A116979 -id:A116979 - cp's OEIS frontend

A351029 Number of integers whose arithmetic derivative is equal to the n-th primorial.

0, 1, 3, 19, 114, 905, 9494, 124181, 2044847, 43755729, 1043468388, 30309948250

Offset: 1

Antti Karttunen, Feb 01 2022

Number of integers k such that A003415(k) = A002110(n).
a(7) = A116979(7) + 1 since 1547371'=510510 and 1547371=7^2*23*1373 and every other example has only two prime factors. a(8) > A116979(8) because there is at least one term k in A327978 for which A003415(k) = 9699690 = A002110(8), which is not semiprime, that k being 79332523 = 17^2 * 277 * 991. - Edited by Craig J. Beisel, Sep 13 2022 and Antti Karttunen, Jan 05 2023
Most such k are semiprimes, i.e., are "Goldbachian solutions", counted by A116979. The non-semiprime solutions (A366890) form a very tiny minority, and are counted by A369000. - Antti Karttunen, Jan 19 2024

			a(1) = 0 because there are no such k that A003415(k) = 2 = A002110(1).
a(2) = 1 because there is only one number, 9, such that A003415(9) = A002110(2) = 6.
a(3) = 3 because there are exactly three numbers, k = 161, 209, 221, for which A003415(k) = A002110(3) = 30. (See A327978). These are all semiprime solutions, generated by the partitions of 30 into 2 primes: 30 = 7 + 23 = 11 + 19 = 13 + 17, and we have 7*23 = 161; 11*19 = 209; 13*17 = 221.

Cf. A002110, A002620, A003415, A099302, A099303, A116979, A327978, A366890 (nonsemiprime solutions), A368703 (the least of solutions), A368704 (the largest of solutions), A369000.

Cf. also A369239.

A002110(n) = prod(i=1,n,prime(i));
A002620(n) = ((n^2)>>2);
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A351029(n) = { my(g=A002110(n)); sum(k=1,A002620(g),A003415(k)==g); }; \\ Very naive and slow. See comments in A327978.

A351029(n) = {v=prod(j=1,n,prime(j)); c=0; for(k=2, v^2/4, d=0; m=factor(k); for(i=1, matsize(m)[1], d+=(m[i,2]/m[i,1])*k; if(d>v, break;); ); if(d==v, c=c+1; ); ); c;} \\ Craig J. Beisel, Sep 13 2022

a(n) = Sum_{k=1..A002620(A002110(n))} [A003415(k) = A002110(n)], where [ ] is the Iverson bracket.

a(n) = A116979(n) + A369000(n). - Antti Karttunen, Jan 19 2024

a(7) from Craig J. Beisel, Sep 13 2022

a(8)..a(12) [the last based on the value of A116979(12)] from Antti Karttunen, Jan 09 2024

A366890 Irregular triangle, wherein row n lists in ascending order all numbers k whose arithmetic derivative k' is equal to the n-th primorial, A002110(n), and that have more than two prime factors with multiplicity. Rows of length zero are simply omitted, i.e., when A369000(n) = 0.

Original entry on oeis.org

1547371, 79332523, 1102527599503, 25336943536819, 25962012375103, 25970380120783, 66702554987143, 526285951027003, 927949814519899, 7777707036642079, 9584173681667203, 13082430772438171, 22101822021783739, 4958985803436403, 32006922970429003, 32076018550175863, 49806227168831659, 84682266449971639, 97995266657958403

Offset: 1

Antti Karttunen, Jan 09 2024

For n > 0, numbers k such that A003415(k) = A002110(n) and A001222(k) > 2.
Sequence as a whole is not listed in ascending order, even though each batch of solutions for each n for which A369000(n) > 0 are. For example, we have a(14) < a(13) because A003415(22101822021783739) = A002110(12), while A003415(4958985803436403) = A002110(13). See the examples.
Question: Are there any common terms with A036785, that is, with A368697?

