A006094 -id:A006094 - cp's OEIS frontend

A261079 Sum of index differences between prime factors of n, summed over all unordered pairs of primes present (with multiplicity) in the prime factorization of n.

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

Offset: 1

Antti Karttunen, Sep 23 2015

nonn

			For n = 1 the prime factorization is empty, thus there is nothing to sum, so a(1) = 0.
For n = 6 = 2*3 = prime(1) * prime(2), a(6) = 1 because the (absolute value of) difference between prime indices of 2 and 3 is 1.
For n = 10 = 2*5 = prime(1) * prime(3), a(10) = 2 because the difference between prime indices of 2 and 5 is 2.
For n = 12 = 2*2*3 = prime(1) * prime(1) * prime(2), a(12) = 2 because the difference between prime indices of 2 and 3 is 1, and the pair (2,3) occurs twice as one can pick either one of the two 2's present in the prime factorization to be a pair of a single 3. Note that the index difference between 2 and 2 is 0, thus the pair (2,2) of prime divisors does not contribute to the sum.
For n = 36 = 2*2*3*3, a(36) = 4 because the index difference between 2 and 3 is 1, and the prime factor pair (2,3) occurs 2^2 = four times in total. As the index difference is zero between 2 and 2 as well as between 3 and 3, the pairs (2,2) and (3,3) do not contribute to the sum.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Cf. A000720.
Cf. A000961 (positions of zeros), A006094 (positions of ones).
Cf. A243055, A242411.
Cf. also A260737.
A055396 gives minimum prime index, maximum A061395.
A112798 list prime indices, length A001222, sum A056239.
A304818 adds up partial sums of reversed prime indices, row sums of A359361.
A318283 adds up partial sums of prime indices, row sums of A358136.
Cf. A029931, A316413, A325362, A326836, A326844, A355536, A358137, A359362.

Table[Function[p, Total@ Map[Function[b, Times @@ {First@ Differences@ PrimePi@ b, Count[Subsets[p, {2}], c_ /; SameQ[c, b]]}], Subsets[Union@ p, {2}]]][Flatten@ Replace[FactorInteger@ n, {p_, e_} :> ConstantArray[p, e], 2]], {n, 120}] (* Michael De Vlieger, Mar 08 2017 *)

a(n) = A304818(n) - A318283(n). - Gus Wiseman, Jan 09 2023

a(n) = 2*A304818(n) - A359362(n). - Gus Wiseman, Jan 09 2023

A046325 Product of 7 successive primes.

Original entry on oeis.org

510510, 4849845, 37182145, 215656441, 955049953, 3212440751, 10131543907, 25626846353, 63392725189, 146078888479, 297194980009, 584803025179, 1058967640189, 1833822011059, 3113232716449, 5232880523393, 8194888366823

Offset: 1

Patrick De Geest, Jun 15 1998

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A002110.

Cf. product of n successive primes: A006094, A046301, A046302, A046303, A046324, A046326, A046327.

[&*[ NthPrime(n+k): k in [0..6] ]: n in [1..17]];  // Bruno Berselli, Feb 25 2011

Times@@@Partition[Prime[Range[50]],7,1] (* Harvey P. Dale, Oct 21 2011 *)

Offset changed from 0 to 1 by Vincenzo Librandi, Jan 16 2012

A097889 Numbers that are products of (at least two) consecutive primes.

Original entry on oeis.org

6, 15, 30, 35, 77, 105, 143, 210, 221, 323, 385, 437, 667, 899, 1001, 1147, 1155, 1517, 1763, 2021, 2310, 2431, 2491, 3127, 3599, 4087, 4199, 4757, 5005, 5183, 5767, 6557, 7387, 7429, 8633, 9797, 10403, 11021, 11663, 12317, 12673, 14351, 15015, 16637, 17017

