A014342 -id:A014342 - cp's OEIS frontend

A299053 Minimum value of the cyclic autocorrelation of first n primes.

4, 12, 31, 62, 133, 224, 377, 558, 865, 1304, 1805, 2462, 3337, 4280, 5389, 6726, 8449, 10264, 12663, 15294, 18061, 21200, 24961, 29166, 34173, 39508, 45017, 50870, 57141, 63788, 72299, 81234, 91365, 101732, 113327, 125166, 138355, 152348, 167179, 182862

Offset: 1

Andres Cicuttin, Feb 01 2018

nonn

Maximum values of the cyclic autocorrelation of first n primes are in A024450.

If we use this definition with integers instead of primes it is obtained A088003.

			For n = 4 the four possible cyclic autocorrelations of first four primes are:
(2,3,5,7).(2,3,5,7) = 2*2 + 3*3 + 5*5 + 7*7 = 4 + 9 + 25 + 49 = 87,
(2,3,5,7).(7,2,3,5) = 2*7 + 3*2 + 5*3 + 7*5 = 14 + 6 + 15 + 35 = 70,
(2,3,5,7).(5,7,2,3) = 2*5 + 3*7 + 5*2 + 7*3 = 10 + 21 + 10 + 21 = 62,
(2,3,5,7).(3,5,7,2) = 2*3 + 3*5 + 5*7 + 7*2 = 6 + 15 + 35 + 14 = 70,
then a(4)=62 because 62 is the minimum among the four values.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A024450, A014342, A299111, A088003.

a:= n-> min(seq(add(ithprime(i)*ithprime(irem(i+k, n)+1), i=1..n), k=1..n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Feb 06 2018

p[n_]:=Prime[Range[n]];
Table[Table[p[n].RotateRight[p[n],j],{j,0,n-1}]//Min,{n,1,36}]

a(n) = vecmin(vector(n, k, sum(i=1, n, prime(i)*prime(1+(i+k)%n)))); \\ Michel Marcus, Feb 07 2018

a(n) = Min_{k=1..n} Sum_{i=1..n} prime(i)*prime(1 + (i+k) mod n).

A333370 Convolution of primorial numbers (A002110) with themselves.

Original entry on oeis.org

1, 4, 16, 84, 576, 5820, 72720, 1181460, 21984480, 493882620, 13996733520, 430612001820, 15742074348000, 641147559872820, 27488197348531920, 1286344285877911260, 67817877972050366160, 3984226025421591129180, 242703493548359285922480, 16211176424801583698573100

Offset: 0

Ilya Gutkovskiy, Mar 17 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..350
Eric Weisstein's World of Mathematics, Primorial
Index entries for sequences related to primorial numbers

Cf. A002110, A003149, A014342, A333371.

p:= proc(n) option remember; `if`(n<1, 1, ithprime(n)*p(n-1)) end:
a:= n-> add(p(i)*p(n-i), i=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 17 2020

primorial[n_] := Product[Prime[k], {k, 1, n}]; a[n_] := Sum[primorial[k] primorial[n - k], {k, 0, n}]; Table[a[n], {n, 0, 19}]

G.f.: (Sum_{k>=0} prime(k)# * x^k)^2, where prime()# = A002110.

a(n) = Sum_{k=0..n} prime(k)# * prime(n-k)#.

A337327 Maximum value of the cyclic self-convolution of the first n terms of the characteristic function of primes.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 4, 3, 3, 3, 4, 4, 6, 5, 4, 4, 6, 5, 8, 7, 6, 6, 8, 7, 8, 7, 6, 6, 8, 7, 10, 9, 8, 8, 8, 8, 10, 9, 8, 8, 10, 10, 12, 12, 10, 11, 12, 12, 12, 13, 12, 12, 14, 13, 14, 13, 12, 12, 12, 12, 14, 13, 12, 13, 12, 14, 14, 15, 12, 14, 14, 16, 16, 18

Offset: 1

Andres Cicuttin, Aug 23 2020

			The primes among the first 5 positive integers (1,2,3,4,5) are 2, 3, and 5, then the corresponding characteristic function of primes is (0,1,1,0,1) (see A010051) and the corresponding five possible cyclic self-convolutions are the dot products between (0,1,1,0,1) and the rotations of its mirrored version as shown below:
  (0,1,1,0,1).(1,0,1,1,0) = 0*1 + 1*0 + 1*1 + 0*1 + 1*0 = 1,
  (0,1,1,0,1).(0,1,0,1,1) = 0*0 + 1*1 + 1*0 + 0*1 + 1*1 = 2,
  (0,1,1,0,1).(1,0,1,0,1) = 0*1 + 1*0 + 1*1 + 0*0 + 1*1 = 2,
  (0,1,1,0,1).(1,1,0,1,0) = 0*1 + 1*1 + 1*0 + 0*1 + 1*0 = 1,
  (0,1,1,0,1).(0,1,1,0,1) = 0*0 + 1*1 + 1*1 + 0*0 + 1*1 = 3.
Then a(5)=3 because 3 is the maximum among the five values.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Andres Cicuttin, Graph of first 2^10 terms