			For rows n=1..6, 9 & 10 nothing is listed, as those rows are empty.
Row for n=7 has just one term: 1547371 (= 7^2 * 23 * 1373). Note that A003415(1547371) = 510510 = A002110(7).
Row for n=8 has just one term: 79332523 (= 17^2 * 277 * 991).
Row for n=11 has two terms:
  1102527599503 (= 11^2 * 11071 * 823033),
  25336943536819 (= 157 * 743 * 5749 * 37781).
Row for n=12 has nine terms:
  25962012375103 (= 7^2 * 8597 * 61630451),
  25970380120783 (= 7^2 * 41387 * 12806141),
  66702554987143 (= 19^2 * 167 * 1106416889),
  526285951027003 (= 73 * 3919 * 7013 * 262313),
  927949814519899 (= 269 * 271 * 1697 * 7501033),
  7777707036642079 (= 2203 * 2791 * 7349 * 172127),
  9584173681667203 (= 2131 * 5953 * 7901 * 95621),
  13082430772438171 (= 3109 * 5861 * 24421 * 29399),
  22101822021783739 (= 8783 * 11777 * 13921 * 15349).
Row for n=13 has 18 terms, and begins with:
  4958985803436403 (= 37^2 * 137 * 26440450451),
and ends with:
  3206697143570677543 (= 36899 * 41983 * 45233 * 45763).
Note that A003415(3206697143570677543) = 304250263527210 = A002110(13).

Antti Karttunen, Table of n, a(n) for n = 1..31
Antti Karttunen, PARI program

When the whole sequence is sorted into ascending order, equal to A327978 without any semiprime solutions (solutions in A001358), and also a subsequence of following sequences: A004709, A327862, A328234.
Cf. A001222, A002110, A003415, A116979, A351029, A368703.
Cf. also A036785, A368697, A328243, A369240.

PARI
```
\\ See the attached PARI-program
```

A369000 a(n) is the number of integers, not semiprimes, whose arithmetic derivative is equal to the n-th primorial.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 9, 18

Offset: 1

Antti Karttunen, Jan 16 2024

a(n) = number of solutions k to equation A003415(k) = A002110(n) with A001222(k) > 2.

Antti Karttunen, PARI-program
Index entries for sequences related to primorial numbers

Cf. A001222, A002110, A003415, A116979, A351029, A366890 (solutions themselves).

PARI
```
\\ See the attached PARI-program.
```

a(n) = A351029(n) - A116979(n).

A369245 Number of representations of the n-th Euclid number, A002110(n) + 1, as a sum (pq + pr + q*r) with three odd primes p <= q <= r. (Definition implies that p=3 and q > 3).

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 2, 1, 0, 0, 1, 1, 0, 1, 2, 0, 1

Offset: 0

Antti Karttunen, Jan 22 2024

Number of representations of the n-th Euclid number, A002110(n) + 1, as a sum of the form 3*(p+q) + p*q, where p and q are odd primes.
Question: Will there be an eventual growth spurt for this sequence? Even though all solutions must be multiples of 3 (but not of 9), because A006862(n) == 1 (mod 3), for n > 1, and the solutions belong to a set listed by A369461.
Similar sequence A369242 grows more vigorously because A033312(n) == -1 (mod 3) for n >= 3, thus allowing non-multiples of 3 as solutions. See comments in A369252.

			a(4) = 1 as there exists a natural number 399 = 3 * 7 * 19, whose arithmetic derivative (indicated with 399', see A003415) is computed as ((3*7) + (3*19) + (7*19)) = 211 = 1 + prime(4)# = A006862(4), and because 399 is the unique term in A046316 that satisfies the condition.
a(17) >= 1 because there exists (at least one) solution k = 4903038892893242229501 = 3 * 17 * 96138017507710631951 with A003415(k) = 1+A002110(17).
For other cases, see examples in A369246.

Cf. A002110, A003415, A006862, A046316, A369054, A369246 (the solutions), A369252, A369461.

Cf. also A116979, A369000, A369239 for similar counts, also A369241, A369242 and A369247.

