A328243
Numbers whose arithmetic derivative (A003415) is larger than 1 and one of the terms of A143293 (partial sums of primorials).
Original entry on oeis.org
14, 45, 74, 198, 5114, 10295, 65174, 1086194, 20485574, 40354813, 465779078, 12101385979, 15237604243, 18046312939, 29501083259, 52467636437, 65794608773, 86725630997, 87741700037, 131833085077, 168380217557, 176203950283, 177332276971, 226152989747, 292546582253
Offset: 1
Sequence
A369240 sorted into ascending order.
-
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]));
A143293(n) = if(n==0, 1, my(P=1, s=1); forprime(p=2, prime(n), s+=P*=p); (s)); \\ From A143293.
A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
isA328243(n) = { my(u=A003415(n)); ((u>1)&&(1==A276150(A276086(u)))); }; \\ This is very slow program!
k=0; for(n=1,A002620(A143293(6)),if(isA328243(n), k++; print1(n,", ")));
A366890
Irregular triangle, wherein row n lists in ascending order all numbers k whose arithmetic derivative k' is equal to the n-th primorial, A002110(n), and that have more than two prime factors with multiplicity. Rows of length zero are simply omitted, i.e., when A369000(n) = 0.
Original entry on oeis.org
1547371, 79332523, 1102527599503, 25336943536819, 25962012375103, 25970380120783, 66702554987143, 526285951027003, 927949814519899, 7777707036642079, 9584173681667203, 13082430772438171, 22101822021783739, 4958985803436403, 32006922970429003, 32076018550175863, 49806227168831659, 84682266449971639, 97995266657958403
Offset: 1
For rows n=1..6, 9 & 10 nothing is listed, as those rows are empty.
Row for n=7 has just one term: 1547371 (= 7^2 * 23 * 1373). Note that A003415(1547371) = 510510 = A002110(7).
Row for n=8 has just one term: 79332523 (= 17^2 * 277 * 991).
Row for n=11 has two terms:
1102527599503 (= 11^2 * 11071 * 823033),
25336943536819 (= 157 * 743 * 5749 * 37781).
Row for n=12 has nine terms:
25962012375103 (= 7^2 * 8597 * 61630451),
25970380120783 (= 7^2 * 41387 * 12806141),
66702554987143 (= 19^2 * 167 * 1106416889),
526285951027003 (= 73 * 3919 * 7013 * 262313),
927949814519899 (= 269 * 271 * 1697 * 7501033),
7777707036642079 (= 2203 * 2791 * 7349 * 172127),
9584173681667203 (= 2131 * 5953 * 7901 * 95621),
13082430772438171 (= 3109 * 5861 * 24421 * 29399),
22101822021783739 (= 8783 * 11777 * 13921 * 15349).
Row for n=13 has 18 terms, and begins with:
4958985803436403 (= 37^2 * 137 * 26440450451),
and ends with:
3206697143570677543 (= 36899 * 41983 * 45233 * 45763).
Note that A003415(3206697143570677543) = 304250263527210 = A002110(13).
When the whole sequence is sorted into ascending order, equal to
A327978 without any semiprime solutions (solutions in
A001358), and also a subsequence of following sequences:
A004709,
A327862,
A328234.
A369239
Number of integers whose arithmetic derivative is larger than 1 and equal to the n-th partial sum of primorial numbers.
Original entry on oeis.org
0, 1, 2, 1, 2, 1, 2, 1, 27, 0, 319, 1
Offset: 1
a(12) = 1 as there is a unique solution k such that k' = A143293(12) = 7628001653829, that k being 318745032938881 = 71*173*307*1259*67139. It's also the first solution with more than four prime factors.
a(14) >= 1, because as A143293(14)-2 = 13394639596851069-2 = 13394639596851067 is a prime, we have at least one solution, with A003415(2*13394639596851067) = A003415(26789279193702134) = 2+13394639596851067 = A143293(14).
For more examples, see A369240.
A377992
Irregular triangle giving on row n all antiderivatives of A024451(n), for n >= 2.
Original entry on oeis.org
6, 30, 58, 210, 435, 507, 2310, 8435, 21827, 29233, 30030, 39030, 62762, 69914, 76442, 78874, 510510, 1342785, 1958673, 9699690, 28235362, 223092870, 975351895, 1527890095, 1885679383, 2189118743, 2329696457, 2338611863, 3485765789, 4586671213, 5453593183, 5472849253, 5674340053, 8071055747, 8931775397, 9332889127
Offset: 2
The initial rows of the triangle:
Row n terms
2 6;
3 30, 58;
4 210, 435, 507;
5 2310, 8435, 21827, 29233;
6 30030, 39030, 62762, 69914, 76442, 78874;
7 510510, 1342785, 1958673;
8 9699690, 28235362;
9 223092870, 975351895, 1527890095, ..., , 1167539981207, 1171314743479; (row 9 has 330 terms that are given separately in A378209)
10 6469693230, 27623935255, 37262208055;
11 200560490130, 345634019382, 440192669882;
etc.