Cf. A010051, A299111, A014342.

b[n_]:=Table[If[PrimeQ[i],1,0],{i,1,n}];
Table[Max@Table[b[n].RotateRight[Reverse[b[n]],j],{j,0,n-1}],{n,1,100}]

a(n) = vecmax(vector(n, k, sum(i=1, n, isprime(n-i+1)*isprime(1+(i+k)%n)))); \\ Michel Marcus, Aug 26 2020

A337802 Minimum value of the cyclic self-convolution of the first n terms of the characteristic function of primes.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Andres Cicuttin, Sep 22 2020

nonn

In the first 1000 terms, a(n) = 1 only for n = 3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 47, 61, 73, 107, 109, 113, 181, 199, and 467.

Is there an index k such that a(n) = 0 for n > k ?

			The primes among the first 5 positive integers (1,2,3,4,5) are 2, 3, and 5, then the corresponding characteristic function of primes is (0,1,1,0,1) (see A010051) and the corresponding five possible cyclic self-convolutions are the dot products between (0,1,1,0,1) and the rotations of its mirrored version as shown below:
  (0,1,1,0,1).(1,0,1,1,0) = 0*1 + 1*0 + 1*1 + 0*1 + 1*0 = 1,
  (0,1,1,0,1).(0,1,0,1,1) = 0*0 + 1*1 + 1*0 + 0*1 + 1*1 = 2,
  (0,1,1,0,1).(1,0,1,0,1) = 0*1 + 1*0 + 1*1 + 0*0 + 1*1 = 2,
  (0,1,1,0,1).(1,1,0,1,0) = 0*1 + 1*1 + 1*0 + 0*1 + 1*0 = 1,
  (0,1,1,0,1).(0,1,1,0,1) = 0*0 + 1*1 + 1*1 + 0*0 + 1*1 = 3.
Then a(5)=1 because 1 is the minimum among the five values.

Cf. A337327, A010051, A299111, A014342.

b[n_] := Table[If[PrimeQ[i], 1, 0], {i, 1, n}];
Table[Min@Table[b[n].RotateRight[Reverse[b[n]], j], {j, 0, n - 1}], {n, 1, 100}]

a(n) = vecmin(vector(n, k, sum(i=1, n, isprime(n-i+1)*isprime(1+(i+k)%n)))); \\ Michel Marcus, Sep 23 2020

A084370 Convolution of odd primes with themselves.

Original entry on oeis.org

9, 30, 67, 136, 237, 386, 587, 852, 1213, 1658, 2227, 2932, 3765, 4766, 5939, 7324, 8917, 10746, 12851, 15200, 17845, 20794, 24083, 27748, 31785, 36250, 41107, 46376, 52113, 58350, 65111, 72444, 80353, 88858, 98003, 107744, 118201, 129410, 141355, 154080

Offset: 1

Jon Perry, Jun 23 2003

nonn

			a(4) = 11*3 + 7*5 + 5*7 + 3*11 = 136.

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

Cf. A065091, A014342, A209403.

Table[With[{c=Prime[Range[2,n]]},ListConvolve[c,c]],{n,2,40}]//Flatten (* Harvey P. Dale, Apr 25 2016 *)

conv(n)=local(v,s); v=primes(n+1); s=0; for(i=2, length(v), s+=v[i]*v[length(v)-i+2]); s
for(n=1,40,print1(conv(n)","))

from numpy import convolve
from sympy import prime, primerange
def aupton(terms):
    p = list(primerange(3, prime(terms+1)+1))
    return list(convolve(p, p))[:terms]
print(aupton(40)) # Michael S. Branicky, Sep 30 2021

a(n) = Sum_{k=1..n} A065091(k) * A065091(n-k+1).

A338816 a(0) = 1; for n > 0, a(n) = Sum_{k=0..n-1} prime(a(k)) * prime(a(n-k-1)).

