A351074 -id:A351074 - cp's OEIS frontend

A327860 Arithmetic derivative of the primorial base exp-function: a(n) = A003415(A276086(n)).

0, 1, 1, 5, 6, 21, 1, 7, 8, 31, 39, 123, 10, 45, 55, 185, 240, 705, 75, 275, 350, 1075, 1425, 3975, 500, 1625, 2125, 6125, 8250, 22125, 1, 9, 10, 41, 51, 165, 12, 59, 71, 247, 318, 951, 95, 365, 460, 1445, 1905, 5385, 650, 2175, 2825, 8275, 11100, 30075, 4125, 12625, 16750, 46625, 63375, 166125, 14, 77, 91, 329, 420

Offset: 0

Antti Karttunen, Sep 30 2019

Are there any other fixed points after 0, 1, 7, 8 and 2556? (A328110, see also A351087 and A351088).
Out of the 30030 initial terms, 19220 are multiples of 5. (See A327865).
Proof that a(n) is even if and only if n is a multiple of 4: Consider Charlie Neder's Feb 25 2019 comment in A235992. As A276086 is never a multiple of 4, and as it toggles the parity, we only need to know when A001222(A276086(n)) = A276150(n) is even. The condition for that is given in the latter sequence by David A. Corneth's Feb 27 2019 comment. From this it also follows that A166486 gives similarly the parity of terms of A342002, A351083 and A345000. See also comment in A327858. - Antti Karttunen, May 01 2022

			2556 has primorial base expansion [1,1,1,1,0,0] as 1*A002110(5) + 1*A002110(4) + 1*A002110(3) + 1*A002110(2) = 2310 + 210 + 30 + 6 = 2556. That in turn is converted by A276086 to 13^1 * 11^1 * 7^1 * 5^1 = 5005, whose arithmetic derivative is 5' * 1001 + 1001' * 5 = 1*1001 + 311*5 = 2556, thus 2556 is one of the rare fixed points (A328110) of this sequence.

Antti Karttunen, Table of n, a(n) for n = 0..2310
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..30030
Index entries for sequences related to primorial base

Cf. A002110 (positions of 1's), A003415, A048103, A276086, A327858, A327859, A327865, A328110 (fixed points), A328233 (positions of primes), A328242 (positions of squarefree terms), A328388, A328392, A328571, A328572, A329031, A329032, A329041, A342002.
Cf. A345000, A351074, A351075, A351076, A351077, A351080, A351083, A351084, A351087 (numbers k such that a(k) is a multiple of k), A351088.
Coincides with A329029 on positions given by A276156.
Cf. A166486 (a(n) mod 2), A353630 (a(n) mod 4).
Cf. A267263, A276150, A324650, A324653, A324655 for omega, bigomega, phi, sigma and tau applied to A276086(n).
Cf. also A351950 (analogous sequence).

Block[{b = MixedRadix[Reverse@ Prime@ Range@ 12]}, Array[Function[k, If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]] ] &@ Abs[Times @@ Power @@@ # &@ Transpose@{Prime@ Range@ Length@ k, Reverse@ k}]]@ IntegerDigits[#, b] &, 65, 0]] (* Michael De Vlieger, Mar 12 2021 *)

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
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; };
A327860(n) = A003415(A276086(n));

A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); }; \\ (Standalone version) - Antti Karttunen, Nov 07 2019

a(n) = A003415(A276086(n)).
a(A002110(n)) = 1 for all n >= 0.
From Antti Karttunen, Nov 03 2019: (Start)
Whenever A329041(x,y) = 1, a(x + y) = A003415(A276086(x)*A276086(y)) = a(x)*A276086(y) + a(y)*A276086(x). For example, we have:
a(n) = a(A328841(n)+A328842(n)) = A329031(n)*A328572(n) + A329032(n)*A328571(n).
A051903(a(n)) = A328391(n).
A328114(a(n)) = A328392(n).
(End)
From Antti Karttunen, May 01 2022: (Start)
a(n) = A328572(n) * A342002(n).
For all n >= 0, A000035(a(n)) = A166486(n). [See comments]
(End)

Verbal description added to the definition by Antti Karttunen, May 01 2022

A350074 Difference between the maximal digit in the primorial base expansion of n and the maximal exponent in the prime factorization of n.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, -2, -1, 1, 1, 0, 1, 1, 1, -2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 3, 1, 2, 3, 0, 0, -4, 0, 1, 1, -1, 0, 0, 0, -1, 1, 1, 1, 0, 0, 1, 1, -1, 1, 1, 2, 1, 2, 1, 3, 1, 3, 3, 3, 0, 1, 1, 0, -4, 1, 1, 1, 0, 1, 1, 1, -1, 1, 1, 0, 0, 1, 2, 2, -1, -1, 2, 2, 2, 3, 3, 3, 1, 3, 1, 2, 1, 2, 2, 2, -2, 2, 1, 1, 1, 2, 2, 2, 0, 2

Offset: 1

Antti Karttunen, Feb 01 2022

Antti Karttunen, Table of n, a(n) for n = 1..30030
Index entries for sequences related to primorial base

Cf. A051903, A276086, A328114.
Cf. A350075 (positions of negative terms), A350076 (of terms >= 0), A350070 (their characteristic function), A351038 (positions of terms <= 0), A351039 (their characteristic function).
Cf. also A351074.

