A090881 -id:A090881 - cp's OEIS frontend

A054841 If n = 2^a * 3^b * 5^c * 7^d * ... then a(n) = a + 10 * b + 100 * c + 1000 * d + ... .

0, 1, 10, 2, 100, 11, 1000, 3, 20, 101, 10000, 12, 100000, 1001, 110, 4, 1000000, 21, 10000000, 102, 1010, 10001, 100000000, 13, 200, 100001, 30, 1002, 1000000000, 111, 10000000000, 5, 10010, 1000001, 1100, 22, 100000000000, 10000001, 100010

Offset: 1

Henry Bottomley, Apr 11 2000

Are there any other numbers besides n=12 for which n=a(n) ? - Ctibor O. Zizka, Oct 08 2008
The sequence is a morphism from (N*,*) into (N,+), cf. formula. Up to n=1023, the digit sum A007953(a(n)) equals Omega(n) = A001222(n). This holds whenever A051903(n)<10. Restricted to these n, the sequence is also injective. However, when n is a multiple of 2^10, 3^10, 5^10 etc, then a(n) is equal to some a(m) with mM. F. Hasler, Nov 16 2008
This has been called the "Exponential Prime Power Representation" of n by W. Nissen in a post to the sci.math newsgroup (where probably some more sophisticated convention for representing digits > 10 would be used). - M. F. Hasler, Jul 03 2016

			a(25) = 200 because 25 = 5^2 * 3^0 * 2^0.
a(1024) = 10 = a(3), because 1024 = 2^10; but this two-digit multiplicity overflows into the 10^1 position, which encodes for powers of three.

Robert Israel, Table of n, a(n) for n = 1..10000 (n=1..1023 from Michael De Vlieger)
Evans A Criswell, A Sequence Puzzle (Posted to rec.puzzles Jan 01 1997)
Walter Nissen, Exponential Prime Power Representation, sci.math newsgroup, May 23 1995.

Row 10 of A104244.
Left inverse of A054842.
Cf. A001222, A048675, A090880, A090881, A090882, A276075, A276085 (analogous constructions for other bases), A090883, A090884, A049084, A027748, A124010, A056239.

a054841 1 = 0
a054841 n = sum $ zipWith (*)
                  (map ((10 ^) . subtract 1 . a049084) $ a027748_row n)
                  (map fromIntegral $ a124010_row n)
-- Reinhard Zumkeller, Aug 03 2015

A:= n -> add(t[2]*10^(numtheory:-pi(t[1])-1),t= ifactors(n)[2]);
seq(A(n), n=1..1000); # Robert Israel, Jul 24 2014

a054841[n_Integer] := Catch[FromDigits[IntegerDigits[Apply[Plus,
     Which[n == 0, Throw["undefined"],
        n == 1, 0,
        Max[Last /@ FactorInteger @ n] > 9, Throw["overflow"],
        True, Power[10, PrimePi[Abs[#]] - 1]] & /@
      Flatten[ConstantArray @@@ FactorInteger[n]]]]]] (* Michael De Vlieger, Jul 24 2014 *)

A054841(n)=sum(i=1,#n=factor(n)~,10^primepi(n[1,i])*n[2,i])/10 \\ M. F. Hasler, Nov 16 2008

from sympy import factorint, primepi
def a(n): return sum(e*10**(primepi(p)-1) for p, e in factorint(n).items())
print([a(n) for n in range(1, 41)]) # Michael S. Branicky, Mar 17 2024

a(m*n) = a(m) + a(n) for all m,n > 0. A007953(a(n))=A001222(n) for all n such that A051903(n) < 10. - M. F. Hasler, Nov 16 2008
a(n) = sum(10^(A049084(A027748(k))-1) * A124010(k): k = 1..A001221(n)). - Reinhard Zumkeller, Aug 03 2015
a(A054842(n)) = n for all n >= 0. - Antti Karttunen, Aug 29 2016
a(n) = Sum_{i>0} e_i*10^(i-1) when n = Product_{i>0} prime(i)^e_i. - M. F. Hasler, Mar 14 2018

A090880 Suppose n=(p1^e1)(p2^e2)... where p1,p2,... are the prime numbers and e1,e2,... are nonnegative integers. Then a(n) = e1 + (e2)3 + (e3)9 + (e4)27 + ... + (ek)(3^(k-1)) + ...

