A351029 - OEIS

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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

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OFFSET

1,3

COMMENTS

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

LINKS

Table of n, a(n) for n=1..12.

Antti Karttunen, PARI-program

Index entries for sequences related to Goldbach conjecture

Index entries for sequences related to primorial numbers

FORMULA

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

EXAMPLE

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.

PROG

(PARI)

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.

(PARI) 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

CROSSREFS

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.

Sequence in context: A037774 A037662 A116979 * A267802 A229928 A309183

Adjacent sequences: A351026 A351027 A351028 * A351030 A351031 A351032

KEYWORD

nonn,hard,more

AUTHOR

Antti Karttunen, Feb 01 2022

EXTENSIONS

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

STATUS

approved