A124268 (prime(3-almost prime(n))), A124269 (3-almost prime(prime(n))).
Cf. A106349 (prime(semiprime(n))), A106350 (semiprime(prime(n))), A122824 (prime(semiprime(n)) - semiprime(prime(n))).

lista(nn) = {p = primes(nn); pp = select(x->bigomega(x)==3, vector(nn, n, n)); for (n=1, nn, print1(p[pp[n]] - pp[p[n]], ", "););} \\ Michel Marcus, Oct 15 2014

a(n) = A000040(A014612(n)) - A014612(A000040(n)).

a(n) = A124268(n) - A124269(n).

A246362 Numbers n such that if 2n-1 = Product_{k >= 1} (p_k)^(c_k), then n < Product_{k >= 1} (p_{k-1})^(c_k), where p_k indicates the k-th prime, A000040(k).

Original entry on oeis.org

4, 6, 7, 9, 10, 12, 15, 16, 19, 20, 21, 22, 24, 27, 29, 30, 31, 34, 35, 36, 37, 40, 42, 44, 45, 46, 47, 48, 49, 51, 52, 54, 55, 56, 57, 60, 62, 64, 65, 66, 67, 69, 70, 71, 72, 75, 76, 78, 79, 80, 81, 82, 84, 85, 87, 89, 90, 91, 92, 96, 97, 99, 100, 101, 102, 103, 105, 106, 107, 108, 109, 110, 111, 112, 114, 115

Offset: 1

Antti Karttunen, Aug 24 2014

nonn

Numbers n such that A064216(n) > n.
Numbers n such that A064989(2n-1) > n.
The sequence grows as:
      a(100) = 148
     a(1000) = 1449
    a(10000) = 14264
   a(100000) = 141259
  a(1000000) = 1418197
and the powers of 10 occur at:
        a(5) = 10
       a(63) = 100
      a(701) = 1000
     a(6973) = 10000
    a(70845) = 100000
   a(705313) = 1000000
suggesting that the ratio a(n)/n is converging to a constant and an arbitrary natural number is more than twice as likely to be here than in the complement A246361. Compare this to the ratio present in the "inverse" case A246282.

			4 is present, as 2*4 - 1 = 7 = p_4, and p_{4-1} = p_3 = 5 > 4.
5 is not present, as 2*5 - 1 = 9 = p_2 * p_2, and p_1 * p_1 = 4, with 4 < 5.
6 is present, as 2*6 - 1 = 11 = p_5, and p_{5-1} = p_4 = 7 > 6.
35 is present, as 2*35 - 1 = 69 = 3*23 = p_2 * p_9, and p_1 * p_8 = 2*19 = 38 > 35.

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

Complement: A246361.
Setwise difference of A246372 and A048674.
Cf. A000040, A064216, A064989, A246282.

default(primelimit, 2^30);
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A064216(n) = A064989((2*n)-1);
isA246362(n) = (A064216(n) > n);
n = 0; i = 0; while(i < 10000, n++; if(isA246362(n), i++; write("b246362.txt", i, " ", n)));

;; With Antti Karttunen's IntSeq-library.
(define A246362 (MATCHING-POS 1 1 (lambda (n) (> (A064216 n) n))))

A065858 m-th composite number c(m) = A002808(m), where m is the n-th prime number: a(n) = A002808(A000040(n)).

Original entry on oeis.org

6, 8, 10, 14, 20, 22, 27, 30, 35, 44, 46, 54, 58, 62, 66, 75, 82, 85, 92, 96, 99, 108, 114, 120, 129, 134, 136, 142, 144, 148, 166, 171, 178, 182, 194, 196, 204, 210, 215, 221, 230, 232, 245, 247, 252, 254, 268, 285, 289, 291, 296, 302, 304, 318, 324, 330, 338

Offset: 1

Labos Elemer, Nov 26 2001

nonn

Composites (A002808) with prime (A000040) subscripts. a(n) U A175251(n) = A002808(n). Subsequence of A022449 (composites (A002808) with noncomposite (A008578) subscripts), a(n) = A022449(n+1). - Jaroslav Krizek, Mar 14 2010

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002808, A000040, A008578, A022449, A065856, A065857, A175251.