Original entry on oeis.org

0, 1, 3, 2, 9, 4, 27, 3, 6, 10, 81, 5, 243, 28, 12, 4, 729, 7, 2187, 11, 30, 82, 6561, 6, 18, 244, 9, 29, 19683, 13, 59049, 5, 84, 730, 36, 8, 177147, 2188, 246, 12, 531441, 31, 1594323, 83, 15, 6562, 4782969, 7, 54, 19, 732, 245, 14348907, 10, 90, 30, 2190

Offset: 1

Sam Alexander, Dec 12 2003

Replace "3" 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.

For examples of such evaluations at x=3, see "Other identities" in the Formula section. - Antti Karttunen, Jul 31 2015

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

Antti Karttunen, Table of n, a(n) for n = 1..512
Sam Alexander, Post to sci.math.

Row 3 of A104244.

Cf. A001222, A006190, A048675, A054841, A090881, A090882, A090883, A090884, A178590, A206296, A260443.

(define (A090880 n) (if (= 1 n) (- n 1) (+ (A000244 (- (A055396 n) 1)) (A090880 (A032742 n))))) ;; Antti Karttunen, Jul 29 2015

a(1) = 0; for n > 1, a(n) = 3^(A055396(n)-1) + a(A032742(n)). [Where A055396(n) gives the index of the smallest prime dividing n and A032742(n) gives the largest proper divisor of n.] - Antti Karttunen, Jul 29 2015
Other identities. For all n >= 0:
a(A206296(n)) = A006190(n).
a(A260443(n)) = A178590(n).

More terms from Ray Chandler, Dec 20 2003

A104244 Suppose m = Product_{i=1..k} p_i^e_i, where p_i is the i-th prime number and each e_i is a nonnegative integer. Then we can define P_m(x) = Sum_{i=1..k} e_i*x^(i-1). The sequence is the square array A(n,m) = P_m(n) read by descending antidiagonals.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 1, 2, 3, 1, 0, 2, 4, 2, 4, 1, 0, 1, 3, 9, 2, 5, 1, 0, 3, 8, 4, 16, 2, 6, 1, 0, 2, 3, 27, 5, 25, 2, 7, 1, 0, 2, 4, 3, 64, 6, 36, 2, 8, 1, 0, 1, 5, 6, 3, 125, 7, 49, 2, 9, 1, 0, 3, 16, 10, 8, 3, 216, 8, 64, 2, 10, 1, 0, 1, 4, 81, 17, 10, 3, 343, 9, 81, 2, 11, 1, 0, 2, 32, 5

Offset: 1

Olaf Voß, Feb 26 2005

From Antti Karttunen, Jul 29 2015: (Start)
The square array A(row,col) is read by downwards antidiagonals as: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
A(n,m) (entry at row=n, column=m) gives the evaluation at x=n of the polynomial (with nonnegative integer coefficients) bijectively encoded in the prime factorization of m. See A206284, A206296 for the details of that encoding. (The roles of variables n and m were accidentally swapped in this description, corrected by Antti Karttunen, Oct 30 2016)
(End)
Each row is a completely additive sequence, row n mapping prime(m) to n^(m-1). - Peter Munn, Apr 22 2022

