OFFSET
1,3
COMMENTS
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
FORMULA
EXAMPLE
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
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