A000040 -id:A000040 - cp's OEIS frontend

A108548 Fully multiplicative with a(prime(j)) = A108546(j), where A108546 is the lexicographically earliest permutation of primes such that after 2 the forms 4k+1 and 4k+3 alternate, and prime(j) is the j-th prime in A000040.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 12, 11, 14, 15, 16, 17, 18, 19, 20, 21, 26, 29, 24, 25, 22, 27, 28, 23, 30, 37, 32, 39, 34, 35, 36, 31, 38, 33, 40, 41, 42, 43, 52, 45, 58, 53, 48, 49, 50, 51, 44, 47, 54, 65, 56, 57, 46, 61, 60, 59, 74, 63, 64, 55, 78, 73, 68, 87, 70, 67, 72

Offset: 1

Reinhard Zumkeller, Jun 10 2005

Multiplicative with a(2^e) = 2^e, else if p is the m-th prime then a(p^e) = q^e where q is the m/2-th prime of the form 4*k + 3 (A002145) for even m and a(p^e) = r^e where r is the (m-1)/2-th prime of the form 4*k + 1 (A002144) for odd m. - David A. Corneth, Apr 25 2022

Permutation of the natural numbers with fixed points A108549: a(A108549(n)) = A108549(n).

Antti Karttunen, Table of n, a(n) for n = 1..26927
Index entries for sequences that are permutations of the natural numbers

Cf. A002144, A002145, A049084, A108546, A108549 (fixed points), A332808 (inverse permutation).
Cf. also A332815, A332817 (this permutation applied to Doudna tree and its mirror image), also A332818, A332819.
Cf. also A267099, A332212 and A348746 for other similar mappings.

terms = 72;
A111745 = Module[{prs = Prime[Range[2 terms]], m3, m1, min},
     m3 = Select[prs, Mod[#, 4] == 3&];
     m1 = Select[prs, Mod[#, 4] == 1&];
     min = Min[Length[m1], Length[m3]];
     Riffle[Take[m3, min], Take[m1, min]]];
A108546[n_] := If[n == 1, 2, A111745[[n - 1]]];
A049084[n_] := PrimePi[n]*Boole[PrimeQ[n]];
a[n_] := If[n == 1, 1, Module[{p, e}, Product[{p, e} = pe; A108546[A049084[p]]^e, {pe, FactorInteger[n]}]]];
Array[a, terms] (* Jean-François Alcover, Nov 19 2021, using Harvey P. Dale's code for A111745 *)

up_to = 26927; \\ One of the prime fixed points.
A108546list(up_to) = { my(v=vector(up_to), p,q); v[1] = 2; v[2] = 3; v[3] = 5; for(n=4,up_to, p = v[n-2]; q = nextprime(1+p); while(q%4 != p%4, q=nextprime(1+q)); v[n] = q); (v); };
v108546 = A108546list(up_to);
A108546(n) = v108546[n];
A108548(n) = { my(f=factor(n)); f[,1] = apply(A108546,apply(primepi,f[,1])); factorback(f); }; \\ Antti Karttunen, Apr 25 2022

Name edited by Antti Karttunen, Apr 25 2022

A211000 Coordinates (x,y) of the endpoint of a structure (or curve) formed by Q-toothpicks in which the inflection points are the prime numbers A000040.

Original entry on oeis.org

0, 0, 1, 1, 2, 0, 3, -1, 4, -2, 3, -3, 2, -4, 3, -5, 4, -6, 3, -7, 2, -6, 3, -5, 4, -4, 3, -3, 2, -2, 3, -1, 4, -2, 3, -3, 2, -4, 3, -5, 4, -6, 3, -7, 2, -6, 3, -5, 4, -4, 3, -3, 2, -4, 3, -5, 4, -4, 3, -3, 2, -2, 3, -1, 4, 0, 3, 1, 2, 0, 3, -1, 4, 0

Offset: 0

Omar E. Pol, Mar 30 2012

On the infinite square grid the structure looks like a column of tangent circles of radius 1. The structure arises from the prime numbers A000040. The behavior seems to be as modular arithmetic but in a growing structure. The values on the axis "x" are easy to predict (see A211010). On the other hand the values on the axis "y" do not seem to be predictable (see A211011). This is a member of the family of the structures or curves mentioned in A210838. The odd numbers > 1 are located on the main axis of the structure. Note that here the Q-toothpicks can be superposed. For the definition of Q-toothpicks see A187210. A211021 gives the number of stage where a new circle appears in the structure. For the number of circles after the n-th stage see A211020. For the location of the centers of the circles see A211022. For the sums of the visible nodes after the n-th stage see A211024.