The only terms that occur on row 4 are k = 210, 435, 507 ( = 2*3*5*7, 3*5*29, 3 * 13^2) as they are only numbers for which A003415(k) = 247 = A024451(4) = A003415(A002110(4)), as we have (2*3*5*7)' = (3*5)'*(2*7) + (2*7)'*3*5 = (8*14) + (9*15) = (3*5*29)' = (3*5)'*29 + (3*5)*29' = (8*29 + 15*1) = (3 * 13 * 13)' = (3*13)'*13 + (3*13)*13' = 16*13 + 3*13*1 = 19*13 = 247.
Note that 507 is so far the only known term in this triangle that is not squarefree (in A005117).
Subsequence of
A048103, conjectured also to be a subsequence of
A004709.
Cf.
A002110 (left edge, from its term a(2)=6 onward),
A378209 (row 9).
A377987
Irregular triangle giving on row n all those antiderivatives k of the n-th factorial, for which bigomega(k) > 2.
Original entry on oeis.org
20, 116, 716, 2512, 5036, 40316, 84672, 176364, 1390500, 1782108, 3628773, 3628796, 10529953, 12258673, 76944384, 5338541473, 8944397353, 11690698969, 1236868096, 1849666112, 3096111708, 1004929973233, 54465962625, 1657198101073, 6791831913289, 1307674367996, 5739085040351, 21522396453889, 63577408859233, 104747513922049, 287711613106993, 626768279186209
Offset: 4
Row n k such that A003415(k) = n! and A001222(k) > 2.
(no solutions for n = 1..3)
4: 20; (20 = 2*2*5, so 20' = 4'*5 + 5'*4 = 4*5 + 1*4 = 24 = 4!)
5: 116; (116 = 2*2*29, so 116' = 4*29 + 1*4 = 120 = 5!)
6: 716; (716 = 2*2*179, so 716' = 4*179 + 1*4 = 720 = 6!)
7: 2512, 5036;
8: 40316;
9: 84672, 176364; (2^6 * 3^3 * 7^2 and 2^2 * 3^3 * 23 * 71)
10: 1390500, 1782108, 3628773, 3628796, 10529953, 12258673;
11: (no solutions)
12: 76944384, 5338541473, 8944397353, 11690698969;
13: 1236868096, 1849666112, 3096111708, 1004929973233;
14: 54465962625, 1657198101073, 6791831913289;
15: 1307674367996, 5739085040351, 21522396453889, 63577408859233, 104747513922049, 287711613106993, 626768279186209;
etc.
Note that although A003415(9) = 6 = 3!, it is not included in this table as 9 is a semiprime, with A001222(9) = 2.
-
\\ Use the programs given in A377987 and A376410.
\\ the data needs also to be post-processed (sorted) with
\\ sols = sort_solutions_vector(readvec("a_terms_for_A377987_unsorted.txt"));
\\ using these functions:
sort_solutions_vector(v) = vecsort(v,sort_by_A003415_and_magnitude);
sort_by_A003415_and_magnitude(x,y) = { my(s = sign(A003415(x)-A003415(y))); if(!s, sign(x-y), s); };
A369246
Irregular triangle read by rows, where row n lists in ascending order all numbers k in A046316 for which k' = the n-th Euclid number, where k' stands for the arithmetic derivative, and the Euclid numbers are given by A006862. Rows of length zero are simply omitted, i.e., when A369245(n) = 0.
Original entry on oeis.org
399, 4809, 5763, 63021, 76449, 1301673, 19204051701, 421177029231, 908999759928891, 39248269334566041, 39248273246018313, 68437232802099093891, 4903038892893242229501
Offset: 1
Rows 1..3 have no terms.
Row 4 has one term: 399 = 3 * 7 * 19, whose arithmetic derivative (see A003415) 399' is 211 = 1 + prime(4)# [= A006862(4)].
Row 5 has two terms: 4809 = 3 * 7 * 229 and 5763 = 3 * 17 * 113, with 4809' = 5763' = 2311 = 1 + prime(5)#.
Row 6 has two terms: 63021 = 3 * 7 * 3001 and 76449 = 3 * 17 * 1499, with 63021' = 76449' = 30031 = 1 + prime(6)#.
Row 7 has one term: 1301673 = 3 * 17 * 25523, whose arithmetic derivative is 510511 = 1 + prime(7)#.
Rows 8 and 9 have no terms.
Row 10 has one term: 19204051701 = 3 * 281 * 22780607, whose arithmetic derivative is 6469693231 = 1 + prime(10)#.
Row 11 has one term: 421177029231 = 3 * 7 * 20056049011, whose arithmetic derivative is 200560490131 = 1 + prime(11)#.
Row 12 has no terms.
Row 13 has one term: 908999759928891 = 3 * 727 * 416781182911, whose arithmetic derivative is 304250263527211 = 1 + prime(13)#.
Row 14 has two terms: 39248269334566041 = 3 * 8071457 * 1620866771, and 39248273246018313 = 3 * 11056387 * 1183276033, which both have arithmetic derivative 13082761331670031 = 1 + prime(14)#.
Row 15 has no terms.
Row 16 has one term: 68437232802099093891 = 3 * 7 * 3258915847719004471, whose arithmetic derivative is 32589158477190044731 = 1 + prime(16)#.
Row 17 has at least this term: 4903038892893242229501 = 3 * 17 * 96138017507710631951, whose arithmetic derivative is 1922760350154212639071 = 1 + prime(17)#.
Showing 1-6 of 6 results.
Comments