A001748 -id:A001748 - cp's OEIS frontend

A196934 a(n) is the first occurrence of n in sequence A078498.

5, 8, 18, 14, 25, 38, 43, 50, 61, 48, 132, 167, 100, 88, 151, 217, 176, 216, 270, 214, 300, 785, 429, 687, 308, 1083, 374, 644, 713, 320, 840, 608, 654, 577, 1005, 1409, 1631, 1215, 928, 1386, 2304, 1984, 1203, 2336, 853, 1638, 1899, 1806, 1974, 1594, 1228

Offset: 1

Lei Zhou, Oct 07 2011

nonn

Conjecture: Any prime number greater than 11 (p) can be the center term of arithmetic progressions prime chain p-6k, p, p+6k, while k>0.
a(n) is also the maximum number k that is needed to find a p(i)-6k, p(i), p(i)+6k kind of arithmetic progressions prime chain for all i <= n, while p(i) is the i-th prime number.
The Mathematica program gives the first 51 items.

			A078498(5)=1 (take the offset 5),  so a(1)=5;
2 first occurs as A078498(8), so a(2)=8;

Definition of Arithmetic Progressions of Primes

Cf. A078498, A078497, A001748.

max = 51; Array[fa, max]; Do[fa[i] = 0, {i, 1, max}]; ct = 0; i = 4; While[ct < max, i++; p = Prime[i]; j = 0; While[j++; df = 6*j; ! ((PrimeQ[p + df]) && (PrimeQ[p - df]))]; If[j <= max, If[fa[j] == 0, fa[j] = i; ct++]]]; Table[fa[i], {i, 1, max}]

A263648 a(1)=4, a(2)=9: a(n) is the smallest semiprime not yet appearing in the sequence which is coprime to a(n-1) and not coprime to a(n-2).

Original entry on oeis.org

4, 9, 10, 21, 22, 15, 14, 25, 6, 35, 26, 49, 34, 77, 38, 33, 46, 39, 58, 51, 62, 57, 74, 69, 82, 87, 86, 93, 94, 111, 106, 123, 118, 129, 122, 141, 134, 159, 142, 177, 146, 183, 158, 201, 166, 213, 178, 219, 194, 237, 202, 249, 206, 267, 214, 291, 218, 303, 226, 309

Offset: 1

Bob Selcoe, Oct 22 2015

nonn

Contrary to what might be expected (see comment after Proof), a(n) is not a permutation of all semiprimes; it is a permutation of even semiprimes {S_2} and semiprimes with smallest factor 3 {S_3}, plus {25, 35, 49, 77}. Proof (Start):
i. The sequence is infinite: we can always consider p*q for a(n+1), where p is the smallest factor in a(n-1) and q is the smallest prime > than the largest factor of any term already appearing in the sequence;
ii. a(15)=38 = 2*19 = 2*prime(8) and a(16)=33 = 3*11 = 3*prime(5);
iii. all {S_2} <= 38 and {S_3} <= 33 have appeared up to a(16), with 38 and 33 being maximum terms in {S_2} and {S_3}, respectively;
iv. all semiprimes with smallest factor >= 5 which are < 2*prime(9)=46 and 3*prime(6)=39 have appeared up to a(16). Consequently, the terms starting at a(17)=46 alternate between 2*prime(k) and 3*prime(k-3) k=9..infinity.
v. the only other numbers to have appeared are {25, 35, 49, 77}.
(End)
The above behavior is in contrast to A119718 (a permutation of all semiprimes because it lacks the constraint of a(n) being not coprime to a(n-2)). Interestingly, this sequence (A263648) shares the same essential rules as A098550 (the Yellowstone permutation) and many of its variations, while A119718 does not; one therefore might expect the opposite behavior to occur between this sequence and A119718. What observations or generalizations might we draw from this?

Michael De Vlieger, Table of n, a(n) for n = 1..10000
David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, The Yellowstone Permutation, arXiv preprint arXiv:1501.01669 [math.NT], 2015. Also Journal of Integer Sequences, Vol. 18 (2015), Article 15.6.7
Michael De Vlieger, Annotated scatterplot of (n, k), where k is the position of a(n) in A001358, for n = 1..48. Color code: even a(n) in red, 3 | a(n) in blue, 6 | a(n) in purple, else black.
Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^18, showing even terms in red, 3 | a(n) in blue, 6 | a(n) in purple, else black. The first 15 terms are labeled.

