A052147 -id:A052147 - cp's OEIS frontend

A175224 a(n) = prime(n) + 8.

10, 11, 13, 15, 19, 21, 25, 27, 31, 37, 39, 45, 49, 51, 55, 61, 67, 69, 75, 79, 81, 87, 91, 97, 105, 109, 111, 115, 117, 121, 135, 139, 145, 147, 157, 159, 165, 171, 175, 181, 187, 189, 199, 201, 205, 207, 219, 231, 235, 237, 241, 247, 249, 259, 265, 271, 277, 279

Offset: 1

Jaroslav Krizek, Mar 06 2010

nonn

a(n) = A000040(n) + 8 = A008864(n) + 7 = A052147(n) + 6 = A113395(n) + 5 = A175221(n) + 4 = A175222(n) + 3 = A139049(n) + 2 = A175223(n) + 1 = A140353(n) - 1 = A175225(n) - 2.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000040, A008864, A052147, A113395, A175221, A175222, A139049, A175223, A140353, A175225.

Filtered([1..300], k-> IsPrime(k) ) + 8; # G. C. Greubel, May 20 2019

[(p+8): p in PrimesUpTo(500)]; // Vincenzo Librandi, Dec 04 2010

Prime[Range[60]]+8 (* Harvey P. Dale, Mar 02 2015 *)

{a(n) = prime(n) + 8}; \\ G. C. Greubel, May 20 2019

[nth_prime(n) + 8 for n in (1..60)] # G. C. Greubel, May 20 2019

More terms from Vincenzo Librandi, Mar 14 2010

A175225 a(n) = prime(n) + 10.

Original entry on oeis.org

12, 13, 15, 17, 21, 23, 27, 29, 33, 39, 41, 47, 51, 53, 57, 63, 69, 71, 77, 81, 83, 89, 93, 99, 107, 111, 113, 117, 119, 123, 137, 141, 147, 149, 159, 161, 167, 173, 177, 183, 189, 191, 201, 203, 207, 209, 221, 233, 237, 239, 243, 249, 251, 261, 267, 273, 279, 281

Offset: 1

Jaroslav Krizek, Mar 06 2010

nonn

a(n) = A000040(n) + 10 = A008864(n) + 9 = A052147(n) + 8 = A113395(n) + 7 = A175221(n) + 6 = A175222(n) + 5 = A139049(n) + 4 = A175223(n) + 3 = A175224(n) + 2 = A140353(n) + 1.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000040, A008864, A052147, A113395, A139049, A140353, A175221, A175222, A175223, A175224.

Filtered([1..300], k-> IsPrime(k) ) + 10; # G. C. Greubel, May 21 2019

[(p+10):p in PrimesUpTo(500)]; // Vincenzo Librandi, Dec 04 2010

10 + Prime[Range[70]] (* G. C. Greubel, May 21 2019 *)

{a(n) = prime(n) + 10}; \\ G. C. Greubel, May 21 2019

[nth_prime(n) + 10 for n in (1..70)] # G. C. Greubel, May 21 2019

More terms from Vincenzo Librandi, Mar 14 2010

A216620 Square array read by antidiagonals: T(n,k) = Sum_{c|n,d|k} phi(gcd(c,d)) for n>=1, k>=1.

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 3, 4, 4, 3, 2, 6, 5, 6, 2, 4, 4, 6, 6, 4, 4, 2, 8, 4, 10, 4, 8, 2, 4, 4, 10, 6, 6, 10, 4, 4, 3, 8, 4, 12, 7, 12, 4, 8, 3, 4, 6, 8, 6, 8, 8, 6, 8, 6, 4, 2, 8, 8, 14, 4, 20, 4, 14, 8, 8, 2, 6, 4, 8, 9, 8, 8, 8, 8, 9, 8, 4, 6, 2, 12, 4, 12, 6

Offset: 1

Peter Luschny, Sep 12 2012

T(n,n) = A060648(n) = Sum_{d|n} Dedekind_Psi(d).
T(n,1) = T(1,n) = A000005(n) = tau(n).
T(n,2) = T(2,n) = A062011(n) = 2*tau(n).
T(n+1,n) = A092517(n) = tau(n+1)*tau(n).
T(prime(n),1) = A007395(n) = 2.
T(prime(n),prime(n)) = A052147(n) = prime(n)+2.