P,C:= selectremove(isprime,[seq(i,i=2..10^3)]):
seq(C[P[i]],i=1..100); # Robert Israel, Mar 09 2025

Composite[n_] := FixedPoint[n + PrimePi[#] + 1 & , n + PrimePi[n] + 1];
a[n_] := Composite[Prime[n]];
Array[a, 100] (* Jean-François Alcover, Jan 26 2018, after Robert G. Wilson v *)

A332461 a(n) = Product_{d|n, d>1} A000040(A297113(d)), where A000040(n) gives the n-th prime, and A297113(n) = the excess of n plus the index of the largest dividing prime (A046660 + A061395).

Original entry on oeis.org

1, 2, 3, 6, 5, 18, 7, 30, 15, 50, 11, 270, 13, 98, 75, 210, 17, 450, 19, 1050, 147, 242, 23, 9450, 35, 338, 105, 3234, 29, 11250, 31, 2310, 363, 578, 245, 47250, 37, 722, 507, 57750, 41, 43218, 43, 9438, 2625, 1058, 47, 727650, 77, 2450, 867, 17238, 53, 22050, 605, 210210, 1083, 1682, 59, 8268750, 61, 1922, 8085, 30030

Offset: 1

Antti Karttunen, Feb 22 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A002110, A046660, A046523, A048675, A061395, A097248, A156552, A297113, A324202, A332462.

A297113(n) = if(1==n, 0, (primepi(vecmax(factor(n)[, 1])) + (bigomega(n)-omega(n))));
A332461(n) = if(1==n,1, my(m=1); fordiv(n,d,if(d>1, m *= prime(A297113(d)))); (m));

a(n) = Product_{d|n, d>1} A000040(A297113(d)).
a(p) = p for all primes p.
For all n >= 0, a(2^n) = A002110(n).
For all n >= 1:
A046523(a(n)) = A324202(n).
A048675(a(n)) = A156552(n).
A097248(a(n)) = A332462(n).

A182986 Zero together with the prime numbers (A000040).

Original entry on oeis.org

0, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269

Offset: 1

Charles R Greathouse IV, Feb 05 2011

These numbers are the possible characteristics of a field.
First differences are in A054541. - Omar E. Pol, Oct 31 2013
Also A158611 without its second term. - Omar E. Pol, Nov 01 2013
The ideals generated by a(n) form Spec(Z), the set of prime ideals of the ring of integers. Due to its importance in algebraic geometry, algebraic geometers often consider 0 to be an honorary prime. - Keith J. Bauer, Jan 09 2024

Cf. A141468.

Complement of A018252. - Arkadiusz Wesolowski, Sep 15 2011

Prepend[Table[Prime[n], {n, 60}], 0] (* Arkadiusz Wesolowski, Sep 15 2011 *)
NestList[ NextPrime, 0, 57] (* Robert G. Wilson v, Jul 21 2015 *)

a(n)=if(n>1,prime(n-1),0) \\ Charles R Greathouse IV, Jul 15 2013

A251720 a(n) = (p_n)^2 * p_{n+1}, where p_n is the n-th prime, A000040(n).

Original entry on oeis.org

12, 45, 175, 539, 1573, 2873, 5491, 8303, 15341, 26071, 35557, 56129, 72283, 86903, 117077, 165731, 212341, 249307, 318719, 367993, 420991, 518003, 613121, 768337, 950309, 1050703, 1135163, 1247941, 1342553, 1621663, 2112899, 2351057, 2608891, 2878829, 3352351

Offset: 1

Antti Karttunen, Dec 14 2014

nonn

Subsequence of A014612: a(1)=12=A014612(2), a(2)=45=A014612(10) - Zak Seidov, Apr 26 2016

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

Cf. A000040, A001248, A006094, A250477, A014612, A268733.

a251720[n_Integer] := Prime[#]^2*Prime[# + 1] & /@ Range[n]; a251720[35] (* Michael De Vlieger, Dec 14 2014 *)
#[[1]]^2 #[[2]]&/@Partition[Prime[Range[40]],2,1] (* Harvey P. Dale, Mar 12 2015 *)

a(n) = A000040(n) * A000040(n) * A000040(n+1).
a(n) = A000040(n) * A006094(n).
a(n) = A001248(n) * A000040(n+1).