Offset: 1

Bart la Bastide (bart(AT)xs4all.nl), Sep 21 2004

Subsequence of A073485; A073490(a(n)) = 0. - Reinhard Zumkeller, Nov 20 2004
A proper subset of A073485. - Robert G. Wilson v, Jun 11 2010
A192280(a(n)) * (1 - A010051(a(n))) = 1. - Reinhard Zumkeller, Aug 26 2011 [corrected by Jason Yuen, Aug 29 2024]
The Heinz numbers of the partitions into at least 2 consecutive parts. The Heinz number of an integer partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). Examples: (i) 105 (=3*5*7) is in the sequence because it is the Heinz number of the partition [2,3,4]; (ii) 108 (= 2*2*3*3*3) is not in the sequence because it is the Heinz number of the partition [1,1,2,2,2]. - Emeric Deutsch, Oct 02 2015

			1001 = 7 * 11 * 13.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Union of A006094, A046301, A046302, A046303, A046324, A046325, A046326, A046327, etc.
Cf. A050936.
Intersection of A073485 and A002808.
Cf. A151800, A215366.

import Data.Set (singleton, deleteFindMin, insert)
a097889 n = a097889_list !! (n-1)
a097889_list = f $ singleton (6, 2, 3) where
   f s = y : f (insert (w, p, q') $ insert (w `div` p, a151800 p, q') s')
         where w = y * q'; q' = a151800 q
               ((y, p, q), s') = deleteFindMin s
-- Reinhard Zumkeller, May 12 2015, Aug 26 2011

isA097889 := proc(n)
    local plist,p,i ;
    plist := sort(convert(numtheory[factorset](n),list)) ;
    if nops(plist) < 2 then
        return false;
    end if;
    for i from 1 to nops(plist) do
        p := op(i,plist) ;
        if modp(n,p^2) = 0 then
            return false;
        end if;
        if i > 1 then
            if nextprime(op(i-1,plist)) <> p then
                return false;
            end if;
        end if;
    end do:
    true;
end proc:
for n from 1 to 1000 do
    if isA097889(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Jan 12 2016

a = {}; Do[ AppendTo[a, Apply[ Times, (Prime /@ Partition[ Range[30], n, i]), 1]], {n, 2, 6}, {i, n - 1}]; Take[ Union[ Flatten[ a]], 45] (* Robert G. Wilson v, Sep 24 2004 *)

list(lim)=my(v=List(), p, t); for(e=2, log(lim+.5)\log(2), p=1; t=prod(i=1, e-1, prime(i)); forprime(q=prime(e), lim, t*=q/p; if(t>lim, next(2)); listput(v, t); p=nextprime(p+1))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Oct 24 2012

import heapq
from sympy import sieve
sieve.extend(10**6)
primes = list(sieve._list)
def prime(n): return primes[n-1]
def aupton(terms, verbose=False):
    p = prime(1)*prime(2); h = [(p, 1, 2)]; nextcount = 3; alst = []
    while len(alst) < terms:
        (v, s, l) = heapq.heappop(h)
        alst.append(v)
        if verbose: print(f"{v}, [= Prod_{{i = {s}..{l}}} prime(i)]")
        if v >= p:
            p *= prime(nextcount)
            heapq.heappush(h, (p, 1, nextcount))
            nextcount += 1
        v //= prime(s); s += 1; l += 1; v *= prime(l)
        heapq.heappush(h, (v, s, l))
    return alst
print(aupton(45)) # Michael S. Branicky, Jun 15 2021

a(n) ~ n^2 log^2 n. - Charles R Greathouse IV, Oct 24 2012

More terms from Robert G. Wilson v, Sep 24 2004

Data corrected for n > 41 by Reinhard Zumkeller, Aug 26 2011

A255483 Infinite square array read by antidiagonals downwards: T(0,m) = prime(m), m >= 1; for n >= 1, T(n,m) = T(n-1,m)*T(n-1,m+1)/gcd(T(n-1,m), T(n-1,m+1))^2, m >= 1.