			[----1---2---3---4---5---6---7---8---9--10--11--12]
[ 1] 1,  2,  2,  3,  2,  4,  2,  4,  3,  4,  2,  6
[ 2] 2,  4,  4,  6,  4,  8,  4,  8,  6,  8,  4, 12
[ 3] 2,  4,  5,  6,  4, 10,  4,  8,  8,  8,  4, 15
[ 4] 3,  6,  6, 10,  6, 12,  6, 14,  9, 12,  6, 20
[ 5] 2,  4,  4,  6,  7,  8,  4,  8,  6, 14,  4, 12
[ 6] 4,  8, 10, 12,  8, 20,  8, 16, 16, 16,  8, 30
[ 7] 2,  4,  4,  6,  4,  8,  9,  8,  6,  8,  4, 12
[ 8] 4,  8,  8, 14,  8, 16,  8, 22, 12, 16,  8, 28
[ 9] 3,  6,  8,  9,  6, 16,  6, 12, 17, 12,  6, 24
[10] 4,  8,  8, 12, 14, 16,  8, 16, 12, 28,  8, 24
[11] 2,  4,  4,  6,  4,  8,  4,  8,  6,  8, 13, 12
[12] 6, 12, 15, 20, 12, 30, 12, 28, 24, 24, 12, 50
.
Displayed as a triangular array:
   1,
   2, 2,
   2, 4,  2,
   3, 4,  4,  3,
   2, 6,  5,  6, 2,
   4, 4,  6,  6, 4,  4,
   2, 8,  4, 10, 4,  8, 2,
   4, 4, 10,  6, 6, 10, 4, 4,
   3, 8,  4, 12, 7, 12, 4, 8, 3,

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Cf. A216621, A216622, A216623, A216624, A216625, A216626, A216627.

with(numtheory):
T:= (n, k)-> add(add(phi(igcd(c,d)), c=divisors(n)), d=divisors(k)):
seq(seq(T(n, 1+d-n), n=1..d), d=1..14);  # Alois P. Heinz, Sep 12 2012

t[n_, k_] := Outer[ EulerPhi[ GCD[#1, #2]]&, Divisors[n], Divisors[k]] // Flatten // Total; Table[ t[n-k+1, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 26 2013 *)

def A216620(n, k) :
    cp = cartesian_product([divisors(n), divisors(k)])
    return reduce(lambda x,y: x+y, map(euler_phi, map(gcd, cp)))
for n in (1..12): [A216620(n,k) for k in (1..12)]

A048974 Odd numbers that are the sum of 2 primes.

Original entry on oeis.org

5, 7, 9, 13, 15, 19, 21, 25, 31, 33, 39, 43, 45, 49, 55, 61, 63, 69, 73, 75, 81, 85, 91, 99, 103, 105, 109, 111, 115, 129, 133, 139, 141, 151, 153, 159, 165, 169, 175, 181, 183, 193, 195, 199, 201, 213, 225, 229, 231, 235, 241, 243, 253, 259

Offset: 1

N. J. A. Sloane

A048974, A052147, A067187 and A088685 are very similar after dropping terms less than 13. - Eric W. Weisstein, Oct 10 2003

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

Cf. A014092, A065091.

Select[Flatten@Table[Prime[i] + Prime[j], {i, 100}, {j, 1, i}], # < Prime[100] && OddQ[#] &] (* Robert Price, Apr 21 2025 *)

One of the primes must be 2, so this is simply the odd primes + 2.

a(n) = A065091(n) + 2. - Sean A. Irvine, Jul 15 2021

A157931 Numbers that are both the sum and the product of two primes.

