A108605 -id:A108605 - cp's OEIS frontend

A262626 Visible parts of the perspective view of the stepped pyramid whose structure essentially arises after the 90-degree-zig-zag folding of the isosceles triangle A237593.

1, 1, 1, 3, 2, 2, 2, 2, 2, 1, 1, 2, 7, 3, 1, 1, 3, 3, 3, 3, 2, 2, 3, 12, 4, 1, 1, 1, 1, 4, 4, 4, 4, 2, 1, 1, 2, 4, 15, 5, 2, 1, 1, 2, 5, 5, 3, 5, 5, 2, 2, 2, 2, 5, 9, 9, 6, 2, 1, 1, 1, 1, 2, 6, 6, 6, 6, 3, 1, 1, 1, 1, 3, 6, 28, 7, 2, 2, 1, 1, 2, 2, 7, 7, 7, 7, 3, 2, 1, 1, 2, 3, 7, 12, 12, 8, 3, 1, 2, 2, 1, 3, 8, 8, 8, 8, 8, 3, 2, 1, 1

Offset: 1

Omar E. Pol, Sep 26 2015

Also the rows of both triangles A237270 and A237593 interleaved.
Also, irregular triangle read by rows in which T(n,k) is the area of the k-th region (from left to right in ascending diagonal) of the n-th symmetric set of regions (from the top to the bottom in descending diagonal) in the two-dimensional diagram of the perspective view of the infinite stepped pyramid described in A245092 (see the diagram in the Links section).
The diagram of the symmetric representation of sigma is also the top view of the pyramid, see Links section. For more information about the diagram see also A237593 and A237270.
The number of cubes at the n-th level is also A024916(n), the sum of all divisors of all positive integers <= n.
Note that this pyramid is also a quarter of the pyramid described in A244050. Both pyramids have infinitely many levels.
Odd-indexed rows are also the rows of the irregular triangle A237270.
Even-indexed rows are also the rows of the triangle A237593.
Lengths of the odd-indexed rows are in A237271.
Lengths of the even-indexed rows give 2*A003056.
Row sums of the odd-indexed rows gives A000203, the sum of divisors function.
Row sums of the even-indexed rows give the positive even numbers (see A005843).
Row sums give A245092.
From the front view of the stepped pyramid emerges a geometric pattern which is related to A001227, the number of odd divisors of the positive integers.
The connection with the odd divisors of the positive integers is as follows: A261697 --> A261699 --> A237048 --> A235791 --> A237591 --> A237593 --> A237270 --> this sequence.