			We start at stage 0 with no Q-toothpicks.
At stage 1 we place a Q-toothpick centered at (1,0) with its endpoints at (0,0) and (1,1).
At stage 2 we place a Q-toothpick centered at (1,0) with its endpoints at (1,1) and (2,0). Since 2 is a prime number we have that the end of the curve is also an inflection point.
At stage 3 we place a Q-toothpick centered at (3,0) with its endpoints at (2,0) and (3,-1). Since 3 is a prime number we have that the end of the curve is also an inflection point.
At stage 4 we place a Q-toothpick centered at (3,-2) with its endpoints at (3,-1) and (4,-2).
-------------------------------------
.                    The end as
.          Pair      inflection
n        (x    y)      point
-------------------------------
0         0,   0,        -
1         1,   1,        -
2         2,   0,       Yes
3         3,  -1,       Yes
4         4,  -2,        -
5         3,  -3,       Yes
6         2,  -4,        -
7         3,  -5,       Yes
8         4,  -6,        -
9         3,  -7,        -
10        2,  -6,        -
11        3,  -5,       Yes
...
Illustration of the nodes of the structure:
-----------------------------------------------------
After 9 stages    After 10 stages    After 11 stages
-----------------------------------------------------
.
.    1                 1                  1
.  0   2             0   2              0   2
.        3                 3                  3
.          4                 4                  4
.        5                 5                  5
.      6                 6                  6
.        7                 7                 11
.          8            10   8             10   8
.        9                 9                  9
.

Paolo Xausa, Table of n, a(n) for n = 0..9999
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Bisections: A211010, A211011.

Cf. A187210, A210838, A210841, A211001-A211003, A211020-A211024, A355478, A357434.

A211000[nmax_]:=Module[{walk={{0,0}},angle=3/4Pi,turn=Pi/2},Do[If[!PrimeQ[n],If[n>5&&PrimeQ[n-1],turn*=-1];angle-=turn];AppendTo[walk,AngleVector[Last[walk],{Sqrt[2],angle}]],{n,0,nmax-1}];walk];
A211000[100] (* Generates 101 coordinate pairs *) (* Paolo Xausa, Aug 23 2022 *)

A211000(nmax) = my(walk=vector(nmax+1), turn=1, p1, p2); walk[1]=[0,0];if(nmax==0,return(walk));walk[2]=[1,1];for(n=1, nmax-1, p1=walk[n];p2=walk[n+1];if(isprime(n),walk[n+2]=[2*p2[1]-p1[1],2*p2[2]-p1[2]],if(n>5 && isprime(n-1), turn*=-1);walk[n+2]=[p2[1]-turn*(p1[2]-p2[2]),p2[2]+turn*(p1[1]-p2[1])]));walk;
A211000(100) \\ Generates 101 coordinate pairs - Paolo Xausa, Sep 22 2022

from sympy import isprime
def A211000(nmax):
    walk, turn = [(0,0),(1,1)], 1
    for n in range(1,nmax):
        p1, p2 = walk[-2], walk[-1]
        if isprime(n): # Go straight
            walk.append((2*p2[0]-p1[0],2*p2[1]-p1[1]))
        else:          # Turn
            if n>5 and isprime(n-1): turn *= -1
            walk.append((p2[0]-turn*(p1[1]-p2[1]),p2[1]+turn*(p1[0]-p2[0])))
    return walk[:nmax+1]
print(A211000(100)) # Generates 101 coordinate pairs - Paolo Xausa, Sep 22 2022