Original entry on oeis.org

4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 38, 39, 46, 49, 55, 58, 62, 69, 74, 82, 85, 86, 91, 94, 106, 111, 115, 118, 122, 129, 133, 134, 141, 142, 146, 158, 159, 166, 169, 178, 183, 194, 201, 202, 206, 213, 214, 218, 226, 235, 253, 254, 259, 262, 265, 274, 278

Offset: 1

William Weeks (dach(AT)kuci.org), Mar 09 2009

Assuming the Goldbach conjecture, this is A001358 intersect (A005843 union A052147), since an odd number n is the sum of two primes iff n-2 is prime. - N. J. A. Sloane, Mar 14 2009

The first few terms of A001358: Semiprimes, not members of A157931 are: 35, 51, 57, 65, 77, 87, 93, 95, ..., . - Robert G. Wilson v, Mar 15 2009

			For the numbers up to 100, the solutions are 4 = (2+2) = (2*2); 6 = (3+3) = (2*3); 9 = (2+7) = (3*3); 10 = (3+7) = (2*5); 14 = (3+11) = (2*7); 15 = (2+13) = (3*5); 21 = (2+19) = (3*7); 22 = (3+19) = (2*11); 25 = (2+23) = (5*5); 26 = (3+23) = (2*13); 33 = (2+31) = (3*11); 34 = (3+31) = (2*17); 38 = (7+31) = (2*19); 39 = (2+37) = (3*13); 46 = (3+43) = (2*23); 49 = (2+47) = (7*7); 55 = (2+53) = (5*11); 58 = (5+53) = (2*29); 62 = (3+59) = (2*31); 69 = (2+67) = (3*23); 74 = (3+71) = (2*37); 82 = (3+79) = (2*41); 85 = (2+83) = (5*17); 86 = (3+83) = (2*43); 91 = (2+89) = (7*13); 94 = (5+89) = (2*47).

Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 1096 terms from Robert G. Wilson v)

Cf. A001358, A005843, A052147, A062721.
Cf. A043326 Numbers n such that n is a product of two different primes and n - 2 is prime, A062721 Numbers n such that n is a product of two primes and n - 2 is prime. - Zak Seidov, Mar 15 2009
Cf. A014091, A064911, A100962.

a157931 n = a157931_list !! (n-1)
a157931_list = filter ((== 1) . a064911) a014091_list
-- Reinhard Zumkeller, Oct 15 2014

isA014091 := proc(n) for i from 1 do p := ithprime(i) ; if p > n/2 then RETURN(false); fi; if isprime(n-p) then RETURN(true) ; fi; od: end: isA001358 := proc(n) RETURN(numtheory[bigomega](n) = 2) ; end: for n from 4 to 500 do if isA001358(n) and isA014091(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, Mar 15 2009

fQ[n_] := Block[{k = 2}, While[k < n, If[ PrimeQ[n - k], Break[]]; k = NextPrime@k]; k + 1 < n]; semiPrimeQ[n_] := Plus @@ Last /@ FactorInteger@n == 2; Select[ Range@ 295, fQ@# && semiPrimeQ@# &] (* Robert G. Wilson v, Mar 15 2009 *)
Select[Union[Flatten[Table[Prime[i] + Prime[j], {i, 50}, {j, 50}]]], PrimeOmega[#] == 2 &] (* Alonso del Arte, Feb 08 2013 *)
Union[Select[Total/@Tuples[Prime[Range[60]],2],PrimeOmega[#]==2&]] (* Harvey P. Dale, Jul 27 2015 *)

A014091 INTERSECT A001358. - R. J. Mathar, Mar 15 2009

Edited by N. J. A. Sloane, Mar 14 2009

Extended by R. J. Mathar and Robert G. Wilson v, Mar 15 2009

A117530 Triangle read by rows: T(n,k) = k^2 - k + prime(n), 1<=k<=n.

