A090880 -id:A090880 - cp's OEIS frontend

A090884 a(n) is the derivative of n via transport of structure from polynomials. Completely multiplicative with a(2) = 1, a(prime(i+1)) = prime(i)^i for i > 0.

1, 1, 2, 1, 9, 2, 125, 1, 4, 9, 2401, 2, 161051, 125, 18, 1, 4826809, 4, 410338673, 9, 250, 2401, 16983563041, 2, 81, 161051, 8, 125, 1801152661463, 18, 420707233300201, 1, 4802, 4826809, 1125, 4, 25408476896404831, 410338673, 322102, 9

Offset: 1

Sam Alexander, Dec 12 2003

Previous name: There exists an isomorphism from the positive rationals under multiplication to Z[x] under addition, defined by f(q) = e1 + (e2)x + (e3)(x^2) +...+ (ek)(x^(k-1)) + ... (where e_i is the exponent of the i-th prime in q's prime factorization) The a(n) above are calculated by a(n) = f^(-1)[d/dx f(n)] (In other words: differentiate n's image in Z[x] and return to Q).

With primes noted p_0 = 2, p_1 = 3, etc., let f be the function that maps n = Product_{i=0..d} p_i^e_i to P = Sum_{i=0..d} e_i*X^i; and let g be the inverse function of f. a(n) is by definition g(P') = g((f(n))'). - Luc Rousseau, Aug 06 2018

			504 = 2^3 * 3^2 * 7 is mapped to polynomial 3+2X+X^3, whose derivative is 2+3X^2, which is mapped to 2^2 * 5^3 = 500. Then, a(504) = 500. - _Luc Rousseau_, Aug 06 2018

Joseph J. Rotman, The Theory of Groups: An Introduction, 2nd ed. Boston: Allyn and Bacon, Inc. 1973. Page 9, problem 1.26.

Andrew Howroyd, Table of n, a(n) for n = 1..500
Sam Alexander, Post to sci.math.
Wikipedia, Transport of structure

Cf. A001222, A048675, A054841, A090880, A090881, A090882, A090883.

Polynomial multiplication using the same isomorphism: A297845.

a(n)={my(f=factor(n)); prod(i=1, #f~, my([p,e]=f[i,]); if(p==2, 1, precprime(p-1)^(e*primepi(p-1))))} \\ Andrew Howroyd, Jul 31 2018

More terms from Ray Chandler, Dec 20 2003

New name from Peter Munn, Aug 10 2022 using existing formula (Andrew Howroyd, Jul 31 2018) and introductory comment.

A331751 Numbers k such that A048675(sigma(k)) is equal to A048675(2*k).

Original entry on oeis.org

2, 6, 27, 28, 84, 270, 496, 1053, 1120, 1488, 1625, 1638, 3360, 3780, 4875, 8128, 10530, 24384, 66960, 147420, 167400, 406224, 611226, 775000, 872960, 943250, 1097280, 1245699, 1255338, 1303533, 1464320, 1686400, 1740024, 1922375, 1952500, 2011625, 2193408, 2325000, 2611440, 2618880, 2829750, 2941029, 4392960

Offset: 1

Antti Karttunen, Feb 05 2020

nonn

Numbers k such that A097248(sigma(k)) is equal to A097248(2*k).
Numbers k such that A331750(k) is equal to 1+A048675(k), which in turn is equal to A048675(A225546(2*k)) = A048675(2*A225546(k)).
Among the first 60 terms, 15 are odd: 27, 1053, 1625, 4875, 1245699, 1303533, 1922375, 2011625, 2941029, 5767125, 6034875, 12733875, 17137575, 26316675, 29362905, and only 1053 = 3^4 * 13 is in A228058.
Note that the condition A090880(sigma(k)) == A090880(2*k) appears to be much more constrained.

			For n = 1053 = 3^4 * 13^1, A331750(1053) = A331750(81) + A331750(13) = 32+9 = 41, while A048675(2*1053) = A048675(2)+A048675(81)+A048675(13) = 1+8+32 = 41 also, thus 1053 is included in this sequence.
For n = 3360 = 2^5 * 3^1 * 5^1 * 7^1, A331750(3360) = A331750(32)+A331750(3)+A331750(5)+A331750(7) = 12+2+3+3 = 20, while A048675(2*3360) = A048675(2)+A048675(32)+A048675(3)+A048675(5)+A048675(7) = 1+5+2+4+8 = 20 also, thus 3360 is included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..60 (up to the term 33550336).
Index entries for sequences where any odd perfect numbers must occur

Cf. A000203, A048675, A090880, A097246, A097248, A331750, A331752, A332208.

