Giorgio Balzarotti - cp's OEIS frontend

A231864 Partial sums of the second power of arithmetic derivative function A003415.

0, 1, 2, 18, 19, 44, 45, 189, 225, 274, 275, 531, 532, 613, 677, 1701, 1702, 2143, 2144, 2720, 2820, 2989, 2990, 4926, 5026, 5251, 5980, 7004, 7005, 7966, 7967, 14367, 14563, 14924, 15068, 18668, 18669, 19110, 19366, 23990, 23991, 25672, 25673, 27977, 29498

Offset: 1

Giorgio Balzarotti, Nov 14 2013

nonn

a(n)-> ~ 0.4*n^3 as n-> oo (note: 1^2+2^2+3^3+4^4+5^4 ...-> ~ 1/3*n^3)

Note: the partial sums of a power of the arithmetic derivatives of the natural numbers tend to infinity as the partial sums of the natural numbers of the same power. In more general sense: sum(D^d(i)^m, i = 1..n) -> k*n^(m+1) as n-> oo where D^d(i) is the derivative of order d th of the natural number i (d may be = 0, i.e. no derivate).

			(1')^2+(2')^2+(3')^2+(4')^2+(5')^2=0+1+1+16+1=19->a(5)=19.

E. J. Barbeau, Remark on an arithmetic derivative, Canad. Math. Bull., vol. 4, no. 2, May 1961, pp. 117-122.

Cf. A003415, A190121, A231946

der:=n->n*add(op(2, p)/op(1, p), p=ifactors(n)[2]): seq(add(der(i)^2,i=1..j),j=1..45);

dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; Accumulate[Table[dn[n]^2, {n, 100}]] (* T. D. Noe, Nov 20 2013 *)

a(n) = sum((i')^2, i=1..n) where i'=A003415.

A232172 Partial sums of second arithmetic derivative of natural numbers.

Original entry on oeis.org

0, 0, 0, 4, 4, 5, 5, 21, 26, 27, 27, 59, 59, 65, 77, 157, 157, 167, 167, 211, 218, 219, 219, 267, 274, 282, 309, 389, 389, 390, 390, 566, 575, 576, 592, 684, 684, 694, 726, 798, 798, 799, 799, 911, 927, 937, 937, 1177, 1186, 1225, 1249, 1341, 1341, 1449, 1481

Offset: 1

Giorgio Balzarotti, Nov 19 2013

nonn

a(n) = 1''+2''+3''+4''+5''+...+n'' -> ~ constant * n^2 as n -> oo.

Note: a(n) = sum(D^d(i)^m,i=1..n) -> constant * n^(m+1) as n -> oo where D^d(i) is the derivative of order d th of the natural number i (results on arithmetic derivatives descent from Barbeau's paper in References).

			a(5) = 1'' + 2'' + 3'' + 4'' + 5'' = 0+0+0+4+0 = 4.

E. J. Barbeau, Remark on an arithmetic derivative, Canad. Math. Bull. vol. 4, no. 2, May 1961.

Cf. A003415, A068346, A190121, A231946.

der:=n->n*add(op(2, p)/op(1, p), p=ifactors(n)[2]): seq(add(der(der(i)),i=1..j),j=1..55);

a(n) = sum(i'', i=1..n), where i'' is the second arithmetic derivative of i (A068346).

A231946 Partial sums of the third power of the arithmetic derivative function A003415.

Original entry on oeis.org

0, 1, 2, 66, 67, 192, 193, 1921, 2137, 2480, 2481, 6577, 6578, 7307, 7819, 40587, 40588, 49849, 49850, 63674, 64674, 66871, 66872, 152056, 153056, 156431, 176114, 208882, 208883, 238674, 238675, 750675, 753419, 760278, 762006, 978006, 978007, 987268, 991364

Offset: 1

Giorgio Balzarotti, Nov 15 2013

nonn

a(n) grows roughly like 0.66*n^4 as n->oo.

Note: 1^3 + 2^3 + 3^3 + 4^3 + 5^3 ... -> ~ (1/4)*n^4; the asymptotic similarity between the sum of powers of positive integers and the sum of powers of their derivatives stands also with sums in which the terms are higher powers, i.e., Sum_{j=1..n} j'^m -> k*n^(m+1) as Sum_{j=i..n} j^m -> h*n^(m+1) when n->oo, in other words, the ratio of the two sums is a constant.

			(1')^3 + (2')^3 + (3')^3 + (4')^3 + (5')^3 = 0+1+1+64+1 = 67, so a(5)=67.

Robert Israel, Table of n, a(n) for n = 1..10000
E. J. Barbeau, Remark on an arithmetic derivative, Canad. Math. Bull., vol. 4, no. 2, May 1961, pp. 117-122.