Cf. A098550, A119718, A000040 (primes), A100484 (even semiprimes), A001748 (essentially, semiprimes with smallest factor 3).

a[1]=4; a[2]=9; a[n_] := a[n] = Module[{k}, For[k=6, True, k++, If[MatchQ[ FactorInteger[k], {{, 1}, {, 1}}|{{, 2}}] && FreeQ[Array[a, n-1], k] && CoprimeQ[k, a[n-1]] && !CoprimeQ[k, a[n-2]], Return[k]]]]; Array[a, 60] (* _Jean-François Alcover, Oct 06 2018  *)

For n >= 15:
a(n) = 2*prime((n+1)/2) when n is odd;
a(n) = 3*prime(n/2-3) when n is even.

More terms from Jean-François Alcover, Oct 06 2018

A344872 Semiprimes of the form 3m+2.

Original entry on oeis.org

14, 26, 35, 38, 62, 65, 74, 77, 86, 95, 119, 122, 134, 143, 146, 155, 158, 161, 185, 194, 203, 206, 209, 215, 218, 221, 254, 278, 287, 299, 302, 305, 314, 323, 326, 329, 335, 341, 362, 365, 371, 377, 386, 395, 398, 407, 413, 422, 437, 446, 458, 473, 482, 485, 497

Offset: 1

Peter Munn, May 31 2021

There are no square terms, as squares are congruent to 0 or 1 modulo 3.

Products of a prime of the form 3m+1 and a prime of the form 3m+2 (the former necessarily being of the form 6m+1).

			14 = 2 * 7 has 2 prime factors (counting repetitions) so is a semiprime, and 14 = 3*4 + 2, so has the form 3m+2. So 14 is in the sequence.

Intersection of A001358 and A016789.
Disjoint union of A108172 and A112772.
Complement within A001358 of A001748, A112771 and A112774.
Subsequence of A344703.

Select[Range[2,500,3],PrimeOmega@#==2&] (* Giorgos Kalogeropoulos, Jun 02 2021 *)

isok(m) = bigomega(m) == 2 && m % 3 == 2;

A129758 Smallest prime p such that there are primes q and r with the property that p, q and r form an arithmetic progression and their sum is the same as three times the (n+2)-nd prime number.

Original entry on oeis.org

3, 3, 5, 7, 11, 7, 17, 17, 19, 31, 29, 19, 41, 47, 47, 43, 61, 59, 67, 61, 59, 71, 67, 89, 97, 101, 79, 89, 103, 113, 107, 127, 131, 139, 151, 127, 137, 167, 167, 163, 149, 163, 167, 157, 199, 163, 197, 181, 227, 227, 211, 239, 251, 257, 257, 229, 271, 269

Offset: 1

Giovanni Teofilatto, May 15 2007

The same selection rule as in A078497 applies: if there is more than one prime triple (p,q=p+d,r=q+d) with p+q+r=A001748(n), take p from the triple with minimum d. - R. J. Mathar, May 19 2007

			3 + 5 + 7 = 15, which is three times the (1+2)th prime number. Thus a(1) = 3.

Cf. A078497, A071681, A078611.

A129758 := proc(n) local p3, i,d,r,p; p3 := ithprime(n) ; i := n+1 ; while true do r := ithprime(i) ; d := r-p3 ; p := p3-d ; if isprime(p) then RETURN(p) ; fi ; i := i+1 ; od ; RETURN(-1) ; end: for n from 3 to 60 do printf("%d, ",A129758(n)) ; od ; # R. J. Mathar, May 19 2007

a[n_]:=Module[{},k=1; While[Not[PrimeQ[Prime[n+1]-k] && PrimeQ[Prime[n+1]+k]], k++ ]; Prime[n + 1] - k]; Table[a[n], {n, 2, 60}]

A078497(n)-prime(n)=prime(n)-a(n)=d. - R. J. Mathar, May 19 2007

Conjecture: Limit_{N->oo} (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} prime(n+2)) = 1. - Alain Rocchelli, May 01 2024

Edited and extended by R. J. Mathar, Giovanni Teofilatto and Stefan Steinerberger, May 19 2007

A163628 Integers such that the two adjacent integers are a prime and three times a prime.

Original entry on oeis.org

8, 10, 14, 16, 20, 22, 32, 38, 40, 52, 58, 68, 70, 88, 110, 112, 128, 130, 140, 158, 178, 182, 200, 212, 238, 250, 268, 292, 308, 310, 338, 380, 382, 410, 418, 448, 488, 490, 500, 502, 520, 542, 572, 578, 592, 598, 632, 682, 700, 718, 752, 770, 772, 788, 808

Offset: 1

Juri-Stepan Gerasimov, Aug 02 2009

nonn

Union[3*A023208 + 1, 3*A088878 - 1]. [Zak Seidov, Aug 07 2009]

			a(1)=8 which lies between 7=A000040(4) and 9 = A001748(2).
a(2)=10 which lies between 9=A001748(2) and 11 = A000040(5).

