A287646 -id:A287646 - cp's OEIS frontend

A188439 Irregular triangle of odd primitive abundant numbers (A006038) in which row n has numbers with n distinct prime factors.

945, 1575, 2205, 7425, 78975, 131625, 342225, 570375, 3465, 4095, 5355, 5775, 5985, 6435, 6825, 7245, 8085, 8415, 8925, 9135, 9555, 9765, 11655, 12705, 12915, 13545, 14805, 16695, 18585, 19215, 21105, 22365, 22995, 24885, 26145, 28035, 28215, 29835

Offset: 3

T. D. Noe, Mar 31 2011

The initial row has 8 terms. Row n begins with A188342(n). Dickson proves that each row has a finite number of terms. He lists the first two rows in factored form in his paper. However, as Ferrier and Herzog report, Dickson's tables have many errors. There are 576 odd primitive abundant numbers (OPAN) having 4 distinct prime factors, the last of which is 3^10 5^5 17^4 251^2 = 970969744245403125. The next row, for 5 distinct prime factors, has over 100000 terms.

If the prime factors were counted with multiplicity, then the table would start with row 5, having 121 terms: (945, 1575, 2205, 3465, 4095, ..., 430815, 437745, 442365). Row 6 would start (7425, 28215, 29835, 33345, 34155, ...), and row 7, (81081, 121095, 164835, 182655, 189189, ...). - M. F. Hasler, Jul 27 2016 [See A287646.]

			From _M. F. Hasler_, Jul 27 2016: (Start)
Row 3: 945, 1575, 2205, 7425, 78975, 131625, 342225, 570375;
Row 4: 3465, 4095, 5355, ...(571 more)..., 249450402403828125, 970969744245403125;
Row 5: 15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, ...
Row 6: 692835, 838695, 937365, 1057485, 1130415, 1181895, 1225785, 1263405, ...
Row 7: 22309287, 28129101, 30069039, 34051017, 35888853, 36399363, ...
The first column is A188342 = (945, 3465, 15015, 692835, 22309287, ...) (End)

T. D. Noe, Rows n = 3..4, flattened
L. E. Dickson, Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors, American Journal of Mathematics 35 (1913), pp. 413-422.
A. Ferrier, Table errata 176, MTAC 4 (1950), 222.
Fritz Herzog, Table Errata 571, Math. Comp. 34 (1980), 652.
T. D. Noe, 576 odd primitive abundant numbers, factored

Row lengths are A303933.

Cf. A006038 (all OPAN), A188342 (first column of this table), A287646 (variant where row n contains all OPAN with n prime factors counted with multiplicity).

A287581 Largest squarefree odd primitive abundant number with n prime factors.

Original entry on oeis.org

442365, 13455037365, 1725553747427327895, 977844705701880720314685634538055, 29094181301361888360228876470808927597684302024968488289496445

Offset: 5

M. F. Hasler, May 26 2017

nonn

There is no squarefree odd abundant number with fewer than 5 prime factors: the largest abundancy an odd squarefree number with 4 prime factors can have is that of N = 3*5*7*11 with sigma_{-1}(N) = sigma(N)/N = 2 - 2/385.
See A287590 for the number of squarefree odd primitive abundant numbers (A249263) with n prime factors.
The next term, a(10), is too large to display.
It appears that the largest odd primitive abundant number with a given number of prime factors counted with multiplicity (bigomega = A001222), is always squarefree. Whenever this holds for a given n, then a(n) is also equal to the last term in row n of A287646 which lists odd primitive abundant numbers with n prime factors.

			a(5) = 442365 = 3 * 5 * 7 * 11 * 383 is the largest squarefree odd primitive abundant number (SOPAN). Here, 3*5*7*11 is the smallest possibility to produce a squarefree odd deficient number with 4 prime factors, and it is the one with the largest possible abundancy, and 383 is the largest prime by which this can be multiplied to yield an abundant number. One can increase 11 up to 19 to get more SOPAN (for a total of 71 + 12 + 3 + 1 = 87 = A287590(5) SOPAN with 5 factors), none of which is larger. One can see that increasing the 3rd prime factor 7 to 11 yields no further possibilities, and therefore also the second and third factor can't be increased.
a(6) = 13455037365 = 3 * 5 * 7 * 11 * 389 * 29947,
a(7) = 1725553747427327895 = 3 * 5 * 7 * 11 * 389 * 29959 * 128194559,
a(8) = 3 * 5 * 7 * 11 * 389 * 29959 * 128194589 * 566684450325179,
a(9) = a(8)/gpf(a(8)) * 566684450325197 * 29753376105337343078941364893,
a(10) = a(9)/gpf(a(9)) * 29753376105337343078941364947 * 30082232218581187462432471034748868284388270918928732059.

Cf. A287590, A249263.

A287581(n,p=3,P=p,s=2)={forstep(i=n,2,-1,n=max(1\(-1+s/=1+1/p),p+1); P*=p=if(i>2,nextprime(n),precprime(n)));P}

a(n+1) = (a(n)/p(n))*p'(n)*q(n), where p(n) = gpf(a(n)), p'(n) = nextprime(p(n)+1), q(n) = precprime(1/(2/sigma[-1](a(n)/p(n)*p'(n))-1)), sigma[-1](x) = sigma(x)/x; conjectured to hold for all n >= 5.

cp's OEIS Frontend

A188439 Irregular triangle of odd primitive abundant numbers (A006038) in which row n has numbers with n distinct prime factors.

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