A287646 Irregular triangle read by rows where row n lists all odd primitive abundant numbers with n prime factors, counted with multiplicity.

945, 1575, 2205, 3465, 4095, 5355, 5775, 5985, 6435, 6825, 7245, 8085, 8415, 8925, 9135, 9555, 9765, 11655, 12705, 12915, 13545, 14805, 15015, 16695, 18585, 19215, 19635, 21105, 21945, 22365, 22995, 23205, 24885, 25935, 26145, 26565, 28035, 30555, 31395, 31815, 32445, 33495

Offset

5,1

Comments

This triangle is the analog of A188439 for A001222 ("bigomega", total number of prime factors) instead of A001221 ("omega", distinct prime divisors). It starts with row 5, since there is no odd primitive abundant number, N in A006038, with less than A001222(N) = 5 prime factors (counted with multiplicity).

Sequence A287728 gives the row lengths: Row 5 has 121 terms (945, 1575, 2205, 3465, 4095, ..., 430815, 437745, 442365). This mostly equals the initial terms of A006038, except for those with indices {12, 39, 40, 45, 48, 54, ..., 87}. These are in turn mostly (except for the 17th and 18th term) those of the subsequent row 6 which has 15772 terms, (7425, 28215, 29835, 33345, 34155, ..., 13443355695, 13446051465, 13455037365).

Sequences A275449 and A287581 give the smallest and largest* element of each row (*assuming that the largest term in the row is squarefree). Accordingly, row 7 starts with A275449(7) = 81081, and ends with A287581(7) = 1725553747427327895.

Links

Table of n, a(n) for n=5..46.

Prog

(PARI) A287646_row( r, p=3, a=2, n=1/(a-1))={ r>1 || return(if(n>=p, primes([p, n]))); p<n && p=nextprime(n); my(e=1, S=if(p-1/p^r>(p-1)*a && p-1/p^(r-1)<(p-1)*a, [p^r], []), ap=1, np=nextprime(p+1)); until( 0, if( (1+1/np)^(r-e) > (aa = a/ap += 1/p^e) && aa > 1, if(n=A287646_row(r-e, np, aa), if(e>1, my(aaa=a/(ap-1/p^e)); n=select(t->sigma(t, -1)<aaa, n)); S=setunion(S, p^e*n); e++<r && next), n=0); e>1 || n || break; np=nextprime((e=ap=1)+p=np)); S}

Crossrefs

Cf. A188439, A001222, A006038, A287728, A275449, A287581, A338133.

Sequence in context: A174535 A243104 A006038 * A316116 A188439 A275472

Adjacent sequences: A287643 A287644 A287645 * A287647 A287648 A287649

Keyword

nonn,tabf

Author

M. F. Hasler, May 30 2017

Status

approved