			a(13) = 3 because 3 = p_1^0 * p_2^1 * p_3^0 * ..., so P_3(x) = 0*x^(1-1) + 1*x^(2-1) + 0*x^(3-1) + ... = x. Hence a(13) = A(3,3) = P_3(3) = 3. [Elaborated by _Peter Munn_, Aug 13 2022]
The top left corner of the array:
0, 1,  1, 2,   1,  2,   1,  3,  2,   2,     1,  3,      1,    2,   2, 4
0, 1,  2, 2,   4,  3,   8,  3,  4,   5,    16,  4,     32,    9,   6, 4
0, 1,  3, 2,   9,  4,  27,  3,  6,  10,    81,  5,    243,   28,  12, 4
0, 1,  4, 2,  16,  5,  64,  3,  8,  17,   256,  6,   1024,   65,  20, 4
0, 1,  5, 2,  25,  6,  125, 3, 10,  26,   625,  7,   3125,  126,  30, 4
0, 1,  6, 2,  36,  7,  216, 3, 12,  37,  1296,  8,   7776,  217,  42, 4
0, 1,  7, 2,  49,  8,  343, 3, 14,  50,  2401,  9,  16807,  344,  56, 4
0, 1,  8, 2,  64,  9,  512, 3, 16,  65,  4096, 10,  32768,  513,  72, 4
0, 1,  9, 2,  81, 10,  729, 3, 18,  82,  6561, 11,  59049,  730,  90, 4
0, 1, 10, 2, 100, 11, 1000, 3, 20, 101, 10000, 12, 100000, 1001, 110, 4
...

Cf. A000720.
Transpose: A104245.
Main diagonal: A090883.
Row 1: A001222, row 2: A048675, row 3: A090880, row 4: A090881, row 5: A090882, row 10: A054841; and, in the extrapolated table, row 0: A007814, row -1: A195017.
Other completely additive sequences with prime(k) mapped to a function of k include k: A056239, k-1: A318995, k+1: A318994, k^2: A289506, 2^k-1: A293447, k!: A276075, F(k-1): A265753, F(k-2): A265752.
For completely additive sequences with primes p mapped to a function of p, see A001414.
For completely additive sequences where some primes are mapped to 1, the rest to 0 (notably, some ruler functions) see the cross-references in A249344.
For completely additive sequences, s, with primes p mapped to a function of s(p-1) and maybe s(p+1), see A352957.
See the formula section for the relationship to A073133, A206296.
See the comments for the relevance of A206284.
A297845 represents multiplication of the relevant polynomials.
Cf. A090884, A248663, A265398, A265399 for other related sequences.
A167219 lists columns that contain their own column number.

A(n,A206296(k)) = A073133(n,k). [This formula demonstrates how this array can be used with appropriately encoded polynomials. Note that A073133 reads its antidiagonals by ascending order, while here the order is opposite.] - Antti Karttunen, Oct 30 2016
From Peter Munn, Apr 05 2021: (Start)
The sequence is defined by the following identities:
A(n, 3) = n;
A(n, m*k) = A(n, m) + A(n, k);
A(n, A297845(m, k)) = A(n, m) * A(n, k).
(End)

Starting offset changed from 0 to 1 by Antti Karttunen, Jul 29 2015

Name edited (and aligned with rest of sequence) by Peter Munn, Apr 23 2022

A090882 Suppose n=(p1^e1)(p2^e2)... where p1,p2,... are the prime numbers and e1,e2,... are nonnegative integers. Then a(n) = e1 + (e2)5 + (e3)25 + (e4)125 + ... + (ek)(5^(k-1)) + ...

Original entry on oeis.org

0, 1, 5, 2, 25, 6, 125, 3, 10, 26, 625, 7, 3125, 126, 30, 4, 15625, 11, 78125, 27, 130, 626, 390625, 8, 50, 3126, 15, 127, 1953125, 31, 9765625, 5, 630, 15626, 150, 12, 48828125, 78126, 3130, 28, 244140625, 131, 1220703125, 627, 35, 390626, 6103515625, 9, 250

Offset: 1

Sam Alexander, Dec 12 2003

Replace "5" 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.