Original entry on oeis.org

2, 3, 6, 5, 15, 10, 7, 35, 21, 210, 11, 77, 55, 1155, 22, 13, 143, 91, 5005, 39, 858, 17, 221, 187, 17017, 85, 3315, 1870, 19, 323, 247, 46189, 133, 11305, 5187, 9699690, 23, 437, 391, 96577, 253, 33649, 21505, 111546435, 46

Offset: 0

N. J. A. Sloane, Feb 28 2015

The first column of the array is given by A123098; subsequent columns are obtained by applying the function A003961, i.e., replacing each prime factor by the next larger prime. - M. F. Hasler, Sep 17 2016
Interpretation with respect to A329329 from Peter Munn, Feb 08 2020: (Start)
With respect to the ring defined by A329329 and A059897, the first row gives powers of 3, the first column gives powers of 6, both in order of increasing exponent, and the body of the table gives their products. A329049 is the equivalent table in which the first column gives powers of 4.
A099884 is the equivalent table for the ring defined by A048720 and A003987. That ring is an image of the polynomial ring GF(2)[x] using a standard representation of the polynomials as integers. A329329 describes a comparable mapping to integers from the related polynomial ring GF(2)[x,y].
Using these mappings, the tables here and in A099884 are matching images: the first row represents powers of x, the first column represents powers of (x+1) and the body of the table gives their products.
Hugo van der Sanden's formula (see formula section) indicates that A019565 provides a mapping from A099884. In the wider terms described above, A019565 is an injective homomorphism between images of the 2 polynomial rings, and maps the image of each GF(2)[x] polynomial to the image of the equivalent GF(2)[x,y] polynomial.
(End)

			The top left corner of the array, row index 0..5, column index 1..10:
    2,    3,     5,     7,    11,     13,     17,     19,      23,      29
    6,   15,    35,    77,   143,    221,    323,    437,     667,     899
   10,   21,    55,    91,   187,    247,    391,    551,     713,    1073
  210, 1155,  5005, 17017, 46189,  96577, 215441, 392863,  765049, 1363783
   22,   39,    85,   133,   253,    377,    527,    703,     943,    1247
  858, 3315, 11305, 33649, 95381, 198679, 370481, 662929, 1175921, 1816879

Alois P. Heinz, Antidiagonals n = 0..125, flattened
C. Cobeli, A. Zaharescu, A game with divisors and absolute differences of exponents, Journal of Difference Equations and Applications, Vol. 20, #11, 2014.
C. Cobeli, A. Zaharescu, A game with divisors and absolute differences of exponents, arXiv:1411.1334 [math.NT], 2014.
Discussion of SeqFan-mailing list
Index entries for sequences related to Gilbreath conjecture and transform

First two columns = A123098, A276804.
Rows = A000040, A006094, A090076, A046302, ...
A kind of generalization of A036262.
Cf. A003961, A007913, A019565, A048675, A066117, A099884.
Transpose: A276578, terms sorted into ascending order: A276579.
A003987, A048720, A059897, A329049 relate to the A329329 polynomial ring interpretation.

T:= proc(n, m) option remember; `if`(n=0, ithprime(m),
      T(n-1, m)*T(n-1, m+1)/igcd(T(n-1, m), T(n-1, m+1))^2)
    end:
seq(seq(T(n, 1+d-n), n=0..d), d=0..10);  # Alois P. Heinz, Feb 28 2015

T[n_, m_] := T[n, m] = If[n == 0, Prime[m], T[n-1, m]*T[n-1, m+1]/GCD[T[n-1, m], T[n-1, m+1]]^2]; Table[Table[T[n, 1+d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)

T=matrix(N=15,N);for(j=1,N,T[1,j]=prime(j));(f(x,y)=x*y/gcd(x,y)^2);for(k=1,N-1,for(j=1,N-k,T[k+1,j]=f(T[k,j],T[k,j+1])));A255483=concat(vector(N,i,vector(i,j,T[j,1+i-j]))) \\ M. F. Hasler, Sep 17 2016

A255483(n,k)=prod(j=0,n,if(bitand(n-j,j),1,prime(j+k))) \\ M. F. Hasler, Sep 18 2016