Original entry on oeis.org

1, 4, 28, 477, 15054, 716619, 46386636, 3829544473, 386956394842, 46498297487679, 6503866348955704, 1041432998153210277, 188367392877258840974

Offset: 0

Ilya Gutkovskiy, Nov 10 2020

Cf. A000040, A000108, A007097, A014342.

a[0] = 1; a[n_] := a[n] = Sum[Prime[a[k]] Prime[a[n - k - 1]], {k, 0, n - 1}]; Table[a[n], {n, 0, 9}]

G.f.: 1 + x * (Sum_{n>=0} prime(a(n)) * x^n)^2.

a(10)-a(11) from Daniel Suteu, Nov 10 2020

a(12) from Jinyuan Wang, Nov 14 2020

A067773 a(n) is the unique positive integer m which has a self-conjugate partition whose parts are the first n primes.

Original entry on oeis.org

4, 8, 17, 29, 53, 77, 117, 157, 217, 289, 369, 469, 585, 713, 849, 1025, 1197, 1393, 1617, 1845, 2113, 2373, 2661, 2973, 3321, 3681, 4045, 4481, 4865, 5285, 5793, 6253, 6785, 7341, 7949, 8513, 9169, 9765, 10473, 11233, 11969, 12733, 13541, 14337

Offset: 1

Naohiro Nomoto, Feb 06 2002

In general, given a finite set of positive integers p(1) < ... < p(n), there's a unique self-conjugate partition using these parts; p(n) occurs p(1) times and p(n-i) occurs p(i+1)-p(i) times for 1<=i

			2+2 = 4; 2+3+3 = 8; 2+2+3+5+5 = 17; ....

Cf. A000700, A014342.

a[n_] := 2Prime[n]+Sum[Prime[n-i](Prime[i+1]-Prime[i]), {i, 1, n-1}]

a(n) = 2 prime(n) + Sum_{i=1..n-1} prime(n-i)*(prime(i+1)-prime(i)) = A014342(n-1) - A014342(n-2).

Edited by Dean Hickerson, Feb 12 2002

A316186 Expansion of e.g.f. P(P(x)), where P(x) = Sum_{k>=1} prime(k)*x^k/k!.

Original entry on oeis.org

4, 18, 104, 687, 5064, 40934, 358083, 3346832, 33123000, 345219919, 3777134694, 43291666298, 518855171115, 6491738816768, 84656365477452, 1148895613585775, 16201725990730392, 237030534528945348, 3591398122456079285, 56254812062478841340, 909319044063443870702

Offset: 1

Ilya Gutkovskiy, Jun 26 2018

nonn

Self-composition of e.g.f. of A000040 (prime numbers).

			E.g.f.: A(x) = 4*x + 18*x^2/2! + 104*x^3/3! + 687*x^4/4! + 5064*x^5/5! + 40934*x^6/6! + ...

N. J. A. Sloane, Transforms

Cf. A000040, A014342, A014345, A030278.

p[x_] := p[x] = Sum[Prime[k] x^k/k!, {k, 21}]; a[n_] := a[n] = SeriesCoefficient[p[p[x]], {x, 0, n}]; Table[n! a[n], {n, 21}]

A363655 a(0) = 1; for n > 0, a(n) = prime( Sum_{k=0..n-1} a(k) * a(n-k-1) ).

Original entry on oeis.org

1, 2, 7, 61, 863, 17569, 472741, 16007419, 659408567, 32231133931, 1833425773489, 119498316410171, 8810846732918257, 727089137774221667

Offset: 0

Ilya Gutkovskiy, Jun 13 2023

Cf. A000040, A000108, A014342, A007097, A093502, A338816.

a[0] = 1; a[n_] := a[n] = Prime[Sum[a[k] a[n - k - 1], {k, 0, n - 1}]]; Table[a[n], {n, 0, 11}]

G.f.: sqrt( Sum_{n>=0} index of prime a(n+1) * x^n ).

a(12)-a(13) from Amiram Eldar, Jun 13 2023

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cp's OEIS Frontend

A299053 Minimum value of the cyclic autocorrelation of first n primes.

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A333370 Convolution of primorial numbers (A002110) with themselves.

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A337327 Maximum value of the cyclic self-convolution of the first n terms of the characteristic function of primes.

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A337802 Minimum value of the cyclic self-convolution of the first n terms of the characteristic function of primes.

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A084370 Convolution of odd primes with themselves.

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A338816 a(0) = 1; for n > 0, a(n) = Sum_{k=0..n-1} prime(a(k)) * prime(a(n-k-1)).

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A067773 a(n) is the unique positive integer m which has a self-conjugate partition whose parts are the first n primes.

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A316186 Expansion of e.g.f. P(P(x)), where P(x) = Sum_{k>=1} prime(k)*x^k/k!.

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