Antti Karttunen, Table of n, a(n) for n = 1..1001
Sam Alexander, Post to sci.math.

Cf. A001222, A048675, A054841, A090880, A090881, A090883, A090884, and also A195017, A248663, A276075, A276085.

A090882(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*5^(primepi(f[k, 1])-1)); }; \\ Antti Karttunen, Apr 28 2022

More terms from Ray Chandler, Dec 20 2003

A090883 Write n as Product_{i=1..k} prime(i)^e_i, where prime(i) is the i-th prime number and e_i is a nonnegative integer. a(n) = Sum_{i=1..k} e_i*n^(i-1).

Original entry on oeis.org

0, 1, 3, 2, 25, 7, 343, 3, 18, 101, 14641, 14, 371293, 2745, 240, 4, 24137569, 37, 893871739, 402, 9282, 234257, 78310985281, 27, 1250, 11881377, 81, 21954, 14507145975869, 931, 819628286980801, 5, 1185954, 1544804417, 44100, 74

Offset: 1

Sam Alexander, Dec 12 2003

In the definition, replace "e_i*n^(i-1)" with "e_i*x^(i-1)" for all i to define a function P:N+ -> N[x]. If we extend this definition to positive rationals by allowing negative e_i, P(.) becomes an isomorphism between positive rationals under multiplication and polynomials over Z under addition. We can use P to generalize A001222, A048675 and A054841: evaluate each term of the sequence of polynomials P(1), P(2), ... at x=1, x=2 and x=10, respectively. [Edited and corrected by Peter Munn, Aug 12 2022]

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. [Broken link]

The main diagonal of A104244 (A104245).

Cf. A001222, A048675, A054841, A090880, A090881, A090882, A090884.

a(n) = my(f = factor(n)); sum(k=1, #f~, f[k,2]*n^(primepi(f[k,1])-1)); \\ Michel Marcus, Nov 01 2016

Name edited by Peter Munn, Aug 12 2022

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.

Original entry on oeis.org

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.

cp's OEIS Frontend

A054841 If n = 2^a * 3^b * 5^c * 7^d * ... then a(n) = a + 10 * b + 100 * c + 1000 * d + ... .

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A090880 Suppose n=(p1^e1)(p2^e2)... where p1,p2,... are the prime numbers and e1,e2,... are nonnegative integers. Then a(n) = e1 + (e2)*3 + (e3)*9 + (e4)*27 + ... + (ek)*(3^(k-1)) + ...

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A104244 Suppose m = Product_{i=1..k} p_i^e_i, where p_i is the i-th prime number and each e_i is a nonnegative integer. Then we can define P_m(x) = Sum_{i=1..k} e_i*x^(i-1). The sequence is the square array A(n,m) = P_m(n) read by descending antidiagonals.

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A090882 Suppose n=(p1^e1)(p2^e2)... where p1,p2,... are the prime numbers and e1,e2,... are nonnegative integers. Then a(n) = e1 + (e2)*5 + (e3)*25 + (e4)*125 + ... + (ek)*(5^(k-1)) + ...

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A090883 Write n as Product_{i=1..k} prime(i)^e_i, where prime(i) is the i-th prime number and e_i is a nonnegative integer. a(n) = Sum_{i=1..k} e_i*n^(i-1).

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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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A090880 Suppose n=(p1^e1)(p2^e2)... where p1,p2,... are the prime numbers and e1,e2,... are nonnegative integers. Then a(n) = e1 + (e2)3 + (e3)9 + (e4)27 + ... + (ek)(3^(k-1)) + ...

A090882 Suppose n=(p1^e1)(p2^e2)... where p1,p2,... are the prime numbers and e1,e2,... are nonnegative integers. Then a(n) = e1 + (e2)5 + (e3)25 + (e4)125 + ... + (ek)(5^(k-1)) + ...