\\ Needs also program from A369054.
A002110(n) = prod(i=1,n,prime(i));
A369245(n) = A369054(A002110(n)+1);

\\ Optimized version of above, employs the fact that solutions must all be multiples of 3. Outputs also terms for A369246.
search_for_3k1_cases(n) = if(3!=(n%4),0, my(p = 5, q, c=0); while(1, q = (n-(3*p)) / (3+p); if(q < p, return(c), if(1==denominator(q) && isprime(q),c++; write("b369246_by_search_order_to.txt", n, " ", 3*p*q))); p = nextprime(1+p)));
A002110(n) = prod(i=1,n,prime(i));
A369245(n) = search_for_3k1_cases(A002110(n)+1);

a(n) = A369054(A006862(n)).

A082917 Numbers that can be expressed as the sum of two odd primes in more ways than any smaller even number.

Original entry on oeis.org

6, 10, 22, 34, 48, 60, 78, 84, 90, 114, 120, 168, 180, 210, 300, 330, 390, 420, 510, 630, 780, 840, 990, 1050, 1140, 1260, 1470, 1650, 1680, 1890, 2100, 2310, 2730, 3150, 3570, 3990, 4200, 4410, 4620, 5250, 5460, 6090, 6510, 6930, 7980, 8190, 9030, 9240

Offset: 1

Hugo Pfoertner, Apr 15 2003

nonn

The terms up to 114 are identical with A001172. The record-setting number of decompositions is given by A082918.

It appears that every primorial number (A002110) greater than 30 is in this sequence. Sequence A116979 gives the number of decompositions for n equal to a primorial number. - T. D. Noe, Mar 15 2010

			a(1) = 6 = 3 + 3.
a(2) = 10 because 10 is the smallest number that can be written in two ways: 10 = 3 + 7 = 5 + 5.

Bill Hannaford, Table of n, a(n) for n = 1..420 (first 244 terms from T. D. Noe)

Cf. A002375, A001172, A082918. A109679 is another version of the same sequence.

kmax = 40000;
ip[k_] := IntegerPartitions[k, {2}, Select[Range[3, k-1], PrimeQ]];
seq = Module[{k, lg, record = 0, n = 0}, Reap[For[k = 6, k <= kmax, k = k+2, lg = Length[ip[k]]; If[lg > record, record = lg; n = n+1; Print["a(", n, ") = ", k]; Sow[k]]]][[2, 1]]] (* Jean-François Alcover, Jun 04 2022 *)

A368704 a(n) is the greatest integer k whose arithmetic derivative is equal to the n-th primorial, and 0 if no such k exists.

Original entry on oeis.org

0, 9, 221, 11021, 1333349, 225450221, 65155115009, 23520996509141, 12442607161209161, 10464232622576957201, 10056127550296456854221, 13766838616355849433396389, 23142055714094182897602595769, 42789661015360144177667200022669, 94522361182930558488466844910827309, 265513312562851938794103367354849976069

Offset: 1

Antti Karttunen, Jan 16 2024

nonn

a(n) is the greatest integer k for which A003415(k) = A002110(n), and 0 if no such k exists.

A147517 Number of pairs of primes p < q such that (p+q)/2 = A002110(n), the n-th primorial.

Original entry on oeis.org

0, 1, 6, 30, 190, 1564, 17075, 226758, 3792532, 82116003, 1975662890

Offset: 1

Bill McEachen, Nov 05 2008

The sequence is infinite and illustrates the number of primes expected to be centered around a given primorial.
Given ever-increasing primorial P, one can expect to find the highest symmetrical prime just below 2P.
Using a limited dataset, the approximate relation is the quadratic Y=Ax^2+Bx+C (A,B,C)=(0.12267, 0.75758, -1.592) where Y = log(number of prime pairs) (each > the prime factors) and x is number of prime factors of the seed primorial.
Standard heuristics give a(n) ~ exp(gamma)*log(p)*p#/p^2 where p is the n-th prime and gamma is A001620. - Charles R Greathouse IV, Jul 13 2022

			There are 6 pairs centered at primorial=30: (29,31),(23,37),(19,41),(17,43),(13,47),(7,53). As they are symmetrical, each prime pair sums to twice the primorial center.