A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A350074(n) = (A051903(A276086(n)) - A051903(n));

a(n) = A328114(n) - A051903(n) = A051903(A276086(n)) - A051903(n).

A351075 Numbers k such that the maximal exponent in the prime factorization of A327860(k) is less than A051903(k), the maximal exponent in the prime factorization of k.

Original entry on oeis.org

2, 4, 6, 9, 12, 30, 32, 40, 48, 60, 63, 64, 68, 75, 76, 81, 96, 104, 108, 112, 128, 160, 192, 210, 212, 220, 224, 225, 240, 242, 243, 245, 248, 250, 256, 270, 272, 275, 276, 279, 280, 284, 288, 304, 312, 320, 324, 336, 352, 384, 420, 423, 424, 425, 428, 436, 448, 456, 459, 464, 472, 480, 484, 486, 488, 490, 492, 495, 496

Offset: 1

Antti Karttunen, Feb 01 2022

Numbers k for which A328391(k) < A051903(k).

Antti Karttunen, Table of n, a(n) for n = 1..26720; terms <= 9699690
Index entries for sequences related to primorial base

Cf. A003415, A276086, A051903, A327860, A328391, A351076 (complement), A351077 (and its characteristic function).
Positions of negative terms in A351074.
Subsequence of A351038.
Cf. also A350075.

A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
isA351075(n) = (A051903(A327860(n)) < A051903(n));

A351076 Numbers k such that the maximal exponent in the prime factorization of A327860(k) is greater than or equal to A051903(k), the maximal exponent in the prime factorization of k.

Original entry on oeis.org

1, 3, 5, 7, 8, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 72, 73, 74, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98

Offset: 1

Antti Karttunen, Feb 01 2022

nonn

Numbers k for which A328391(k) >= A051903(k).

Index entries for sequences related to primorial base

Cf. A003415, A276086, A051903, A327860, A328391, A351075 (complement), A351077 (the characteristic function).
Positions of nonnegative terms in A351074.
Cf. also A350076.

A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
isA351076(n) = (A051903(A327860(n)) >= A051903(n));

A351097 Difference between {the maximal digit value in primorial base expansion of the arithmetic derivative of n} and {the maximal exponent in the prime factorization of n}.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, -1, -1, 0, 0, 0, 0, 0, 0, -3, 0, 1, 0, 2, 1, 1, 0, -1, 0, 1, 1, -1, 0, 0, 0, -2, 1, 2, 1, 0, 0, 2, 1, -1, 0, 1, 0, 1, -1, 3, 0, -1, 0, 0, 2, 2, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 0, 3, 3, 0, 2, 0, 0, 2, 1, 2, 1, 0, 1, -1, 1, 0, 2, 2, 1, 0, 1, 0, 2, 2, 1, 1, 2, 3, -3, 0, 0, 0, 2, 0, 2, 0, 2, 1

Offset: 1

Antti Karttunen, Feb 03 2022

sign

Antti Karttunen, Table of n, a(n) for n = 1..30030

Cf. A003415, A051903, A276086, A327859, A328390, A351098 (positions of negative terms), A351099.

Cf. also A350074, A351074.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A327859(n) = A276086(A003415(n));
A351097(n) = (A051903(A327859(n)) - A051903(n));

a(n) = A328390(n) - A051903(n) = A051903(A327859(n)) - A051903(n).

cp's OEIS Frontend

A327860 Arithmetic derivative of the primorial base exp-function: a(n) = A003415(A276086(n)).

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A350074 Difference between the maximal digit in the primorial base expansion of n and the maximal exponent in the prime factorization of n.

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A351075 Numbers k such that the maximal exponent in the prime factorization of A327860(k) is less than A051903(k), the maximal exponent in the prime factorization of k.

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A351076 Numbers k such that the maximal exponent in the prime factorization of A327860(k) is greater than or equal to A051903(k), the maximal exponent in the prime factorization of k.

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A351097 Difference between {the maximal digit value in primorial base expansion of the arithmetic derivative of n} and {the maximal exponent in the prime factorization of n}.

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