(define (A255483 n) (A255483bi (A002262 n) (+ 1 (A025581 n))))
;; Then use either an almost standalone version (requiring only A000040):
(define (A255483bi row col) (if (zero? row) (A000040 col) (let ((a (A255483bi (- row 1) col)) (b (A255483bi (- row 1) (+ col 1)))) (/ (lcm a b) (gcd a b)))))
;; Or one based on M. F. Hasler's new recurrence:
(define (A255483bi row col) (if (= 1 col) (A123098 row) (A003961 (A255483bi row (- col 1)))))
;; Antti Karttunen, Sep 18 2016

T(n,1) = A123098(n), T(n,m+1) = A003961(T(n,m)), for all n >= 0, m >= 1. - M. F. Hasler, Sep 17 2016
T(n,m) = Prod_{k=0..n} prime(k+m)^(!(n-k & k)) where !x is 1 if x=0 and 0 else, and & is binary AND. - M. F. Hasler, Sep 18 2016
From Antti Karttunen, Sep 18 2016: (Start)
For n >= 1, m >= 1, T(n,m) = lcm(T(n-1,m),T(n-1,m+1)) / gcd(T(n-1,m),T(n-1,m+1)).
T(n,k) = A007913(A066117(n+1,k)).
T(n,k) = A019565(A099884(n,k-1)) [After Hugo van der Sanden's observations on SeqFan-list].
(End)
From Peter Munn, Jan 08 2020: (Start)
T(0,1) = 2, and for n >= 0, k >= 1, T(n+1,k) = A329329(T(n,k), 6), T(n,k+1) = A329329(T(n,k), 3).
T(n,k) = A329329(T(n,1), T(0,k)).
(End)

A345452 Positive integers with an even number of prime factors (counting repetitions) that sum to an even number.

Original entry on oeis.org

1, 4, 9, 15, 16, 21, 25, 33, 35, 36, 39, 49, 51, 55, 57, 60, 64, 65, 69, 77, 81, 84, 85, 87, 91, 93, 95, 100, 111, 115, 119, 121, 123, 129, 132, 133, 135, 140, 141, 143, 144, 145, 155, 156, 159, 161, 169, 177, 183, 185, 187, 189, 196, 201, 203, 204, 205, 209, 213, 215

Offset: 1

Peter Munn, Jun 20 2021

Numbers with an even number of even prime factors and an even number of odd prime factors.
The representation (as defined in A206284) of polynomials with nonnegative integer coefficients that are in the ideal of the polynomial ring Z[x] generated by x^2+x and 2.
The above property arises because the sequence lists the integers in the multiplicative subgroup of positive rational numbers generated by the squares of primes (A001248) and the products of two consecutive odd primes (A006094\{6}).
The sequence is closed under multiplication, prime shift (A003961), and - where the result is an integer - under division. Using these closures, all the terms can be derived from the presence of 4 and 15. For example, A003961(4) = 9, A003961(9) = 25, A003961(15) = 35, 15 * 35 = 525, 525/25 = 21. Alternatively, the sequence may be defined as the closure of A046337 under multiplication by 4.
From the properties of subgroups of the positive rationals we know that if we take an absent positive integer m and divide all terms that are multiples of m by m, we get all the integers in the same subgroup coset as m, and we can expect some of the nice properties here to carry over to the resulting set. Specifically, dividing the even terms by 2 gives all numbers with an odd number of prime factors that sum to an even number; dividing all terms divisible by an odd prime p by p, gives all numbers with an odd number of prime factors that sum to an odd number. The positive integers satisfying the 4th of the 4 possibilities are generated similarly, dividing by 6 (for example).
Numbers whose squarefree part is in A056913.
Term by term, the sequence is one half of its complement within A036349.

			The definition specifies that we count repeated prime factors.
6 = 2 * 3; the sum of these prime factors is 2 + 3 = 5, an odd number; so 6 is not in the sequence.
50 = 2 * 5 * 5 has 3 prime factors and 3 is an odd number; so 50 is not in the sequence.
60 = 2 * 2 * 3 * 5 has 4 prime factors and 4 is an even number; the sum of these factors is 2 + 2 + 3 + 5 = 12, also an even number; so 60 is in the sequence.
1 has 0 prime factors, which sum to 0 (the empty sum). 0 is even, so 1 is in the sequence.