Bill McEachen, PARI script, Jun 03 2010

Cf. A002110, A002375, A116979.

f = Compile[{{n, Integer}}, Block[{p = 2, c = 0, pn = Times @@ Prime@ Range@ n}, While[p < pn, If[PrimeQ[ 2pn -p], c++]; p = NextPrime@ p]; c]]; Array[f, 10] (* _Robert G. Wilson v, Feb 08 2018 *)

a(n) = pn = prod(k=1, n, prime(k)); nb = 0; forprime(p=2, pn-1, if (isprime(2*pn-p), nb++)); nb; \\ Michel Marcus, Jul 09 2017

a(n) = A002375(A002110(n)). - T. D. Noe, Nov 07 2008

Better description by T. D. Noe, Nov 09 2008
Typo corrected typo by T. D. Noe, Nov 10 2008
Edited by Michel Marcus, Jul 09 2017
a(10)-a(11) from Bill McEachen, Jan 30 2018

A228943 Number of decompositions of highly composite numbers (A002182) into unordered sums of two primes.

Original entry on oeis.org

0, 0, 1, 1, 1, 3, 4, 5, 6, 12, 14, 18, 22, 39, 51, 68, 83, 112, 184, 251, 315, 431, 527, 652, 768, 1011, 1128, 1305, 1836, 2344, 3240, 4082, 4955, 5725, 8023, 8723, 10260, 11945, 16771, 21466, 30280, 38583, 46645, 54789, 77430, 85067, 99199, 120742, 154753

Offset: 1

Jaycob Coleman, Sep 08 2013

nonn

a(n) = A045917(A002182(n)/2) for n>1.

Conjecture: (a) This sequence is strictly increasing beginning with n=5. (b) For all n>2, if p is the greatest prime with p<A002182(n)-1, then A002182(n)-p is prime. This is a strengthening of a conjecture regarding A117825. - Jaycob Coleman, Sep 08 2013

			a(6)=3, since 24=5+19=7+17=11+13.

Cf. A116979, A117825.

nbd(n) = my(s); forprime(p=2, n\2, s+=isprime(n-p)); s;
lista(nn) = {last = 1; print1(nbd(last), ", "); forstep(n=2, nn, 2, if(numdiv(n)> last, last=numdiv(n); print1(nbd(n), ", ")););} \\ Michel Marcus, Sep 10 2013

More terms from Michel Marcus, Sep 10 2013

A286424 Number of partitions of p_n# into parts (q, k) both coprime to p_n#, with q prime and k nonprime, where p_n# = A002110(n).

Original entry on oeis.org

0, 0, 1, 1, 4, 110, 1432, 23338, 397661, 8193828, 212858328, 5941706227

Offset: 0

Michael De Vlieger, May 08 2017

Number of totative pairs (q, k) such that prime q + k nonprime = p_n# and both gcd(q, p_n#) = 1 and gcd(k, p_n#) = 1, with p_n < q <= pi(p_n#), where pi(p_n#) = A000849(n) - n = A048862(n).
Primes p_n < q <= pi(p_n#) are greater than the greatest prime factor of p_n# = p_n, and are thus coprime to p_n#. By the definition of primorial, we need not consider p >= p_n, as these p are divisors of p_n#, i.e., gcd(p, p_n#) = p. Since the totatives of m can be paired such that a + b = m, we need only determine if (p_n# - q) is not prime in order to count pairs (q, k).
a(n) < floor(A005867(n)/2).
a(n) <= A048862(n).
The totative pair (q,1) = (p_n# - 1, 1) is counted by a(n) for n in A057704, with (p_n# - 1) appearing in A057705.

			a(0) = 0 by definition. A002110(0) = 1; 1 is coprime to all numbers; the only possible totative pair is (1,1) and this does not include both a prime and a nonprime.
a(1) = 0 since, of the floor(A005867(1)/2) = 1 totative pair (1,1) of A002110(1) = 2, none include a both a prime and a nonprime.
a(2) = 1 since, the only totative pair (1,5) of A002110(1) = 6 includes both a prime and a nonprime.
a(3) = 1 since only (1,29) includes both a prime and a nonprime.
a(4) = 4 since (23,187), (41,169), (67,143), (89,121) include a both a prime and a nonprime.