A246371 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

5, 8, 11, 13, 14, 17, 18, 23, 28, 32, 38, 39, 41, 43, 50, 53, 58, 59, 61, 63, 68, 73, 74, 77, 83, 86, 88, 94, 95, 98, 104, 113, 116, 122, 123, 128, 131, 137, 138, 140, 143, 149, 158, 163, 167, 172, 173, 176, 179, 182, 185, 188, 193, 194, 200, 203, 212, 213, 215, 218, 221, 228, 230, 233, 238, 239, 242, 248, 254, 257

Offset: 1

Antti Karttunen, Aug 24 2014

nonn

Numbers n such that A064216(n) < n.
Numbers n such that A064989(2n-1) < n.
Note: This sequence has remarkable but possibly merely coincidental overlap with A053726. On Dec 22 2014, Matthijs Coster mistakenly attached a comment intended for that sequence to this one. On Apr 17 2015, Antti Karttunen noted the error. I have moved the comment to the correct sequence, and have removed Karttunen's note. - Allan C. Wechsler, Aug 01 2022

Antti Karttunen, Table of n, a(n) for n = 1..10000
Matthijs Coster, Oplossing Zomerprijsvraag, Pythagoras 54/2 (2014) 4-7.

Complement: A246372.
Setwise difference of A246361 and A048674.
Subsequence of A104275 and A053726 (20 is the first term > 1 which is not in this sequence).
Subsequence: A246374 (the primes present in this sequence).
Cf. A000040, A064216, A064989, A246351.

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);
isA246371(n) = (A064216(n) < n);
n = 0; i = 0; while(i < 10000, n++; if(isA246371(n), i++; write("b246371.txt", i, " ", n)));
(Scheme, with Antti Karttunen's IntSeq-library)
(define A246371 (MATCHING-POS 1 1 (lambda (n) (< (A064216 n) n))))

A376681 Row sums of the absolute value of the array A095195(n, k) = n-th term of the k-th differences of the prime numbers (A000040).

Original entry on oeis.org

2, 4, 8, 10, 22, 36, 72, 134, 266, 500, 874, 1418, 2044, 2736, 4626, 15176, 41460, 95286, 196368, 372808, 660134, 1092790, 1682198, 2384724, 3147706, 4526812, 11037090, 36046768, 93563398, 214796426, 452129242, 885186658, 1619323680, 2763448574, 4368014812

Offset: 1

Gus Wiseman, Oct 15 2024

nonn

			The fourth row of A095195 is: (7, 2, 0, -1), so a(4) = 10.

For firsts instead of row-sums we have A007442 (modern version of A030016).
This is the absolute version of A140119.
If 1 is considered prime (A008578) we get A376684, absolute version of A376683.
For first zero-positions we have A376678 (modern version of A376855).
For composite instead of prime we have A377035.
For squarefree instead of prime we have A377040, nonsquarefree A377048.
A000040 lists the modern primes, differences A001223, seconds A036263.
A008578 lists the noncomposites, differences A075526, seconds A036263 with 0 prepended.
Cf. A064113, A065890, A084758, A173390, A233671, A258025, A333214, A333254, A376602, A376651, A376652, A377033.

nn=15;
t=Table[Take[Differences[NestList[NestWhile[#+1&, #+1,!PrimeQ[#]&]&,2,2*nn],k],nn],{k,0,nn}]
Total/@Abs/@Table[t[[j,i-j+1]],{i,nn},{j,i}]

More terms from Pontus von Brömssen, Oct 17 2024

A080192 Complement of A080191 relative to A000040. Prime p is a term iff there is no prime between 2p and 2q, where q is the next prime after p.