Cf. A000396 (a subsequence).

A097246(n) = { my(f=factor(n)); prod(i=1, #f~, (nextprime(f[i, 1]+1)^(f[i, 2]\2))*((f[i, 1])^(f[i, 2]%2))); };
A097248(n) = { my(k=A097246(n)); while(k<>n, n = k; k = A097246(k)); k; };
isA331751(n) = (A097248(2*n)==A097248(sigma(n)));

A090881 Suppose n=(p1^e1)(p2^e2)... where p1,p2,... are the prime numbers and e1,e2,... are nonnegative integers. Then a(n) = e1 + (e2)4 + (e3)16 + (e4)64 + ... + (ek)(4^(k-1)) + ...

Original entry on oeis.org

0, 1, 4, 2, 16, 5, 64, 3, 8, 17, 256, 6, 1024, 65, 20, 4, 4096, 9, 16384, 18, 68, 257, 65536, 7, 32, 1025, 12, 66, 262144, 21, 1048576, 5, 260, 4097, 80, 10, 4194304, 16385, 1028, 19, 16777216, 69, 67108864, 258, 24, 65537, 268435456, 8, 128, 33, 4100, 1026

Offset: 1

Sam Alexander, Dec 12 2003

Replace "4" with "x" and extend the definition of a to positive rationals and a becomes an isomorphism between positive rationals under multiplication and polynomials over Z under addition. This remark generalizes A001222, A048675 and A054841: evaluate said polynomial at x=1, x=2 and x=10, respectively.

Joseph J. Rotman, The Theory of Groups: An Introduction, 2nd ed. Boston: Allyn and Bacon, Inc. 1973. Page 9, problem 1.26.

Sam Alexander, Post to sci.math.

Cf. A001222, A048675, A054841, A090880, A090882, A090883, A090884.

More terms from Ray Chandler, Dec 20 2003

A283483 Sums of distinct nonzero terms of A003462: a(n) = Sum_{k>=0} A030308(n,k)*A003462(1+k).

Original entry on oeis.org

0, 1, 4, 5, 13, 14, 17, 18, 40, 41, 44, 45, 53, 54, 57, 58, 121, 122, 125, 126, 134, 135, 138, 139, 161, 162, 165, 166, 174, 175, 178, 179, 364, 365, 368, 369, 377, 378, 381, 382, 404, 405, 408, 409, 417, 418, 421, 422, 485, 486, 489, 490, 498, 499, 502, 503, 525, 526, 529, 530, 538, 539, 542, 543, 1093, 1094, 1097, 1098, 1106, 1107, 1110, 1111

Offset: 0

Antti Karttunen, Mar 19 2017

nonn

Indexing starts from zero, with a(0) = 0.

Cf. A000244, A003462, A005187, A030308, A070939, A090880, A283477.

A003462(n) = (3^n-1)/2;
A030308(n,k) = bittest(n,k);
A283483(n) = sum(i=0,(#binary(n)-1),A030308(n,i)*A003462(1+i));

(define (A283483 n) (A090880 (A283477 n)))

a(n) = Sum_{i=0..A070939(n)} A030308(n,i)*A003462(1+i).
a(n) = A090880(A283477(n)).
Other identities. For all n >= 0:
a(2^n) = A003462(n+1).

cp's OEIS Frontend

A090884 a(n) is the derivative of n via transport of structure from polynomials. Completely multiplicative with a(2) = 1, a(prime(i+1)) = prime(i)^i for i > 0.

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A331751 Numbers k such that A048675(sigma(k)) is equal to A048675(2*k).

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A090881 Suppose n=(p1^e1)(p2^e2)... where p1,p2,... are the prime numbers and e1,e2,... are nonnegative integers. Then a(n) = e1 + (e2)*4 + (e3)*16 + (e4)*64 + ... + (ek)*(4^(k-1)) + ...

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A283483 Sums of distinct nonzero terms of A003462: a(n) = Sum_{k>=0} A030308(n,k)*A003462(1+k).

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A090881 Suppose n=(p1^e1)(p2^e2)... where p1,p2,... are the prime numbers and e1,e2,... are nonnegative integers. Then a(n) = e1 + (e2)4 + (e3)16 + (e4)64 + ... + (ek)(4^(k-1)) + ...