Original entry on oeis.org

2, 3, 5, 5, 7, 11, 7, 9, 13, 19, 11, 13, 17, 23, 31, 13, 15, 19, 25, 33, 43, 17, 19, 23, 29, 37, 47, 59, 19, 21, 25, 31, 39, 49, 61, 75, 23, 25, 29, 35, 43, 53, 65, 79, 95, 29, 31, 35, 41, 49, 59, 71, 85, 101, 119, 31, 33, 37, 43, 51, 61, 73, 87, 103, 121, 141, 37, 39, 43, 49, 57

Offset: 1

Reinhard Zumkeller, Mar 25 2006

A117531 gives the number of primes in the n-th row;

if T(n,1) is a Lucky Number of Euler then A117531(n)=n, see A014556.

			T(5,k)=A048058(k)=A048059(k), 1<=k<=5: T(5,1)=A014556(4)=11;
T(7,k)=A007635(k), 1<=k<=7: T(7,1)=A014556(5)=17;
T(13,k)=A005846(k), 1<=k<=13: T(13,1)=A014556(6)=41.

Robert Price, Table of n, a(n) for n = 1..10011
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial
Eric Weisstein's World of Mathematics, Lucky Number of Euler

Cf. A005846, A007635, A014556, A048058, A048059, A117531.

Table[k^2 - k + Prime[n], {n, 141}, {k, n}] // Flatten (* Robert Price, Apr 19 2025 *)

T(n,k) = k^2 - k + prime(n) \\ Charles R Greathouse IV, Apr 24 2015

T(n,1) = A000040(k).
T(n,2) = A052147(k) for k>1.
For 1

A049234 Number of divisors of prime(n) + 2.

Original entry on oeis.org

3, 2, 2, 3, 2, 4, 2, 4, 3, 2, 4, 4, 2, 6, 3, 4, 2, 6, 4, 2, 6, 5, 4, 4, 6, 2, 8, 2, 4, 4, 4, 4, 2, 4, 2, 6, 4, 8, 3, 6, 2, 4, 2, 8, 2, 4, 4, 9, 2, 8, 4, 2, 6, 4, 4, 4, 2, 8, 6, 2, 8, 4, 4, 2, 12, 4, 6, 4, 2, 8, 4, 3, 6, 8, 4, 8, 4, 8, 4, 4, 2, 6, 2, 8, 9, 4, 4, 8, 2, 8, 4, 4, 4, 4, 4, 4, 4, 2, 12, 4, 6, 4, 4

Offset: 1

Labos Elemer

			a(10) = d(prime(10)+2) = d(29+2) = d(29) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A052147.

Table[DivisorSigma[0,Prime[n]+2],{n,1,103}]

a(n) = numdiv(prime(n)+2); \\ Michel Marcus, Dec 15 2013

a(n) = A000005(A052147(n)). - Amiram Eldar, Feb 16 2025

A065342 Triangle of sum of two primes: prime(n)+prime(k) with n >= k >= 1.

Original entry on oeis.org

4, 5, 6, 7, 8, 10, 9, 10, 12, 14, 13, 14, 16, 18, 22, 15, 16, 18, 20, 24, 26, 19, 20, 22, 24, 28, 30, 34, 21, 22, 24, 26, 30, 32, 36, 38, 25, 26, 28, 30, 34, 36, 40, 42, 46, 31, 32, 34, 36, 40, 42, 46, 48, 52, 58, 33, 34, 36, 38, 42, 44, 48, 50, 54, 60, 62, 39, 40, 42, 44, 48

Offset: 1

Henry Bottomley, Oct 30 2001

			Sequence starts 2+2; 3+2, 3+3; 5+2, 5+3, 5+5; etc. i.e. 4; 5,6; 7,8,10; ...
Triangle begins:
   4;
   5,  6;
   7,  8, 10;
   9, 10, 12, 14;
  13, 14, 16, 18, 22;
  ...

Reinhard Zumkeller, First 125 rows of triangle, flattened
Index entries for sequences related to Goldbach conjecture

Cf. A052147 (left edge), A100484 (right edge), A000040.
Cf. A087112.
Cf. A065305.

import Data.List (inits)
a065342 n k = a065342_tabl !! (n-1) !! (k-1)
a065342_row n = a065342_tabl !! (n-1)
a065342_tabl = zipWith (map . (+)) a000040_list $ tail $ inits a000040_list
-- Reinhard Zumkeller, Aug 02 2015, Jan 30 2012

row(n) = vector(n, k, prime(n)+prime(k)); \\ Michel Marcus, Sep 10 2021

T(n, k) = 2*A065305(n, k) [but note different offset].