Cf. A003415, A190121, A231864.

der:=n->n*add(op(2, p)/op(1, p), p=ifactors(n)[2]): seq(add(der(i)^3,i=1..j),j=1..60);

dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; Accumulate[Table[dn[n]^3, {n, 100}]] (* T. D. Noe, Nov 20 2013 *)

a(n) = Sum_{j=1..n} (j')^3, where j' = A003415(j).

A190273 Numbers n such that n' = m+1, with n and m semiprimes and gcd(m,n)>1, where n' is the arithmetic derivative of n.

Original entry on oeis.org

6, 10, 21, 26, 39, 55, 57, 74, 93, 111, 122, 146, 155, 201, 203, 253, 301, 305, 314, 327, 381, 386, 417, 471, 497, 543, 554, 597, 626, 633, 689, 737, 755, 791, 794, 842, 889, 905, 914, 921, 1011, 1027, 1055, 1081, 1082, 1137, 1226, 1227, 1322, 1346, 1379, 1461, 1466, 1477, 1497, 1514, 1623, 1655, 1703, 1711, 1713, 1731, 1751, 1754, 1893, 1967, 1994

Offset: 1

Giorgio Balzarotti, May 07 2011

nonn

The sequence is related to the Rassias Conjecture ("for any prime p there are two primes p1 and p2 such that p*p1=p1+p2+1, p>2, p2>p1", see A190272-A190275), because n = p1*p2, m=p1*p -> p1*p = p1+p2-1. The sequence includes the cases with p=p1 (or p2). Generalization can be achieved by removing semiprimarity condition or accepting gcd(n,m)=1. The differential equation in its general form n'=m+1 includes Giuga Numbers, i.e., n'=b*n+1, or n'=n+1 (A007850).

These are semiprimes n = p*q such that 1/p + 1/q - 1/n = P/Q, where P <> Q are primes. Cf. A326690. - Amiram Eldar and Thomas Ordowski, Jul 25 2019

			n=6, 6'=5, m=5+1=6, gcd(6,6)=6 -> a(1)=6

For Rassias conjecture: Preda Mihăilescu, Review of Problem Solving and Selected Topics in Number Theory, Newsletter of the European Mathematical Society, March 2011, p. 46.

Cf. A001358 (semiprimes), A003415 (arithmetic derivative), A007850 (Giuga numbers), A190272 (n'=m-1), A190273, A190274, A190275.

der:=n->n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
seq(`if`(bigomega(i)=2 and bigomega(der(i)-1)=2 and gcd(i,der(i)-1)>1,i,NULL),i=1..2000);

A190275 Semiprimes of the form p*(p^2 - p + 1).

Original entry on oeis.org

6, 21, 301, 2041, 296341, 486877, 2666437, 3420301, 4304341, 7152001, 38159521, 42387097, 54296677, 95235601, 158048281, 229971241, 265434901, 383712781, 454166017, 775307917, 972261181, 1063290841, 1304557801, 1392422041, 1730882401, 1863895261, 2631883561, 2879450461, 3714274297, 3845297341, 4070454361, 4256780041, 4849695001, 5328809461, 5722533337, 5838483601, 7218898681, 7841065621

Offset: 1

Giorgio Balzarotti, May 07 2011

nonn

This sequence is infinite, assuming Schinzel's Hypothesis H.
Related to Rassias Conjecture ("for any odd prime p there are primes q < r such that p*q = q + r + 1") setting p = q. Generalization can be achieved by removing semiprimality condition and accepting p^e, e >= 2.
These are semiprimes m = p*q such that 1/p + 1/q - 1/m = p/q. Cf. A326690. - Amiram Eldar and Thomas Ordowski, Jul 22 2019

			a(1) = 6 = 2*3 = 2*(2^2-2+1).
a(2) = 21 = 3*7 = 3*(3^2-3+1).
a(3) = 301 = 7*43 = 7*(7^2-7+1).

Giovanni Resta, Table of n, a(n) for n = 1..10000
For Rassias conjecture: Preda Mihăilescu, Review of Problem Solving and Selected Topics in Number Theory, Newsletter of the European Mathematical Society, March 2011, p. 46.

Cf. A065508 (primes p such that p^2-p+1 is prime).

Cf. A001358 (semiprime), A003415 (arithmetic derivative), A164643, A190272 (n'=a-1), A190273 (n'=a+1), A190274 (n'=p^2-1).

seq(`if`(isprime((ithprime(i)^2-ithprime(i)+1))=true,(ithprime(i)^2-ithprime(i)+1)*ithprime(i),NULL),i=1..300);

p = Select[Prime@ Range@ 500, PrimeQ[#^2 - # + 1] &]; p (p^2 - p + 1) (* Giovanni Resta, Jul 22 2019 *)

forprime(p=2,1e4,if(isprime(k=p^2-p+1),print1(p*k", "))) \\ Charles R Greathouse IV, May 08 2011

A190272 Numbers n such that n' = a -1, with n and a semiprimes and gcd(a,n) > 1, where n' is the arithmetic derivative of n.