			Irregular triangle begins:
  1;
  1, 1;
  3;
  2, 2;
  2, 2;
  2, 1, 1, 2;
  7;
  3, 1, 1, 3;
  3, 3;
  3, 2, 2, 3;
  12;
  4, 1, 1, 1, 1, 4;
  4, 4;
  4, 2, 1, 1, 2, 4;
  15;
  5, 2, 1, 1, 2, 5;
  5, 3, 5;
  5, 2, 2, 2, 2, 5;
  9, 9;
  6, 2, 1, 1, 1, 1, 2, 6;
  6, 6;
  6, 3, 1, 1, 1, 1, 3, 6;
  28;
  7, 2, 2, 1, 1, 2, 2, 7;
  7, 7;
  7, 3, 2, 1, 1, 2, 3, 7;
  12, 12;
  8, 3, 1, 2, 2, 1, 3, 8;
  8, 8, 8;
  8, 3, 2, 1, 1, 1, 1, 2, 3, 8;
  31;
  9, 3, 2, 1, 1, 1, 1, 2, 3, 9;
  ...
Illustration of the odd-indexed rows of triangle as the diagram of the symmetric representation of sigma which is also the top view of the stepped pyramid:
.
   n  A000203    A237270    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
   1     1   =      1      |_| | | | | | | | | | | | | | | |
   2     3   =      3      |_ _|_| | | | | | | | | | | | | |
   3     4   =    2 + 2    |_ _|  _|_| | | | | | | | | | | |
   4     7   =      7      |_ _ _|    _|_| | | | | | | | | |
   5     6   =    3 + 3    |_ _ _|  _|  _ _|_| | | | | | | |
   6    12   =     12      |_ _ _ _|  _| |  _ _|_| | | | | |
   7     8   =    4 + 4    |_ _ _ _| |_ _|_|    _ _|_| | | |
   8    15   =     15      |_ _ _ _ _|  _|     |  _ _ _|_| |
   9    13   =  5 + 3 + 5  |_ _ _ _ _| |      _|_| |  _ _ _|
  10    18   =    9 + 9    |_ _ _ _ _ _|  _ _|    _| |
  11    12   =    6 + 6    |_ _ _ _ _ _| |  _|  _|  _|
  12    28   =     28      |_ _ _ _ _ _ _| |_ _|  _|
  13    14   =    7 + 7    |_ _ _ _ _ _ _| |  _ _|
  14    24   =   12 + 12   |_ _ _ _ _ _ _ _| |
  15    24   =  8 + 8 + 8  |_ _ _ _ _ _ _ _| |
  16    31   =     31      |_ _ _ _ _ _ _ _ _|
  ...
The above diagram arises from a simpler diagram as shown below.
Illustration of the even-indexed rows of triangle as the diagram of the deployed front view of the corner of the stepped pyramid:
.
.                                 A237593
Level                               _ _
1                                 _|1|1|_
2                               _|2 _|_ 2|_
3                             _|2  |1|1|  2|_
4                           _|3   _|1|1|_   3|_
5                         _|3    |2 _|_ 2|    3|_
6                       _|4     _|1|1|1|1|_     4|_
7                     _|4      |2  |1|1|  2|      4|_
8                   _|5       _|2 _|1|1|_ 2|_       5|_
9                 _|5        |2  |2 _|_ 2|  2|        5|_
10              _|6         _|2  |1|1|1|1|  2|_         6|_
11            _|6          |3   _|1|1|1|1|_   3|          6|_
12          _|7           _|2  |2  |1|1|  2|  2|_           7|_
13        _|7            |3    |2 _|1|1|_ 2|    3|            7|_
14      _|8             _|3   _|1|2 _|_ 2|1|_   3|_             8|_
15    _|8              |3    |2  |1|1|1|1|  2|    3|              8|_
16   |9                |3    |2  |1|1|1|1|  2|    3|                9|
...
The number of horizontal line segments in the n-th level in each side of the diagram equals A001227(n), the number of odd divisors of n.
The number of horizontal line segments in the left side of the diagram plus the number of the horizontal line segment in the right side equals A054844(n).
The total number of vertical line segments in the n-th level of the diagram equals A131507(n).
The diagram represents the first 16 levels of the pyramid.
The diagram of the isosceles triangle and the diagram of the top view of the pyramid shows the connection between the partitions into consecutive parts and the sum of divisors function (see also A286000 and A286001). - _Omar E. Pol_, Aug 28 2018
The connection between the isosceles triangle and the stepped pyramid is due to the fact that this object can also be interpreted as a pop-up card. - _Omar E. Pol_, Nov 09 2022

Robert Price, Table of n, a(n) for n = 1..16048 (n=1..412 rows)
Omar E. Pol, An infinite stepped pyramid (A237593, A237270, A262626)
Omar E. Pol, Diagram of the isosceles triangle A237593 before the 90-degree-zig-zag folding (rows: 1..28)
Omar E. Pol, Perspective view of the stepped pyramid (levels: 1..16)
Index entries for sequences related to sigma(n)