A088685 Records for the sum-of-primes function sopfr(n) if sopfr(prime) is taken to be 0.

Original entry on oeis.org

0, 4, 5, 6, 7, 9, 10, 13, 15, 19, 21, 25, 31, 33, 39, 43, 45, 49, 55, 61, 63, 69, 73, 75, 81, 85, 91, 99, 103, 105, 109, 111, 115, 129, 133, 139, 141, 151, 153, 159, 165, 169, 175, 181, 183, 193, 195, 199, 201, 213, 225, 229, 231, 235, 241, 243, 253, 259, 265, 271

Offset: 1

Eric W. Weisstein, Oct 05 2003

nonn

A048974, A052147 and A067187 are very similar after dropping terms less than 13. - Eric W. Weisstein, Oct 10 2003

Robert Price, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Sum of Prime Factors

Cf. A001414, A048974, A052147, A067187, A088686.

Union@ FoldList[Max, Table[Total@ Flatten@ Map[ConstantArray[#1, #2] /. 1 -> 0 & @@ # &, FactorInteger@ n] - n Boole[PrimeQ@ n], {n, 540}]] (* Michael De Vlieger, Jun 29 2017 *)

sopfr(k) = my(f=factor(k)); sum(j=1, #f~, f[j, 1]*f[j, 2]);
lista(nn) = {my(record = -1); for (n=1, nn, if (! isprime(n), if ((x=sopfr(n)) > record, record = x; print1(record, ", "));););} \\ Michel Marcus, Jun 29 2017

from sympy import factorint, isprime
def sopfr(n):
    f=factorint(n)
    return sum([i*f[i] for i in f])
l=[]
record=-1
for n in range(1, 501):
    if not isprime(n):
        x=sopfr(n)
        if x>record:
            record=x
            l.append(record)
print(l) # Indranil Ghosh, Jun 29 2017

A139690 a(n) = A109611(n) + 2.

Original entry on oeis.org

4, 5, 7, 9, 13, 15, 19, 21, 25, 31, 33, 39, 43, 49, 55, 61, 69, 73, 85, 91, 103, 109, 111, 115, 129, 133, 139, 141, 151, 159, 169, 181, 183, 193, 199, 201, 213, 229, 235, 241, 253, 259, 265, 271, 283, 295, 309, 313, 319, 339, 349, 355, 361, 381, 391, 403, 411

Offset: 1

Reinhard Zumkeller, Apr 29 2008

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Chen Prime

Intersection of A052147 and A037143; A006512 is a subsequence.

Cf. A109611.

Cases[Import["https://oeis.org/A109611/b109611.txt", "Table"], {, }][[All, 2]] + 2 (* Robert Price, Apr 19 2025 *)

list(lim)=my(v=List(),t); forprime(p=2,lim\2, forprime(q=2,min(p,lim\p), if(isprime(t=p*q-2), listput(v,t+2)))); t=2; forprime(p=3,lim, if(p-t==2, listput(v,p)); t=p); Set(v) \\ Charles R Greathouse IV, Jan 19 2017

A010051(a(n)) = A139689(n); A064911(a(n)) = 1 - A139689(n);

A001222(a(n)) = 2 - A139689(n).

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cp's OEIS Frontend

A175224 a(n) = prime(n) + 8.

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A175225 a(n) = prime(n) + 10.

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A216620 Square array read by antidiagonals: T(n,k) = Sum_{c|n,d|k} phi(gcd(c,d)) for n>=1, k>=1.

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A048974 Odd numbers that are the sum of 2 primes.

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A157931 Numbers that are both the sum and the product of two primes.

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A117530 Triangle read by rows: T(n,k) = k^2 - k + prime(n), 1<=k<=n.

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A049234 Number of divisors of prime(n) + 2.

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