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

%I A287646 #26 Jan 17 2023 16:30:48
%S A287646 945,1575,2205,3465,4095,5355,5775,5985,6435,6825,7245,8085,8415,8925,
%T A287646 9135,9555,9765,11655,12705,12915,13545,14805,15015,16695,18585,19215,
%U A287646 19635,21105,21945,22365,22995,23205,24885,25935,26145,26565,28035,30555,31395,31815,32445,33495
%N A287646 Irregular triangle read by rows where row n lists all odd primitive abundant numbers with n prime factors, counted with multiplicity.
%C A287646 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).
%C A287646 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).
%C A287646 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.
%o A287646 (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}
%Y A287646 Cf. A188439, A001222, A006038, A287728, A275449, A287581, A338133.
%K A287646 nonn,tabf
%O A287646 5,1
%A A287646 _M. F. Hasler_, May 30 2017