Original entry on oeis.org

59, 71, 101, 107, 149, 263, 311, 347, 461, 499, 521, 569, 673, 757, 821, 823, 857, 881, 883, 907, 967, 977, 1009, 1061, 1091, 1093, 1151, 1213, 1279, 1283, 1297, 1301, 1319, 1433, 1487, 1489, 1493, 1549, 1571, 1597, 1619, 1667, 1697, 1721, 1787, 1871, 1873

Offset: 1

Klaus Brockhaus, Feb 10 2003

nonn

From Peter Munn, Oct 19 2017: (Start)
This is also a list of the leaf node labels in the tree of primes described in A290183.
For k > 0, the earliest run of k adjacent primes in this sequence starts with the least prime greater than A215238(k+1)/2. Thus we see that A215238(3) = 1637 corresponds to 821 followed by 823 being the first run of 2 adjacent primes in this sequence.
(End)
From Peter Munn, Nov 02 2017: (Start)
If p is in A005384 (a Sophie Germain prime), 2p+1 is therefore a prime, so p cannot be in this sequence. Similarly, any prime p in A023204 has a corresponding prime 2p+3, which (if p>2) likewise implies its absence (and if p=2 it is in A005384).
If p is the lesser of twin primes it is in this sequence if it is neither Sophie Germain nor in A023204.
Conjecture: a(n)/A000040(n) is asymptotic to 3. Reason: I expect the distribution of terms in A102820 to converge to a geometric distribution with mean value 2.
(End)

			59 is a term since 113 is the prime preceding 2*59, 127 is the next prime and 61 is the largest of all prime factors of 114, ..., 122 = 2*61, ..., 126.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Michel Marcus)

A080191 is the complement of this sequence relative to A000040.
Sequences with related analysis: A005384, A023204, A052248, A102820, A215238, A290183.
Sequences with similar definitions: A195270, A195271, A195325, A195377.

Select[Prime[Range[300]],NextPrime[2#]>2NextPrime[#]&] (* Harvey P. Dale, Jul 07 2011 *)

¯1↓b/⍨(1⌽a)<1πa←2×b←¯2π⍳1E4 ⍝ Michael Turniansky, Dec 29 2020

{forprime(k=2,1873,p=precprime(2*k); q=nextprime(p+1); m=0; for(j=p+1,q-1,f=factor(j); a=f[matsize(f)[1],1]; if(m

isok(p) = isprime(p) && (primepi(2*p) == primepi(2*nextprime(p+1)));
forprime(p=2, 2000, if (isok(p), print1(p, ", "))) \\ Michel Marcus, Sep 22 2017

first(n) = my(res = vector(n), i = 0); {n==0&&return([]); forprime(p = 2, , if(nextprime(2*p) > 2*nextprime(p + 1), i++; res[i] = p; if(i == n, return(res))))} \\ David A. Corneth, Oct 25 2017

For all k, prime(k) = A000040(k) is a term if and only if A102820(k) = 0. - Peter Munn, Oct 24 2017

A008478 Integers of the form Product p_j^k_j = Product k_j^p_j; p_j in A000040.

Original entry on oeis.org

1, 4, 16, 27, 72, 108, 432, 800, 3125, 6272, 12500, 21600, 30375, 50000, 84375, 121500, 169344, 225000, 247808, 337500, 486000, 750141, 823543, 1350000, 1384448, 3000564, 3294172, 6690816, 12002256, 13176688, 19600000, 22235661, 37380096, 37879808, 59295096, 88942644

Offset: 1

Olivier Gérard

nonn

Fixed points of A008477.

a(3) = 16 is the only term of the form p^q with p <> q. - Bernard Schott, Mar 28 2021

			16 = 2^4 = 4^2.
27 = 3^3.
108 = 2^2*3^3.
6272 = 2^7*7^2.
121500 = 2^2 * 3^5*5^3.

Cf. A000040, A008477.

Some subsequences: p_i^p_i (A051674), Product_i {p_i^p_i} (A048102), Product_(j,k)(p_j^p_k * p_k^p_j) with p_j < p_k (A082949) (see examples).

f[n_] := Product[{p, e} = pe; e^p, {pe, FactorInteger[n]}];
Reap[For[n = 1, n <= 10^8, n++, If[f[n] == n, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Mar 29 2021 *)

for(n=2,10^8,if(n==prod(i=1,omega(n), component(component(factor(n),2),i)^component(component(factor(n),1),i)),print1(n,",")))

More terms from David W. Wilson

a(34)-a(36) from Jean-François Alcover, Mar 29 2021

A080147 Positions of primes of the form 4*k+1 (A002144) among all primes (A000040).