Famous sequences that are visible in the stepped pyramid:
Cf. A000040 (prime numbers)......., for the characteristic shape see A346871.
Cf. A000079 (powers of 2)........., for the characteristic shape see A346872.
Cf. A000203 (sum of divisors)....., total area of the terraces in the n-th level.
Cf. A000217 (triangular numbers).., for the characteristic shape see A346873.
Cf. A000225 (Mersenne numbers)...., for a visualization see A346874.
Cf. A000384 (hexagonal numbers)..., for the characteristic shape see A346875.
Cf. A000396 (perfect numbers)....., for the characteristic shape see A346876.
Cf. A000668 (Mersenne primes)....., for a visualization see A346876.
Cf. A001097 (twin primes)........., for a visualization see A346871.
Cf. A001227 (# of odd divisors)..., number of subparts in the n-th level.
Cf. A002378 (oblong numbers)......, for a visualization see A346873.
Cf. A008586 (multiples of 4)......, perimeters of the successive levels.
Cf. A008588 (multiples of 6)......, for the characteristic shape see A224613.
Cf. A013661 (zeta(2))............., (area of the horizontal faces)/(n^2), n -> oo.
Cf. A014105 (second hexagonals)..., for the characteristic shape see A346864.
Cf. A067742 (# of middle divisors), # cells in the main diagonal in n-th level.
Other sequences that are visible in the stepped pyramid: A000096, A001065, A001359, A001747, A002939, A002943, A003056, A004125, A004277, A004526, A005279, A006512, A007606, A007607, A082647, A008438, A008578, A008864, A010814, A014106, A014107, A014132, A014574, A016945, A019434, A024206, A024916, A028552, A028982, A028983, A034856, A038550, A047836, A048050, A052928, A054735, A054844, A062731, A065091, A065475, A071561, A071562, A071904, A092506, A100484, A108605, A139256, A139257, A144396, A152677, A152678, A153485, A155085, A161680, A161983, A162917, A174905, A174973, A175254, A176810, A224880, A235791, A237270, A237271, A237591, A237593, A238005, A238524, A244049, A245092, A259176, A259177, A261348, A278972, A317302, A317303, A317304, A317305, A317307, A319529, A319796, A319801, A319802, A327329, A336305, (and several others).
Apart from zeta(2) other constants that are related to the stepped pyramid are A072691, A353908, A354238.
Cf. A054844, A131507, A196020, A236104, A237048, A239660, A244050, A259179, A261350, A261697, A261699, A262612, A280850, A286000, A286001, A296508.

A157037 Numbers with prime arithmetic derivative A003415.

Original entry on oeis.org

6, 10, 22, 30, 34, 42, 58, 66, 70, 78, 82, 105, 114, 118, 130, 142, 154, 165, 174, 182, 202, 214, 222, 231, 238, 246, 255, 273, 274, 282, 285, 286, 298, 310, 318, 345, 357, 358, 366, 370, 382, 385, 390, 394, 399, 418, 430, 434, 442, 454, 455, 465, 474, 478

Offset: 1

Reinhard Zumkeller, Feb 22 2009

nonn

Equivalently, solutions to n'' = 1, since n' = 1 iff n is prime. Twice the lesser of the twin primes, 2*A001359 = A108605, are a subsequence. - M. F. Hasler, Apr 07 2015

All terms are squarefree, because if there would be a prime p whose square p^2 would divide n, then A003415(n) = (A003415(p^2) * (n/p^2)) + (p^2 * A003415(n/p^2)) = p*[(2 * (n/p^2)) + (p * A003415(n/p^2))], which certainly is not a prime. - Antti Karttunen, Oct 10 2019

			A003415(42) = A003415(2*3*7) = 2*3+3*7+7*2 = 41 = A000040(13), therefore 42 is a term.