Original entry on oeis.org

6, 14, 15, 22, 33, 38, 46, 51, 62, 86, 87, 91, 95, 118, 141, 142, 145, 158, 159, 166, 206, 249, 262, 267, 278, 287, 295, 321, 326, 382, 395, 398, 411, 422, 445, 446, 473, 502, 519, 537, 542, 545, 581, 591, 622, 662, 695, 699, 703, 718, 745, 758, 766, 789, 838, 886, 895, 926, 951, 958, 995, 998, 1046, 1126, 1139, 1145, 1167, 1199, 1238, 1262, 1318, 1329, 1347, 1382, 1401, 1441, 1486, 1678, 1707, 1717, 1718, 1726, 1745, 1757, 1761, 1766

Offset: 1

Giorgio Balzarotti, May 07 2011

nonn

This sequence is infinite, assuming Dickson's conjecture. In fact, the conjecture implies that there are infinitely many terms of this sequence divisible by any fixed prime p. - Charles R Greathouse IV, May 08 2011

Related to the Rassias Conjecture ("for any odd prime p there are primes q < r such that p*q = q+r+1") setting n = q*r, a = q+r+1. The sequence includes the cases with p = q (or p = r). Generalization can be achieved by removing the semiprimality condition or accepting gcd(n,a)=1. The differential equation in its general form n' = a + 1 includes Primary Pseudoperfect numbers, i.e., n' = n-1 (A054377).

			For n=6, 6' = 5, a = 5-1 = 4, gcd(4,6)=2, so 6 is a term.

For Rassias conjecture: Preda Mihăilescu, Review of Problem Solving and Selected Topics in Number Theory, Newsletter of the European Mathematical Society, March 2011, p. 46.

Cf. A001358 (semiprimes), A003415 (arithmetic derivative), A054377 (Primary Pseudoperfect).

der:=n->n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
# for quick reference only
seq(`if`(bigomega(i)=2 and bigomega(der(i)+1)=2 and gcd(i,der(i)+1)>1,i,NULL),i=1..2000);

find(lim)=my(v=List());forprime(p=2,sqrtint(lim\2),forstep(q=2*p-1,lim\p,p+p,if(isprime(q\p+2)&isprime(q),listput(v,p*q))));vecsort(Vec(v)) \\ Charles R Greathouse IV, May 08 2011

Semiprimes pq with (p+q+1)/p prime. - Charles R Greathouse IV, May 08 2011

A191216 Arithmetic derivative of prime(n) * prime(n+1) * prime(n+2) * prime(n+3) * prime(n+4) * prime(n+5).

Original entry on oeis.org

361, 230456, 1005768, 3462570, 11006128, 25925028, 61456764, 127697940, 249379116, 448408452, 740850012, 1263239320, 1914568816, 2884222410, 4371191782, 6287341056, 8758591370, 11640682466, 15938770638, 21721208748, 29153150298, 38784336168, 49888704100, 62506263054, 76188213990, 95511276660, 118760260290, 150724895476, 187405610004, 243040520764

Offset: 1

Giorgio Balzarotti, May 26 2011

nonn

Extension to 5 primes of concepts in A001043 A127489 A127349 A127345

Cf. A001043, A127489, A127349, A127345.

der:=n->n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
seq(dif(ithprime(i)*ithprime(i+1)*ithprime(i+2)*ithprime(i+3)*ithprime(i+4)*ithprime(i+5)),i=1..30);

a(n) = (prime(n) * prime(n+1) * prime(n+2) * prime(n+3) * prime(n+4) * prime(n+5))' where f' is the arithmetic derivative (see A003415) of f.

A190119 a(n) = Sum_{k=1..n} lcm(k,k')/k, where k' is arithmetic derivative of k.

Original entry on oeis.org

0, 1, 2, 3, 4, 9, 10, 13, 15, 22, 23, 27, 28, 37, 45, 47, 48, 55, 56, 62, 72, 85, 86, 97, 99, 114, 115, 123, 124, 155, 156, 161, 175, 194, 206, 211, 212, 233, 249, 266, 267, 308, 309, 321, 334, 359, 360, 367, 369, 378, 398, 412, 413, 416, 432, 455, 477, 508, 509, 532, 533, 566, 583, 586, 604, 665, 666, 684, 710, 769

Offset: 1

Giorgio Balzarotti, May 04 2011

nonn

Use lcm(1,0)=0.

			lcm(1,1')/1+lcm(2,2')/2+lcm(3,3')/3=0+2/2+3/3=2 ->a(3)=2.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A003415.