Original entry on oeis.org

3, 6, 7, 10, 12, 13, 16, 18, 21, 24, 25, 26, 29, 30, 33, 35, 37, 40, 42, 44, 45, 50, 51, 53, 55, 57, 59, 60, 62, 65, 66, 68, 70, 71, 74, 77, 78, 79, 80, 82, 84, 87, 88, 89, 97, 98, 100, 102, 104, 106, 108, 110, 112, 113, 116, 119, 121, 122, 123, 126, 127, 130, 134, 135

Offset: 1

Antti Karttunen, Feb 11 2003

The asymptotic density of this sequence is 1/2 (by Dirichlet's theorem). - Amiram Eldar, Mar 01 2021

			7 is in the sequence because the 7th prime, 17, is of the form 4k+1.
4 is not in the sequence because the 4th prime, 7, is not of the form 4k+1.

Zak Seidov, Table of n, a(n) for n = 1..10000

Almost complement of A080148 (1 is excluded from both).

Cf. A000040, A002144.

with(numtheory,ithprime); pos_of_primes_k_mod_n(300,1,4);
pos_of_primes_k_mod_n := proc(upto_i,k,n) local i,a; a := []; for i from 1 to upto_i do if(k = (ithprime(i) mod n)) then a := [op(a),i]; fi; od; RETURN(a); end;
with(Bits): for n from 1 to 135 do if (And(ithprime(n),2)=0) then print(n) fi od; # Gary Detlefs, Dec 26 2011

Select[Range[135], Mod[Prime[#], 4] == 1 &] (* Amiram Eldar, Mar 01 2021 *)

k=0;forprime(p=2,1e4,k++;if(p%4==1,print1(k", "))) \\ Charles R Greathouse IV, Dec 27 2011

A002144(n) = A000040(a(n)).

Numbers k such that prime(k) AND 2 = 0. - Gary Detlefs, Dec 26 2011

A122824 Prime(semiprime(n)) - semiprime(prime(n)). Commutator [A000040,A001358] at n.

Original entry on oeis.org

1, 4, 9, 8, 10, 12, 24, 24, 32, 15, 46, 24, 27, 34, 25, 40, 44, 46, 51, 54, 53, 46, 54, 60, 70, 70, 98, 105, 104, 91, 64, 72, 45, 48, 95, 118, 120, 120, 116, 108, 100, 96, 101, 118, 102, 144, 123, 86, 76, 81, 136, 138, 143, 112, 132, 131, 153, 160, 171, 169

Offset: 1

Jonathan Vos Post, Oct 23 2006

			a(1) = prime(semiprime(1)) - semiprime(prime(1)) = prime(4) - semiprime(2) = 7 - 6 = 1.
a(2) = prime(semiprime(2)) - semiprime(prime(2)) = prime(6) - semiprime(3) = 13 - 9 = 4.
a(3) = prime(semiprime(3)) - semiprime(prime(3)) = prime(9) - semiprime(5) = 23 - 14 = 9.
a(4) = prime(semiprime(4)) - semiprime(prime(4)) = prime(10) - semiprime(7) = 29 - 21 = 8.

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

Cf. A000040, A001358, A106349, A106350.

sp = Select[Range[1000], PrimeOmega[#] == 2 &]; Table[ Prime[ sp[[i]]] - sp[[Prime[i]]], {i, PrimePi@ Length@ sp}] (* Giovanni Resta, Jun 13 2016 *)

a(n) = A106349(n) - A106350(n).

a(33)-a(54) corrected by and a(55)-a(60) from Giovanni Resta, Jun 13 2016

A243056 If n is the i-th prime, p_i = A000040(i), then a(n) = i, otherwise the difference between the indices of the smallest and the largest prime dividing n: for n = p_i * ... * p_k, where p_i <= ... <= p_k, a(n) = (k-i); a(1) = 0 by convention.