Antti Karttunen, Table of n, a(n) for n = 1..10001 (first 1000 terms from Reinhard Zumkeller)

Cf. A003415, A010051, A038554, A192082, A192189, A192190,  A327978, A328233, A328240, A328384, A328385.
Cf. A189441 (primes produced by these numbers), A241859.
Cf. A192192, A328239 (numbers whose 2nd and numbers whose 3rd arithmetic derivative is prime).
Cf. A108605, A256673 (subsequences).
Subsequence of following sequences: A005117, A099308, A235991, A328234 (A328393), A328244, A328321.

a157037 n = a157037_list !! (n-1)
a157037_list = filter ((== 1) . a010051' . a003415) [1..]
-- Reinhard Zumkeller, Apr 08 2015

dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Total[n*f[[2]]/f[[1]]]]]; Select[Range[500], dn[dn[#]] == 1 &] (* T. D. Noe, Mar 07 2013 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA157037(n) = isprime(A003415(n)); \\ Antti Karttunen, Oct 19 2019

from itertools import count, islice
from sympy import isprime, factorint
def A157037_gen(): # generator of terms
    return filter(lambda n:isprime(sum(n*e//p for p,e in factorint(n).items())), count(2))
A157037_list = list(islice(A157037_gen(),20)) # Chai Wah Wu, Jun 23 2022

A010051(A003415(a(n))) = 1; A068346(a(n)) = 1; A099306(a(n)) = 0.

A003415(a(n)) = A328385(a(n)) = A241859(n); A327969(a(n)) = 3. - Antti Karttunen, Oct 19 2019

A075860 a(n) is the fixed point reached when the map x -> A008472(x) is iterated, starting from x = n, with the convention a(1)=0.

Original entry on oeis.org

0, 2, 3, 2, 5, 5, 7, 2, 3, 7, 11, 5, 13, 3, 2, 2, 17, 5, 19, 7, 7, 13, 23, 5, 5, 2, 3, 3, 29, 7, 31, 2, 3, 19, 5, 5, 37, 7, 2, 7, 41, 5, 43, 13, 2, 5, 47, 5, 7, 7, 7, 2, 53, 5, 2, 3, 13, 31, 59, 7, 61, 3, 7, 2, 5, 2, 67, 19, 2, 3, 71, 5, 73, 2, 2, 7, 5, 5, 79, 7, 3, 43, 83, 5, 13, 2, 2, 13, 89

Offset: 1

Joseph L. Pe, Oct 15 2002

For n>1, the sequence reaches a fixed point, which is prime.
From Robert Israel, Mar 31 2020: (Start)
a(n) = n if n is prime.
a(n) = n/2 + 2 if n is in A108605.
a(n) = n/4 + 2 if n is in 4*A001359. (End)

			Starting with 60 = 2^2 * 3 * 5 as the first term, add the prime factors of 60 to get the second term = 2 + 3 + 5 = 10. Then add the prime factors of 10 = 2 * 5 to get the third term = 2 + 5 = 7, which is prime. (Successive terms of the sequence will be equal to 7.) Hence a(60) = 7.

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

Cf. A008472 (sum of prime divisors of n), A029908.

f:= proc(n) option remember;
  if isprime(n) then n
  else procname(convert(numtheory:-factorset(n), `+`))
  fi
end proc:
f(1):= 0:
map(f, [$1..100]); # Robert Israel, Mar 31 2020

f[n_] := Module[{a}, a = n; While[ !PrimeQ[a], a = Apply[Plus, Transpose[FactorInteger[a]][[1]]]]; a]; Table[f[i], {i, 2, 100}]
(* Second program: *)
a[n_] := If[n == 1, 0, FixedPoint[Total[FactorInteger[#][[All, 1]]]&, n]];
Array[a, 100] (* Jean-François Alcover, Apr 01 2020 *)

fp(n, pn) = if (n == pn, n, fp(vecsum(factor(n)[, 1]), n));
a(n) = if (n==1, 0, fp(n, 0)); \\ Michel Marcus, Sep 02 2023

from sympy import primefactors
def a(n, pn):
    if n == pn:
        return n
    else:
        return a(sum(primefactors(n)), n)
print([a(i, None) for i in range(1, 100)]) # Gleb Ivanov, Nov 05 2021

Better description from Labos Elemer, Apr 09 2003

Name clarified by Michel Marcus, Sep 02 2023

A114522 Numbers n such that sum of distinct prime divisors of n is prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 22, 23, 24, 25, 27, 29, 31, 32, 34, 36, 37, 40, 41, 43, 44, 47, 48, 49, 50, 53, 54, 58, 59, 61, 64, 67, 68, 71, 72, 73, 79, 80, 81, 82, 83, 88, 89, 96, 97, 100, 101, 103, 107, 108, 109, 113, 116, 118, 121, 125, 127

Offset: 1

Leroy Quet, Dec 05 2005

nonn

Sequence is the union of the primes and sequence A047820.