C. K. Caldwell, The Prime Glossary, Primorial.
Eric Weisstein's World of Mathematics, Totative.

Cf. A000849, A002110, A005867, A048862, A048863, A057704, A057705, A116979, A117929.

Table[Function[P, Count[Prime@ Range[n + 1, PrimePi[P]], q_ /; ! PrimeQ[P - q]]]@ Product[Prime@ i, {i, n}], {n, 0, 9}] (* Michael De Vlieger, May 08 2017 *)

a(n) = (A000010(A002110(n)) - A048863(n)) - 2*A117929(A002110(n))
     = (A005867(n) - A048863(n)) - 2*A117929(A002110(n))
     = A048862(n) - 2*A117929(A002110(n)).

a(11) from Giovanni Resta, May 09 2017

A336093 Let P(n) = primorial(n) = A002110(n); a(n) is the number of primes q < P(n) such that P(n) - q is also prime and q^2==1 (mod P(n)).

Original entry on oeis.org

0, 0, 2, 4, 2, 8, 6, 28, 36, 40, 56, 106, 192, 304, 526, 926, 1644, 2756, 4944, 8840, 15958, 28402, 51102, 92372

Offset: 1

David James Sycamore, Michael De Vlieger Jul 09 2020

A totative of k is a number <= k and relatively prime to k.

A336016(n) gives the number of prime totatives p of P(n) for which p^2==1 (mod P(n)). Whereas P(n) - p also has this property, it is not always prime. This sequence gives the number of prime totatives q of P(n) such that q^2==1 mod P(n) and P(n) - q is also prime. All terms are even; a(n) <= A336016(n) for all n. The number of distinct representations of P(n) as the sum of two primes each having multiplicative order 2 (mod P(n)) is given by a(n)/2, for which A116979(n) is an upper bound.

			P(4)=210; all totatives 29,41,71,139,181 are prime. However 210 - 41 = 169 is not prime, whereas 210-29 = 181, 210-71 = 139. Therefore the totatives we count in this case are 29,71,139,181, so a(4) = 4.

Cf. A000010, A336015, A336016, A116979.

with(NumberTheory):
P := proc (k)
local n, v, W, H;
n := 1; v := 0;
W := product(ithprime(j), j = 1 .. k);
H := PrimeCounting(W);
for n from 1 to H do
if mod(ithprime(n)^2, W) = 1 and isprime(W-ithprime(n)) then v := v+1 else v := v end if:
end do:
v;
end proc:
seq(P(k), k = 1 .. 8);

{0, 0}~Join~Table[Block[{P = #, k = 0}, Do[If[AllTrue[{#, P - #}, And[PrimeQ@ #, MultiplicativeOrder[#, P] == 2] &], k++] &@ Prime[i], {i, PrimePi[n + 1], PrimePi[P/2]}]; 2 k ] &@ Product[Prime@ j, {j, n}], {n, 3, 8}]

cp's OEIS Frontend

A351029 Number of integers whose arithmetic derivative is equal to the n-th primorial.

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A366890 Irregular triangle, wherein row n lists in ascending order all numbers k whose arithmetic derivative k' is equal to the n-th primorial, A002110(n), and that have more than two prime factors with multiplicity. Rows of length zero are simply omitted, i.e., when A369000(n) = 0.

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PARI

A369000 a(n) is the number of integers, not semiprimes, whose arithmetic derivative is equal to the n-th primorial.

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Formula

A369245 Number of representations of the n-th Euclid number, A002110(n) + 1, as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r. (Definition implies that p=3 and q > 3).

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Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

PARI

Formula

A082917 Numbers that can be expressed as the sum of two odd primes in more ways than any smaller even number.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A368704 a(n) is the greatest integer k whose arithmetic derivative is equal to the n-th primorial, and 0 if no such k exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A147517 Number of pairs of primes p < q such that (p+q)/2 = A002110(n), the n-th primorial.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A369245 Number of representations of the n-th Euclid number, A002110(n) + 1, as a sum (pq + pr + q*r) with three odd primes p <= q <= r. (Definition implies that p=3 and q > 3).