der:=n->n*add(op(2,p)/op(1,p),p=ifactors(n)[2]):
seq(add(lcm(der(i),i)/i,i=1..n),n=1..50);

A003415[n_]:= If[Abs@n < 2, 0, n Total[#2/#1 & @@@FactorInteger[Abs@n]]]; Table[Sum[LCM[k, A003415[k]]/k, {k, 1, n}], {n,1,50}] (* G. C. Greubel, Dec 29 2017 *)

{A003415(n, f)=sum(i=1, #f=factor(n)~, n/f[1, i]*f[2, i])};
for(n=1,20, print1(sum(k=1,n,lcm(k,A003415(k))/k), ", ")) \\ G. C. Greubel, Dec 29 2017

A190120 a(n) = Sum_{k=1..n} lcm(k,k')/gcd(k,k'), where n' is arithmetic derivative of n.

Original entry on oeis.org

0, 2, 5, 6, 11, 41, 48, 54, 60, 130, 141, 153, 166, 292, 412, 414, 431, 473, 492, 522, 732, 1018, 1041, 1107, 1117, 1507, 1508, 1564, 1593, 2523, 2554, 2564, 3026, 3672, 4092, 4107, 4144, 4942, 5566, 5736, 5777, 7499, 7542, 7674, 7869, 9019, 9066, 9087, 9101, 9191

Offset: 1

Giorgio Balzarotti, May 04 2011

nonn

Use lcm(1,0)=0 and gcd(1,0)=1.

			lcm(1,1')/gcd(1,1')+lcm(2,2')/gcd(2,2')+lcm(3,3')/gcd(3,3')=0+2/1+3/1=5 ->a(3)=5.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A003415, A190118, A190119.

der:=n->n*add(op(2,p)/op(1,p),p=ifactors(n)[2]):
seq(add(lcm(der(i),i)/gcd(der(i),i),i=1..n),n=1..50);

A003415[n_]:= If[Abs@n < 2, 0, n Total[#2/#1 & @@@FactorInteger[Abs@n]]];
Table[Sum[LCM[k, A003415[k]]/GCD[k, A003415[k]], {k, 1, n}], {n,1,50}] (* G. C. Greubel, Dec 29 2017 *)

{A003415(n, f)=sum(i=1, #f=factor(n)~, n/f[1, i]*f[2, i])};
for(n=1, 50, print1(sum(k=1,n,lcm(k,A003415(k))/gcd(k,A003145(k))), ", ")) \\ G. C. Greubel, Dec 29 2017

A190122 a(n) = Sum_{k=1..n} k*lcm(k,k')/gcd(k,k'), where k' is arithmetic derivative of k.

Original entry on oeis.org

0, 4, 13, 17, 42, 222, 271, 319, 373, 1073, 1194, 1338, 1507, 3271, 5071, 5103, 5392, 6148, 6509, 7109, 11519, 17811, 18340, 19924, 20174, 30314, 30341, 31909, 32750, 60650, 61611, 61931, 77177, 99141, 113841, 114381, 115750, 146074, 170410, 177210, 178891

Offset: 1

Giorgio Balzarotti, May 04 2011

nonn

Use lcm(1,0)=0 and gcd(1,0)=1.

			lcm(1,1')/gcd(1,1')*1+lcm(2,2')/gcd(2,2')*2+lcm(3,3')/gcd(3,3')*3=0+2/1*2+3/1*3=13 ->a(3)=13.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A003415, A189036, A190118, A190119, A190120.

der:=n->n*add(op(2,p)/op(1,p),p=ifactors(n)[2]):
seq(add(lcm(der(i),i)/gcd(der(i),i)*i,i=1..n),n=1..50);

A003415[n_]:= If[Abs@n < 2, 0, n Total[#2/#1 & @@@FactorInteger[Abs@n]]];
Table[Sum[k*LCM[k, A003415[k]]/GCD[k, A003415[k]], {k, 1, n}], {n, 1, 50}] (* G. C. Greubel, Dec 29 2017 *)

a(n) = Sum_{k=1..n} k*A189036(k).

cp's OEIS Frontend

User: Giorgio Balzarotti

A231864 Partial sums of the second power of arithmetic derivative function A003415.

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A232172 Partial sums of second arithmetic derivative of natural numbers.

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A231946 Partial sums of the third power of the arithmetic derivative function A003415.

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A190273 Numbers n such that n' = m+1, with n and m semiprimes and gcd(m,n)>1, where n' is the arithmetic derivative of n.

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A190275 Semiprimes of the form p*(p^2 - p + 1).

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A190272 Numbers n such that n' = a -1, with n and a semiprimes and gcd(a,n) > 1, where n' is the arithmetic derivative of n.

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A191216 Arithmetic derivative of prime(n) * prime(n+1) * prime(n+2) * prime(n+3) * prime(n+4) * prime(n+5).

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