			24 = 2^3 * 3 and 2 + 3 = 5, which is prime. So 24 is included.

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

Cf. A000079, A003586, A003592.

Cf. A008472, A047820, A108605, A114518.

[k:k in [2..150]| IsPrime(&+PrimeDivisors(k))]; // Marius A. Burtea, Oct 06 2019

f[n_] := Plus @@ First /@ FactorInteger[n]; Select[Range[130], PrimeQ[f[ # ]] &] (* Ray Chandler, Dec 07 2005 *)
Select[Range@127, PrimeQ[Plus @@ First /@ FactorInteger@# ] &] (* Robert G. Wilson v, Dec 07 2005 *)

for(n=1, 200, v=factor(n); s=0; for(i=1,matsize(v)[1],s+=v[i,1]); if(isprime(s), print1(n, ", "))) \\ Lambert Herrgesell (zero815(AT)googlemail.com), Dec 07 2005

Extended by Robert G. Wilson v, Ray Chandler and Lambert Herrgesell (zero815(AT)googlemail.com), Dec 07 2005

A347236 a(n) = Sum_{d|n} A061019(d) * A003961(n/d), where A061019 negates the primes in the prime factorization, while A003961 shifts the factorization one step towards larger primes.

Original entry on oeis.org

1, 1, 2, 7, 2, 2, 4, 13, 19, 2, 2, 14, 4, 4, 4, 55, 2, 19, 4, 14, 8, 2, 6, 26, 39, 4, 68, 28, 2, 4, 6, 133, 4, 2, 8, 133, 4, 4, 8, 26, 2, 8, 4, 14, 38, 6, 6, 110, 93, 39, 4, 28, 6, 68, 4, 52, 8, 2, 2, 28, 6, 6, 76, 463, 8, 4, 4, 14, 12, 8, 2, 247, 6, 4, 78, 28, 8, 8, 4, 110, 421, 2, 6, 56, 4, 4, 4, 26, 8, 38, 16

Offset: 1

Antti Karttunen, Aug 24 2021

Dirichlet convolution of A003961 and A061019.
Dirichlet convolution of A003973 and A158523.
Multiplicative because A003961 and A061019 are.
All terms are positive because all terms of A347237 are nonnegative and A347237(1) = 1.
Union of sequences A001359 and A108605 (= 2*A001359) seems to give the positions of 2's in this sequence.

Cf. A000040, A001223, A001359, A003961, A003973, A061019, A108605, A158523, A347237 (Möbius transform), A347238 (Dirichlet inverse), A347239.
Cf. also A347136.
Cf. A151800.

f[p_, e_] := ((np = NextPrime[p])^(e + 1) - (-p)^(e + 1))/(np + p); a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 02 2021 *)

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A061019(n) = (((-1)^bigomega(n))*n);
A347236(n) = sumdiv(n,d,A061019(d)*A003961(n/d));

a(n) = Sum_{d|n} A003961(n/d) * A061019(d).
a(n) = Sum_{d|n} A003973(n/d) * A158523(d).
a(n) = Sum_{d|n} A347237(d).
a(n) = A347239(n) - A347238(n).
For all n >= 1, a(A000040(n)) = A001223(n).
Multiplicative with a(p^e) = (A151800(p)^(e+1)-(-p)^(e+1))/(A151800(p)+p). - Sebastian Karlsson, Sep 02 2021

A108606 Semiprimes with prime sum of digits.