A357983 Second MTF-transform of the primes (A000040). Replace prime(k) with prime(A064988(k)) in the prime factorization of n.

Original entry on oeis.org

1, 2, 5, 4, 11, 10, 23, 8, 25, 22, 31, 20, 47, 46, 55, 16, 59, 50, 103, 44, 115, 62, 97, 40, 121, 94, 125, 92, 137, 110, 127, 32, 155, 118, 253, 100, 197, 206, 235, 88, 179, 230, 233, 124, 275, 194, 257, 80, 529, 242, 295, 188, 419, 250, 341, 184, 515, 274

Offset: 1

Gus Wiseman, Oct 24 2022

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. We define the MTF-transform as shifting a number's prime indices along a function; see the Mathematica program.

			First, we have
- 4 = prime(1) * prime(1),
- A000040(1) = 2,
- A064988(4) = prime(2) * prime(2) = 9.
Similarly, A064988(3) = 5. Next,
- 35 = prime(3) * prime(4),
- A064988(3) = 5,
- A064988(4) = 9,
- a(35) = prime(5) * prime(9) = 253.

Other multiplicative sequences: A003961, A357852, A064989, A357977, A357980.
Applying the transformation only once gives A064988.
The union is A076610 (numbers whose prime indices are themselves prime).
For partition numbers instead of primes we have A357979.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000720, A003964, A063834, A215366, A296150, A299201, A299202, A357975.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mtf[f_][n_]:=Product[If[f[i]==0,1,Prime[f[i]]],{i,primeMS[n]}];
Array[mtf[mtf[Prime]],100]

A070748 a(n) = signum(sin(prime(n))), where signum=A057427, prime=A000040.

Original entry on oeis.org

1, 1, -1, 1, -1, 1, -1, 1, -1, -1, -1, -1, -1, -1, 1, 1, 1, -1, -1, 1, -1, -1, 1, 1, 1, 1, 1, 1, 1, -1, 1, -1, -1, 1, -1, 1, -1, -1, -1, -1, 1, -1, 1, -1, 1, -1, -1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, -1, -1, 1, -1, 1, 1, 1

Offset: 1

Reinhard Zumkeller, May 04 2002

sign

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

Cf. A070747, A070750, A070749, A070753, A070754.

Sign[Sin[Prime[Range[10000]]]] (* Andrey Mitin, Apr 22 2022 *)

a(n)=(-1)^(prime(n)/Pi%2\1) \\ Charles R Greathouse IV, Jun 13 2013

cp's OEIS Frontend

A246372 Numbers n such that 2n-1 = product_{k >= 1} (p_k)^(c_k), then n <= product_{k >= 1} (p_{k-1})^(c_k), where p_k indicates the k-th prime, A000040(k).

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A056221 Image of primes (A000040) under "little Hankel" transform that sends [c_0, c_1, ...] to [d_0, d_1, ...] where d_n = c_n^2 - c_{n+1}*c_{n-1}.

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Formula

A124270 a(n) = prime(A014612(n)) - A014612(prime(n)). Commutator [A000040,A014612] at n.

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A246362 Numbers n such that if 2n-1 = Product_{k >= 1} (p_k)^(c_k), then n < Product_{k >= 1} (p_{k-1})^(c_k), where p_k indicates the k-th prime, A000040(k).

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A065858 m-th composite number c(m) = A002808(m), where m is the n-th prime number: a(n) = A002808(A000040(n)).

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A332461 a(n) = Product_{d|n, d>1} A000040(A297113(d)), where A000040(n) gives the n-th prime, and A297113(n) = the excess of n plus the index of the largest dividing prime (A046660 + A061395).

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Formula

A182986 Zero together with the prime numbers (A000040).

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A251720 a(n) = (p_n)^2 * p_{n+1}, where p_n is the n-th prime, A000040(n).

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