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

Cf. A000040, A163492.

n = 1; A023208 = {}; Do[If[PrimeQ[(Prime[k] - 2 n)/(2 n + 1)], AppendTo[A023208, (Prime[k] - 2 n)/(2 n + 1)]], {k, 1, 1000}]; A023208;
A088878 = {}; Do[p = Prime[n]; If[PrimeQ[3*p - 2], AppendTo[A088878, p]], {n, 5!}]; A088878; Union[3*A023208 + 1, 3*A088878 - 1] (* G. C. Greubel, Jul 30 2017 *)

Many terms like 44, 46, 62 etc. removed by R. J. Mathar, Aug 06 2009

A176551 Products of 2 primes of the form 3*k-+1.

Original entry on oeis.org

4, 10, 14, 22, 25, 26, 34, 35, 38, 46, 49, 55, 58, 62, 65, 74, 77, 82, 85, 86, 91, 94, 95, 106, 115, 118, 119, 121, 122, 133, 134, 142, 143, 145, 146, 155, 158, 161, 166, 169, 178, 185, 187, 194, 202, 203, 205, 206, 209, 214, 215, 217, 218, 221, 226, 235, 247

Offset: 1

Juri-Stepan Gerasimov, Apr 20 2010

nonn

Semiprimes without 3*primes (or triple the primes).

Numbers of the form A045344(i)*A045344(j), any i, j. [From R. J. Mathar, Apr 27 2010]

Cf. A001358, A001748, A100959.

Entries checked by R. J. Mathar, Apr 27 2010

A196935 a(n) is the number of arithmetic progressions prime chains in the form of p(n)-6k, p(n), p(n)+6k, while k > 0 and p(n) > 6k.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 3, 3, 3, 4, 4, 5, 3, 4, 6, 5, 4, 4, 6, 5, 7, 6, 6, 6, 5, 7, 8, 9, 6, 10, 8, 7, 6, 9, 8, 9, 6, 8, 10, 10, 6, 9, 10, 11, 8, 11, 10, 9, 13, 13, 13, 13, 9, 10, 13, 11, 12, 14, 15, 11, 12, 12, 14, 17, 13, 18, 14, 14, 16, 14, 16, 14, 16, 15, 16

Offset: 5

Lei Zhou, Oct 07 2011

Conjecture: a(n) > 0 for all n >= 5.

The Mathematica program gives term 5 through 80.

			n = 5, p(5) = 11; {5, 11, 17} forms a difference 6 Arithmetic Progressions Prime chain. And this is the only occurrence for 11.  So a(5) = 1;
n = 6, p(6) = 13; {7, 13, 19} forms a difference 6 Arithmetic Progressions Prime chain. And this is the only occurrence for 11.  So a(6) = 1;
...
n = 10, p(10) = 29; {17, 29, 41}, {11, 29, 47}, {5, 29, 53} form Arithmetic Progressions Prime chains with difference 12, 18, 24 respectively.  So a(10) = 3;

Definition of Arithmetic Progressions of Primes

Cf. A196934, A078498, A078497, A001748.

Table[ct = 0; p = Prime[i]; j = 0; While[j++; df = 6*j; df < p, If[(PrimeQ[p + df]) && (PrimeQ[p - df]), ct++]]; ct, {i, 5, 80}]

A279389 3 times Mersenne primes A000668.

Original entry on oeis.org

9, 21, 93, 381, 24573, 393213, 1572861, 6442450941, 6917529027641081853, 1856910058928070412348686333, 486777830487640090174734030864381, 510423550381407695195061911147652317181

Offset: 1

Omar E. Pol, Dec 20 2016

nonn

Also sum of n-th Mersenne prime and the radical of n-th even perfect number.

The binary representation of a(n) has only two zeros, starting with "10" and ending with "01". The sequence begins: 1001, 10101, 1011101, 101111101, 101111111111101,...

Cf. A000010, A000203, A000396, A000668, A051027, A147757.
Subsequence of A001748, and of A147758, and of A174055, and possibly of other sequences, see below:
Cf. A060482, A068156, A072087, A136252, A144482, A144856, A249780, A269633.

a(n) = 3*A000668(n) = A000668(n) + A139257(n).

a(n) = phi(M(n)) + sigma(sigma(M(n))) = A000010(A000668(n)) + A000203(A000203(A000668(n))) = A000010(A000668(n)) + A051027(A000668(n)).