Original entry on oeis.org

14, 21, 25, 34, 38, 49, 58, 65, 74, 85, 94, 106, 111, 115, 119, 122, 133, 142, 146, 155, 166, 201, 203, 205, 209, 214, 218, 221, 247, 254, 265, 274, 278, 287, 289, 298, 302, 319, 326, 335, 346, 355, 362, 371, 377, 382, 386, 391, 395, 403, 407, 427, 445, 454

Offset: 1

Zak Seidov, Jun 12 2005

34 is the smallest term in common with A108605.

			34 = 2*17 (semiprime) and 2 + 17 = 19 is prime.

J.W.L. (Jan) Eerland, Table of n, a(n) for n = 1..10000

Cf. A001358 (semiprimes), A101605 (3-almost primes), A108605 (semiprimes with prime sum of factors), A108607 (intersection of A108605 and A108606).

A108606=Select[Range[1000], Plus@@(Transpose[FactorInteger[ # ]])[[2]]==2&& PrimeQ[Plus@@IntegerDigits[ # ]]&]
DeleteCases[ParallelTable[If[PrimeOmega[n]==2&&PrimeQ[Total[IntegerDigits[n]]],n,a],{n,0,126181}],a] (* J.W.L. (Jan) Eerland, Dec 21 2021 *)

select(isA108606(n)={bigomega(n)==2&&isprime(sumdigits(n))},[1..1000]) \\ J.W.L. (Jan) Eerland, Dec 23 2021

from sympy import isprime, factorint
def ok(n): return isprime(sum(map(int, str(n)))) and sum(factorint(n).values()) == 2
print([k for k in range(455) if ok(k)]) # Michael S. Branicky, Aug 22 2022

A176810 Semiprimes of the form 2 * (greater of twin primes).

Original entry on oeis.org

10, 14, 26, 38, 62, 86, 122, 146, 206, 218, 278, 302, 362, 386, 398, 458, 482, 542, 566, 626, 698, 842, 866, 926, 1046, 1142, 1202, 1238, 1286, 1322, 1622, 1646, 1658, 1718, 1766, 2042, 2066, 2102, 2126, 2186, 2306, 2462, 2558, 2582, 2606, 2642, 2858, 2906

Offset: 1

Juri-Stepan Gerasimov, Apr 26 2010

nonn

Cf. A001359, A006512, A108605.

a(n)=2*A006512(n).

Corrected (122 inserted, 286 replaced by 386, 2202 replaced by 2102 etc.) by R. J. Mathar, Aug 12 2010

Definition corrected by Zak Seidov, May 06 2013

A288189 a(n) is the smallest composite number whose sum of prime divisors (with multiplicity) is divisible by prime(n).

Original entry on oeis.org

4, 8, 6, 10, 28, 22, 52, 34, 76, 184, 58, 213, 148, 82, 172, 309, 424, 118, 393, 268, 142, 584, 316, 664, 573, 388, 202, 412, 214, 436, 753, 508, 813, 274, 1465, 298, 933, 974, 652, 1336, 1384, 358, 1137, 382, 772, 394, 1257, 1329, 892, 454, 916, 1864, 478, 1497, 1538, 1569

Offset: 1

David James Sycamore, Jul 01 2017

nonn

In most cases a(n) = A288814(prime(n)) but there are exceptions, e.g., a(37)=213, whereas A288814(37)=248. Other exceptions include a(53), a(67), a(127), a(137), etc. These examples occur when there is a number r such that A001414(r*p) is less than A288814(p).

The strictly increasing subsequence of terms (10, 22, 34, 58, 82, 118, 142, 202, 214, 274, 298, ...) where for all m>n, a(m)>a(n) gives the semiprimes with prime sum of prime factors, A108605. The sequence of the indices of this subsequence (5, 7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, ...) gives the greater of twin primes, A006512.

			a(5)=6 because 6 = 2*3 is the smallest number whose sum of prime divisors (2+3 = 5) is divisible by 5.
a(37) = 213 = A288814(74) = A288814(2*37).

