A368704 -id:A368704 - cp's OEIS frontend

A327978 Numbers whose arithmetic derivative (A003415) is a primorial number (A002110) > 1.

9, 161, 209, 221, 2189, 2561, 3281, 3629, 5249, 5549, 6401, 7181, 7661, 8321, 8909, 9089, 9869, 10001, 10349, 10541, 10961, 11009, 11021, 29861, 38981, 52601, 66149, 84101, 93029, 97481, 132809, 150281, 158969, 163301, 197669, 214661, 227321, 235721, 285449, 321989, 338021, 357881, 369701, 381449, 385349, 416261, 420089, 442889

Offset: 1

Antti Karttunen, Oct 09 2019

nonn

Numbers n such that A327859(n) = A276086(A003415(n)) is an odd prime.
Composite terms in A328232.
Although it first might seem that the numbers whose arithmetic derivative is A002110(k) all appear before any of those whose arithmetic derivative is A002110(k+1), that is not true, as for example, we have a(56) = 570149, and A003415(570149) = 2310, a(57) = 570209, and A003415(570209) = 30030, but then a(58) = 573641 with A003415(573641) = 2310 again.
Because this is a subsequence of A327862 (all primorials > 1 are of the form 4k+2), only odd numbers are present.
Conjecture: No multiples of 5 occur in this sequence, and no multiples of 3 after the initial 9.
Of the first 10000 terms, all others are semiprimes (with 9 the only square one), except 1547371 = 7^2 * 23 * 1373 and 79332523 = 17^2 * 277 * 991, the latter being the only known term whose decimal expansion ends with 3. If all solutions were semiprimes p*q such that p+q = A002110(k) for some k > 1 (see A002375), it would be a sufficient reason for the above conjecture to hold. - David A. Corneth and Antti Karttunen, Oct 11 2019
In any case, the solutions have to be of the form "odd numbers with an even number of prime factors with multiplicity" (see A235992), and terms must also be cubefree (A004709), as otherwise the arithmetic derivative would not be squarefree.
Sequence A366890 gives the non-Goldbachian solutions, i.e., numbers that are not semiprimes. See also A368702. - Antti Karttunen, Jan 17 2024

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1078 terms from Antti Karttunen)
Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003, #03.3.4.
Index entries for sequences related to Goldbach conjecture
Index entries for sequences related to primorial numbers

Cf. A002110, A002375, A002620, A003415, A024451, A143293, A157037, A276150, A327859, A327969, A328233, A328243.
Cf. A351029 (number of k for which k' = A002110(n)).
Cf. A368703, A368704 (the least and the greatest k for which k' = A002110(n)).
Cf. A366890 (terms that are not semiprimes), A368702 (numbers k such that k' is one of the terms of this sequence).
Subsequence of following sequences: A004709, A189553, A327862, A328232, A328234.

ad[n_] := n * Total @ (Last[#]/First[#] & /@ FactorInteger[n]); primQ[n_] := Max[(f = FactorInteger[n])[[;;,2]]] == 1 && PrimePi[f[[-1,1]]] == Length[f]; Select[Range[10^4], primQ[ad[#]] &] (* Amiram Eldar, Oct 11 2019 *)

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]));
A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
isA327978flat(n) = { my(u=A003415(n)); ((u>1)&&(1==A276150(u))); }; \\ Slow!
k=0; for(n=1,A002620(30030),if(isA327978flat(n), k++; write("b327978.txt", k, " ", n)));

A327969(a(n)) = 4 for all n.

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

Original entry on oeis.org

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

A368703 a(n) is the least integer k whose arithmetic derivative is equal to the n-th primorial, or 0 if no such k exists.

Original entry on oeis.org

2, 0, 9, 161, 2189, 29861, 510221, 1547371, 79332523, 9592991561, 265257420749, 1102527599503

Offset: 0

Antti Karttunen, Jan 16 2024

a(n) = the smallest integer k for which A003415(k) = A002110(n), and 0 if no such k exists.
If there are non-Goldbachian solutions (A366890) for some n, i.e., if A369000(n) > 0, then the smallest of them appears here as a value of a(n).
a(12) <= 25962012375103, a(13) <= 4958985803436403, a(14) <= 32442711864461575, a(15) <= 11758779158543465383. - David A. Corneth, Jan 17 2024

			a(0) = 2 as the least number k such that A003415(k) = A002110(0) = 1 is 2.
a(1) = 0 as there is no number k such that A003415(k) = A002110(1) = 2.
a(7) = 1547371 as it is the least number k such that A003415(k) = A002110(7) = 510510. See also A366890.

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

Cf. A002110, A003415, A351029, A368704, A366890, A369000.

Cf. also A369243.

a(n) <= A368704(n).

For n<>1, A003415(a(n)) = A002110(n).

A369244 a(n) is the greatest integer k whose arithmetic derivative is equal to the n-th partial sum of primorials, and 0 if no such k exists.

Original entry on oeis.org

0, 14, 74, 198, 10295, 65174, 40354813, 20485574, 680909375411, 0, 17866904665985941, 318745032938881

Offset: 1

Antti Karttunen, Jan 19 2024

a(n) = the largest integer k for which A003415(k) = A143293(n), and 0 if no such k exists.

Antti Karttunen, PARI program for computing terms of this and related sequences.

Cf. A003415, A143293, A369243.

Cf. also A368704.

a(n) >= A369243(n).

A369059 a(0) = 0, and for n > 0, a(n) is the square of the product of first n-1 odd primes.

Original entry on oeis.org

0, 1, 9, 225, 11025, 1334025, 225450225, 65155115025, 23520996524025, 12442607161209225, 10464232622576958225, 10056127550296456854225, 13766838616355849433434025, 23142055714094182897602596025, 42789661015360144177667200050225, 94522361182930558488466844910947025, 265513312562851938794103367354850193225

Offset: 0

Antti Karttunen, Jan 19 2024

nonn

Terms a(2) .. a(9) are equal to the terms A360435(1..8).

Index entries for sequences related to primorial numbers

Cf. A000005, A000244, A002110, A002620, A061742, A070826, A360435.

An upper bound for A368704, and also for A360435(n-1).

Join[{0}, FoldList[Times, 1, Prime[Range[2, 16]]^2]] (* Amiram Eldar, Jan 19 2024 *)

A002110(n) = prod(i=1,n,prime(i));
A002620(n) = ((n^2)>>2);
A369059(n) = A002620(A002110(n));

a(n) = A002620(A002110(n)).
For n > 0, a(n) = A061742(n)/4 = A070826(n)^2.
For n > 0, a(n) >= A368704(n) and for n > 1, a(n) >= A360435(n-1).
For n > 0, A000005(a(n)) = A000244(n-1) = 3^(n-1).

cp's OEIS Frontend

A327978 Numbers whose arithmetic derivative (A003415) is a primorial number (A002110) > 1.

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

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A368703 a(n) is the least integer k whose arithmetic derivative is equal to the n-th primorial, or 0 if no such k exists.

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A369244 a(n) is the greatest integer k whose arithmetic derivative is equal to the n-th partial sum of primorials, and 0 if no such k exists.

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A369059 a(0) = 0, and for n > 0, a(n) is the square of the product of first n-1 odd primes.

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