Eric Weisstein's World of Mathematics, Group.
Wikipedia, Polynomial ring.

Cf. A001222, A001414, A003961, A059897, A307150.
Intersection of any 2 of A003159, A028260, A036349.
Other lists that have conditions on the number of odd prime factors: A046337, A072978.
Subsequences: A001248, A006094\{6}, A046315, A056913.

{1}~Join~Select[Range@1000,(s=Flatten[Table@@@FactorInteger[#]];And@@EvenQ@{Length@s,Total@s})&] (* Giorgos Kalogeropoulos, Jun 24 2021 *)

iseven(x) = ((x%2) == 0);
isok(m) = my(f=factor(m)); iseven(sum(k=1, #f~, f[k,1]*f[k,2])) && iseven(sum(k=1, #f~, f[k,2])); \\ Michel Marcus, Jun 24 2021

is(n) = bigomega(n)%2 == 0 && valuation(n, 2)%2 == 0 \\ David A. Corneth, Jun 24 2021

from sympy import factorint
def ok(n):
    f = factorint(n)
    return sum(f.values())%2 == 0 and sum(p*f[p] for p in f)%2 == 0
print(list(filter(ok, range(1, 216)))) # Michael S. Branicky, Jun 24 2021

{a(n) : n >= 1} = {m >= 1 : A001222(m) mod 2 = A001414(m) mod 2 = 0}.
{A036349(n) : n >= 1} = {a(n) : n >= 1} U {2 * a(n) : n >= 1}.
{A028260(n) : n >= 1} = {a(n) : n >= 1} U {A307150(a(n)) : n >= 1}.
For odd prime p, {A003159(n) : n >= 1} = {a(n) : n >= 1} U {A059897(a(n), p) : n >= 1}.

A126148 Primes p such that pq+p+q is prime, where q is the next prime after p.

Original entry on oeis.org

2, 3, 5, 11, 13, 17, 19, 23, 41, 43, 47, 59, 79, 83, 89, 101, 109, 113, 137, 163, 167, 173, 223, 229, 257, 311, 383, 389, 409, 419, 439, 443, 479, 521, 547, 557, 577, 593, 613, 643, 647, 683, 773, 797, 809, 811, 853, 953, 983, 1019, 1049, 1097, 1109, 1151, 1171

Offset: 1

J. M. Bergot, Mar 07 2007

			Take p = 13 and q = 17: product is 221 and sum is 30; add them to get 251, a prime. So 13 is a member.

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

Cf. A096342, A000040, A001043, A006094, A126199, A096342.

a:=proc(n) if isprime(ithprime(n)*ithprime(n+1) +ithprime(n) +ithprime(n+1)) then ithprime(n) fi end: seq(a(n), n=1..250); # Emeric Deutsch, Mar 08 2007

Prime@Select[Range[200], PrimeQ[Prime[ # ]Prime[ # + 1] + Prime[ # ] + Prime[ # + 1]] &] (* Ray Chandler, Mar 07 2007 *)

v=List();p=2;forprime(q=3,1e4, if(isprime(p*q+p+q), listput(v,p)); p=q); Vec(v) \\ Charles R Greathouse IV, Jul 26 2012

Extended by Ray Chandler, Emeric Deutsch and Robert G. Wilson v, Mar 07 2007

A066117 Triangle read by rows: T(n,k) = T(n-1,k-1)*T(n,k-1) and T(n,1) = prime(n).