A359059 Numbers k such that phi(k) + rad(k) + psi(k) is a multiple of 3.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 11, 13, 17, 18, 19, 20, 23, 27, 29, 31, 32, 36, 37, 41, 42, 43, 44, 45, 47, 49, 50, 53, 54, 59, 61, 63, 67, 68, 71, 72, 73, 78, 79, 80, 81, 83, 84, 89, 90, 92, 97, 99, 101, 103, 105, 107, 108, 109, 110, 113, 114, 116, 117, 125, 126, 127, 128, 131, 135, 137, 139

Offset: 1

Torlach Rush, Dec 14 2022

nonn

When k is prime (denote as p), phi(p) = p - 1, rad(p) = p, and psi(p) = p + 1, so phi(p) + rad(p) + psi(p) = 3*p. Therefore, A000040 is a subsequence.

When k = p^m (m>=1) with p prime, phi(p^m) = (p-1)*p^(m-1), rad(p^m) = p, and psi(p^m) = (p+1)*p^(m-1), so phi(p^m) + rad(p^m) + psi(p^m) = 2*p^m + p = p * (1+2*p^(m-1)). Then, this expression is a multiple of 3 iff p == 0 or 1 (mod 3), equivalently iff p is a generalized cuban prime of A007645. Therefore, as 1 is also a term, every sequence {p^m, p in A007645, m>=0} is a subsequence. See crossrefs section. - Bernard Schott, Jan 25 2023 after an observation of Alois P. Heinz

			8 is a term because 4+2+12 is divisible by 3.

Cf. A000010 (phi), A000040, A001615 (psi), A007645, A007947 (rad), A001748 (3*p), A000244.

Subsequences of the form {p^n, n>=0}: A000244 (p=3), A000420 (p=7), A001022 (p=13), A001029 (p=19), A009975 (p=31), A009981 (p=37), A009987 (p=43).

q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Divisible[Times @@ ((p - 1)*p^(e - 1)) + Times @@ p + Times @@ ((p + 1)*p^(e - 1)), 3]]; Select[Range[170], q] (* Amiram Eldar, Dec 15 2022 *)

isok(m) = ((eulerphi(m) + factorback(factorint(m)[, 1]) + m*sumdiv(m, d, moebius(d)^2/d)) % 3) == 0; \\ Michel Marcus, Dec 27 2022

from sympy.ntheory.factor_ import totient
from sympy import primefactors, prod
def rad(n): return 1 if n < 2 else prod(primefactors(n))
def psi(n):
    plist = primefactors(n)
    return n*prod(p+1 for p in plist)//prod(plist)
# Output display terms.
for n in range(1,170):
    if(0 == (totient(n) + rad(n) + psi(n)) % 3):
        print(n, end = ", ")

A370009 a(n) is the least prime greater than 3*prime(n).

Original entry on oeis.org

7, 11, 17, 23, 37, 41, 53, 59, 71, 89, 97, 113, 127, 131, 149, 163, 179, 191, 211, 223, 223, 239, 251, 269, 293, 307, 311, 331, 331, 347, 383, 397, 419, 419, 449, 457, 479, 491, 503, 521, 541, 547, 577, 587, 593, 599, 641, 673, 683, 691, 701, 719, 727, 757

Offset: 1

Clark Kimberling, Feb 09 2024

nonn

			5 < 3*2 < 7 < 3*3 < 11 < 13 < 3*5 < 17, so (a(1), a(2), a(3)) = (7,11,17).

Cf. A000040, A001748, A059786, A059788, A151800, A370008.

Table[NextPrime[Prime[PrimePi[3*Prime[n]]]], {n,1,200}]

a(n) = nextprime(3*prime(n)); \\ Michel Marcus, Feb 10 2024

a(n) = A151800(A001748(n)).

cp's OEIS Frontend

A196934 a(n) is the first occurrence of n in sequence A078498.

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A263648 a(1)=4, a(2)=9: a(n) is the smallest semiprime not yet appearing in the sequence which is coprime to a(n-1) and not coprime to a(n-2).

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A344872 Semiprimes of the form 3m+2.

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PARI

A129758 Smallest prime p such that there are primes q and r with the property that p, q and r form an arithmetic progression and their sum is the same as three times the (n+2)-nd prime number.

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A163628 Integers such that the two adjacent integers are a prime and three times a prime.

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A176551 Products of 2 primes of the form 3*k-+1.

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A196935 a(n) is the number of arithmetic progressions prime chains in the form of p(n)-6k, p(n), p(n)+6k, while k > 0 and p(n) > 6k.

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A279389 3 times Mersenne primes A000668.

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A359059 Numbers k such that phi(k) + rad(k) + psi(k) is a multiple of 3.