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

Cf. A000040, A001414, A006512, A056240, A108605, A288814.

With[{s = Array[Boole[CompositeQ@ #] Total@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger@ #] &, 10^4] /. 0 -> ""}, Table[FirstPosition[s, ?(Mod[#, p] == 0 &)][[1]], {p, Prime@ Range@ 56}]] (* _Michael De Vlieger, Apr 14 2018 *)

sopfr(k) = my(f=factor(k)); sum(j=1, #f~, f[j, 1]*f[j, 2]);
a(n) = my(pn=prime(n)); forcomposite(c=pn, , if (sopfr(c) % pn == 0, return(c))); \\ Michel Marcus, Jul 03 2017

A278972 Twice the twin primes.

Original entry on oeis.org

6, 10, 14, 22, 26, 34, 38, 58, 62, 82, 86, 118, 122, 142, 146, 202, 206, 214, 218, 274, 278, 298, 302, 358, 362, 382, 386, 394, 398, 454, 458, 478, 482, 538, 542, 562, 566, 622, 626, 694, 698, 838, 842, 862, 866, 922, 926, 1042, 1046, 1138, 1142, 1198, 1202, 1234, 1238, 1282, 1286

Offset: 1

Omar E. Pol, Dec 02 2016

nonn

This is one of the sequences found in the pyramid described in A245092. For more information about the pyramid see A237593 and A262626.

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

Union of A108605 and A176810.
Subsequence of A100484.
Cf. A001097, A001359, A006512, A114907, A237593, A262626.

2Union[Flatten[Select[Partition[Prime[Range[150]],2,1],#[[2]]-#[[1]]==2&]]] (* Harvey P. Dale, Apr 18 2022 *)

a(n) = 2 * A001097(n).

A339437 Numbers k such that A339436(k) is prime.

Original entry on oeis.org

6, 10, 22, 34, 58, 82, 118, 142, 202, 214, 216, 252, 274, 298, 330, 358, 382, 390, 394, 454, 468, 478, 490, 538, 562, 588, 622, 684, 690, 694, 726, 798, 838, 858, 862, 870, 910, 922, 924, 1042, 1044, 1122, 1138, 1176, 1198, 1210, 1224, 1234, 1254, 1282, 1290, 1318, 1332, 1440, 1482, 1518, 1540

Offset: 1

J. M. Bergot and Robert Israel, Dec 04 2020

nonn

All terms are even.

			a(15)=330 is a member because 330 = 2*3*5*11 and A339436(330) = 2 + 2*3 + 2*3*5 + 3*5*11 + 5*11 + 11 = 269 is prime.

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

Includes A108605. Disjoint from A014612.

Cf. A339436.

A339436:= proc(n) local L,m;
  L:= sort(map(t -> t[1]$t[2],ifactors(n)[2]));
  m:= nops(L);
  add(mul(L[i],i=1..j)+mul(L[i],i=j+1..m),j=1..m-1)
end proc:
select(isprime @ A339436, [seq(i,i=2 .. 10000, 2)]);

cp's OEIS Frontend

A262626 Visible parts of the perspective view of the stepped pyramid whose structure essentially arises after the 90-degree-zig-zag folding of the isosceles triangle A237593.

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A157037 Numbers with prime arithmetic derivative A003415.

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A075860 a(n) is the fixed point reached when the map x -> A008472(x) is iterated, starting from x = n, with the convention a(1)=0.

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A114522 Numbers n such that sum of distinct prime divisors of n is prime.

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A347236 a(n) = Sum_{d|n} A061019(d) * A003961(n/d), where A061019 negates the primes in the prime factorization, while A003961 shifts the factorization one step towards larger primes.

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A108606 Semiprimes with prime sum of digits.

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A176810 Semiprimes of the form 2 * (greater of twin primes).

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