Original entry on oeis.org

2, 3, 6, 5, 15, 90, 7, 35, 525, 47250, 11, 77, 2695, 1414875, 66852843750, 13, 143, 11011, 29674645, 41985913344375, 2806877704512541816406250, 17, 221, 31603, 347980633, 10326201751150285, 433555011900329243987584396875

Offset: 1

Henry Bottomley, Dec 05 2001

As a square array read by descending antidiagonals, A(n, k), n >= 1, k >= 1, gives the encoding defined in A297845 of the polynomial (x+1)^(n-1) * x^(k-1). - Peter Munn, Jul 27 2022

			T(4,3) = T(3,2)*T(4,2) = 15*35 = 525. Rows start
     2;
    3, 6;
  5, 15, 90;
7, 35, 525, 47250;
...
From _Antti Karttunen_, Sep 18 2016: (Start)
Alternatively, this table can be viewed as a square array. Then the top left 5x4 corner looks as:
    2,       3,        5,         7,         11
    6,      15,       35,        77,        143
   90,     525,     2695,     11011,      31603
47250, 1414875, 29674645, 347980633, 2255916949
(End)

Index entries for triangles and arrays related to Pascal's triangle

Cf. A000040, A006094 and A066116 (three leftmost diagonal of triangular table = three topmost rows of square array).
Cf. A007188, A267096 (two rightmost diagonals of the triangular table = two leftmost columns of square array).
Cf. A003961, A055396, A297845.
Cf. A064319, A066119.
Cf. also A099884, A255483, A276586, A276588 (other arrays derived from this one).

T[n_, 1] := Prime[n];
T[n_, k_] := T[n, k] = T[n - 1, k - 1]*T[n, k - 1];
Table[T[n, k], {n, 1, 7}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 13 2017 *)

(define (A066117 n) (A066117bi (A002260 n) (A004736 n)))
;; Compute as a square array, with row >= 1, col >= 1:
(define (A066117bi row col) (if (= 1 row) (A000040 col) (* (A066117bi (- row 1) col) (A066117bi (- row 1) (+ col 1)))))
;; With alternative recurrence:
(define (A066117bi row col) (if (= 1 col) (A007188 (- row 1)) (A003961 (A066117bi row (- col 1)))))
;; Antti Karttunen, Sep 18 2016

From Antti Karttunen, Sep 19 2016: (Start)
When computed as a square array A(row,col), row >= 1, col >= 1:
A(1,col) = A000040(col), for row > 1, A(row,col) = A(row-1,col)*A(row-1,col+1).
A(row,1) = A007188(row-1), for col > 1, A(row,col) = A003961(A(row,col-1)).
For all row >= 1, col >= 1, A055396(A(row,col)) = col.
(End)
A(1,1) = 2; for n > 1, A(n,k) = A297845(A(n-1,k),6); for k > 1, A(n,k) = A297845(A(n,k-1),3). - Peter Munn, Jul 20 2022

A300827 Lexicographically earliest sequence such that a(i) = a(j) => A324193(i) = A324193(j) for all i, j >= 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 5, 6, 2, 7, 2, 8, 9, 10, 2, 11, 2, 12, 13, 14, 2, 15, 16, 17, 9, 18, 2, 19, 2, 20, 21, 22, 23, 24, 2, 25, 26, 27, 2, 28, 2, 29, 30, 31, 2, 32, 33, 34, 35, 36, 2, 37, 38, 39, 40, 41, 2, 42, 2, 43, 44, 45, 46, 47, 2, 48, 49, 50, 2, 51, 2, 52, 53, 54, 55, 56, 2, 57, 58, 59, 2, 60, 61, 62, 63, 64, 2, 65, 66, 67