Original entry on oeis.org

0, 1, 2, 0, 3, 1, 4, 0, 0, 2, 5, 1, 6, 3, 1, 0, 7, 1, 8, 2, 2, 4, 9, 1, 0, 5, 0, 3, 10, 2, 11, 0, 3, 6, 1, 1, 12, 7, 4, 2, 13, 3, 14, 4, 1, 8, 15, 1, 0, 2, 5, 5, 16, 1, 2, 3, 6, 9, 17, 2, 18, 10, 2, 0, 3, 4, 19, 6, 7, 3, 20, 1, 21, 11, 1, 7, 1, 5, 22, 2, 0, 12, 23, 3, 4, 13, 8, 4, 24, 2

Offset: 1

Antti Karttunen, May 31 2014

nonn

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

Useful when computing A243057 or A243059.
A025475 (prime powers that are not primes) gives the positions of zeros.
Differs from A241917 for the first time at n=18.
Cf. A243055, A241919, A242411, A049084, A027748, A055396, A061395.

(define (A243056 n) (if (prime? n) (A049084 n) (A243055 n)))

a(1) = 0, for n>1, if n = A000040(i), a(n) = i, otherwise a(n) = A061395(n) - A055396(n) = A243055(n).

A246361 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

1, 2, 3, 5, 8, 11, 13, 14, 17, 18, 23, 25, 26, 28, 32, 33, 38, 39, 41, 43, 50, 53, 58, 59, 61, 63, 68, 73, 74, 77, 83, 86, 88, 93, 94, 95, 98, 104, 113, 116, 122, 123, 128, 131, 137, 138, 140, 143, 149, 158, 163, 167, 172, 173, 176, 179, 182, 185, 188, 193, 194, 200, 203, 212, 213, 215, 218, 221, 228, 230, 233

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) = 332
     a(1000) = 3207
    a(10000) = 34213
   a(100000) = 340703
  a(1000000) = 3388490
suggesting that overall, less than one third of natural numbers appear in this sequence, and more than two thirds in the complement, A246362. See also comments in the latter.

			1 is present, as 2*1 - 1 = empty product = 1.
12 is not present, as (2*12)-1 = 23 = p_9, and p_8 = 19, with 12 < 19.
14 is present, as (2*14)-1 = 27 = p_2^3 = 8, and 14 >= 8.

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

Complement: A246362.
Union of A246371 and A048674.
Subsequence: A246360.
Cf. A000040, A064216, A064989, A246281.

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);
isA246361(n) = (A064216(n) <= n);
n = 0; i = 0; while(i < 10000, n++; if(isA246361(n), i++; write("b246361.txt", i, " ", n)));
(Scheme, with Antti Karttunen's IntSeq-library)
(define A246361 (MATCHING-POS 1 1 (lambda (n) (<= (A064216 n) n))))

cp's OEIS Frontend

A108548 Fully multiplicative with a(prime(j)) = A108546(j), where A108546 is the lexicographically earliest permutation of primes such that after 2 the forms 4*k+1 and 4*k+3 alternate, and prime(j) is the j-th prime in A000040.

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A211000 Coordinates (x,y) of the endpoint of a structure (or curve) formed by Q-toothpicks in which the inflection points are the prime numbers A000040.

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A246371 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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A376681 Row sums of the absolute value of the array A095195(n, k) = n-th term of the k-th differences of the prime numbers (A000040).

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Mathematica

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A080192 Complement of A080191 relative to A000040. Prime p is a term iff there is no prime between 2*p and 2*q, where q is the next prime after p.

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Mathematica

NARS2000

PARI

PARI

PARI

Formula

A008478 Integers of the form Product p_j^k_j = Product k_j^p_j; p_j in A000040.

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PARI

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A080147 Positions of primes of the form 4*k+1 (A002144) among all primes (A000040).

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Maple

Mathematica

PARI

A108548 Fully multiplicative with a(prime(j)) = A108546(j), where A108546 is the lexicographically earliest permutation of primes such that after 2 the forms 4k+1 and 4k+3 alternate, and prime(j) is the j-th prime in A000040.

A080192 Complement of A080191 relative to A000040. Prime p is a term iff there is no prime between 2p and 2q, where q is the next prime after p.