Offset: 1

Antti Karttunen, Mar 13 2018

nonn

Apart from primes, the sequence contains duplicate values at points p*q and p^3, where p*q are the product of two successive primes, with p < q (sequences A006094, A030078). Question: are there any other cases where a(x) = a(y), with x < y ?
The reason why this is not equal to A297169: Even though A297112 contains only powers of two after the initial zero, as A297112(n) = 2^A033265(A156552(d)) for n > 1, and A297168(n) is computed as Sum_{d|n, dA297112(d), still a single 1-bit in binary expansion of A297168(n) might be formed as a sum of several terms of A297112(d), i.e., could be born of carries.
From Antti Karttunen, Feb 28 2019: (Start)
  A297168(n) = Sum_{d|n, dA297112(d) will not produce any carries (in base-2) if and only if n is a power of prime. Only in that case the number of summands (A000005(n)-1) is equal to the number of prime factors counted with multiplicity, A001222(n) = A000120(A156552(n)). (A notable subset of such numbers is A324201, numbers that are mapped to even perfect numbers by A156552). Precisely because there are so few points with duplicate values (apart from primes), this sequence is not particularly good for filtering other sequences, because the number of false positives is high. Any of the related sequences like A324203, A324196, A324197 or A324181 might work better in that respect. In any case, the following implications hold (see formula section of A324193 for the latter): (End)
For all i, j:
  a(i) = a(j) => A297168(i) = A297168(j). (The same holds for A297169).
  a(i) = a(j) => A324181(i) = A324181(j) => A324120(i) = A324120(j).

			For n = 15, with proper divisors 3 and 5, we have f(n) = prime(1+A297167(3)) * prime(1+A297167(5)) = prime(2)*prime(3) = 3*5 = 15.
For n = 27, with proper divisors 3 and 9, we have f(n) = prime(1+A297167(3)) * prime(1+A297167(9)) = prime(2)*prime(3) = 3*5 = 15.
Because f(15) = f(27), the restricted growth sequence transform allots the same number (in this case 9) for both, so a(15) = a(27) = 9.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000120, A000961, A006094, A030078, A046660, A061395, A101296, A156552, A297112, A297167, A297168, A297169, A324193, A324120, A324201.

Cf. also A324181, A323914, A324196, A324197, A324203.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1]))); \\ After M. F. Hasler's code for A006530.
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) -1));
Aux300827(n) = { my(m=1); if(n<=2, n-1, fordiv(n,d,if((d>1)&(dA297167(d)))); (m)); };
write_to_bfile(1,rgs_transform(vector(up_to,n,Aux300827(n))),"b300827.txt");

Restricted growth sequence transform of sequence f, defined as f(1) = 0, f(2) = 1, and for n > 2, f(n) = Product_{d|n, 1A297167(d)).
a(p) = 2 for all primes p.
a(A006094(n)) = a(A030078(n)), for all n >= 1.

Name changed by Antti Karttunen, Feb 21 2019

A117301 a(n) = prime(n+3)prime(n) - prime(n+1)prime(n+2).

Original entry on oeis.org

-1, -2, -12, -24, -12, -24, 56, -78, -48, 42, -184, -24, 152, 46, -260, -48, 102, -304, 110, 126, -60, 276, -250, -630, -24, -12, -24, 1272, -72, -1156, -294, 476, -24, -676, 580, -374, -60, 286, -740, 644, -24, -1206, -12, 1520, 1942, -1880

Offset: 1

Cino Hilliard, Apr 24 2006

sign

The number of negative values in this sequence appears to be consistently larger than the number of positive values. The following list gives the number of positive terms among the first n terms divided by the number of negative terms among the first n terms for various n.
     n  ratio
  10^2  0.51515151515...
  10^3  0.70940170940...
  10^4  0.80212650928...
  10^5  0.83826908582...
  10^6  0.86339454584...
Cino Hilliard conjectures that this ratio converges and that there are infinitely many elements in the sequence whose absolute value is 12.
It appears that the positions of negative multiples of 12 are given by A064026(n+1) for n >= 1.  If so, then Hilliard's conjecture is true, and a further conjecture is that if k >= 2 then there are infinitely many multiples of -12*k in this sequence. - Clark Kimberling, Jan 01 2014

			a(4) = prime(4)*prime(7) - prime(5)*prime(6) = 7*17 - 11*13 = -24.

Table[Prime[n]*Prime[n + 3] - Prime[n + 1]Prime[n + 2], {n, 1, 100}] (* Stefan Steinerberger, Jun 27 2007 *)
(* The following program is significantly faster: *)
(First[#]Last[#]-#[[2]]#[[3]])&/@Partition[Prime[Range[50]],4,1] (* Harvey P. Dale, May 08 2011 *)

det2cont(n) = {local(m,p,x, D); m=0; p=0; for(x=1,n, D=prime(x)*prime(x+3)-prime(x+1)*prime(x+2); if(D<0,m++,p++); print1(D",") ); print(); print("neg= "m); print("pos= "p); print("pos/neg = "p/m+.) }

a(n) = A090090(n) - A006094(n+1). - Michel Marcus, Oct 07 2013

Edited by Stefan Steinerberger, Jun 27 2007

A127346 Primes in A127345.

Original entry on oeis.org

31, 71, 167, 311, 1151, 3119, 4871, 5711, 6791, 14831, 24071, 33911, 60167, 79031, 101159, 106367, 115631, 158231, 235751, 259751, 366791, 402551, 455471, 565919, 635711, 644951, 1124831, 1347971, 1510799, 1547927, 1743419, 1851671, 2048471

Offset: 1

Artur Jasinski, Jan 11 2007

nonn

Primes of the form prime(k)*prime(k+1) + prime(k)*prime(k+2) + prime(k+1)*prime(k+2).

A prime number n is in the sequence if for some k it is the coefficient of x^1 of the polynomial Product_{j=0..2} (x-prime(k+j)); the roots of this polynomial are prime(k), ..., prime(k+2).

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A127345, A127347, A127351, A006094, A002110, A034962, A034965, A082246, A082251, A127340, A127341, A070934, A046301, A046302, A046303, A046324, A046325, A046326, A046327.

b = {}; a = {}; Do[If[PrimeQ[Prime[x] Prime[x + 1] + Prime[x] Prime[x + 2] + Prime[x + 1] Prime[x + 2]], AppendTo[a, Prime[x] Prime[x + 1] + Prime[x] Prime[x + 2] + Prime[x + 1] Prime[x + 2]], AppendTo[b, Prime[x] Prime[x + 1] + Prime[x] Prime[x + 2] + Prime[x + 1] Prime[x + 2]]], {x, 1, 100}]; Print[a] (* Artur Jasinski, Jan 11 2007 *)
s[li_] := li[[1]]*(li[[2]]+li[[3]])+li[[2]]*li[[3]]; Select[(s[#]&/@Partition[Prime[Range[100]], 3, 1]), PrimeQ] (* Zak Seidov, Jan 13 2012 *)

{m=143;k=2;for(n=1,m,a=sum(i=n,n+k-1,sum(j=i+1,n+k,prime(i)*prime(j)));if(isprime(a),print1(a,",")))} \\ Klaus Brockhaus, Jan 21 2007

{m=143;k=2;for(n=1,m,a=polcoeff(prod(j=0,k,(x-prime(n+j))),1);if(isprime(a),print1(a,",")))} \\ Klaus Brockhaus, Jan 21 2007

p=2; q=3; forprime(r=5, 1e3, if(isprime(t=p*q+p*r+q*r), print1(t", ")); p=q; q=r) \\ Charles R Greathouse IV, Jan 13 2012

a(n) = A127345(A204231(n)). - Zak Seidov, Jan 13 2012

Edited and extended by Klaus Brockhaus, Jan 21 2007

cp's OEIS Frontend

A261079 Sum of index differences between prime factors of n, summed over all unordered pairs of primes present (with multiplicity) in the prime factorization of n.

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A046325 Product of 7 successive primes.

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A097889 Numbers that are products of (at least two) consecutive primes.

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A255483 Infinite square array read by antidiagonals downwards: T(0,m) = prime(m), m >= 1; for n >= 1, T(n,m) = T(n-1,m)*T(n-1,m+1)/gcd(T(n-1,m), T(n-1,m+1))^2, m >= 1.

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A345452 Positive integers with an even number of prime factors (counting repetitions) that sum to an even number.

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A126148 Primes p such that pq+p+q is prime, where q is the next prime after p.

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A066117 Triangle read by rows: T(n,k) = T(n-1,k-1)*T(n,k-1) and T(n,1) = prime(n).

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A117301 a(n) = prime(n+3)prime(n) - prime(n+1)prime(n+2).