A351250 -id:A351250 - cp's OEIS frontend

A324583 Numbers k such that k and A276086(k) are coprime, where A276086 is the primorial base exp-function.

0, 1, 2, 4, 5, 6, 7, 8, 11, 12, 13, 14, 16, 17, 18, 19, 22, 23, 24, 26, 28, 29, 30, 31, 32, 34, 36, 37, 38, 41, 43, 44, 46, 47, 48, 52, 53, 54, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 71, 72, 73, 74, 76, 78, 79, 82, 83, 86, 88, 89, 90, 92, 94, 95, 96, 97, 101, 102, 103, 104, 106, 107, 108, 109, 113, 114, 116, 118, 120, 121

Offset: 1

Antti Karttunen, Mar 10 2019

nonn

Numbers k for which A324198(k) = 1.
For terms k > 0 it holds that:
  A000005(A324580(k)) = A000005(k) * A324655(k),
  A000010(A324580(k)) = A000010(k) * A324650(k),
  A000203(A324580(k)) = A000203(k) * A324653(k),
and similarly for any multiplicative function.

Cf. A324584 (complement), A356162 (characteristic function).
Cf. A276086, A324580, A324650, A324653, A324655.
Some subsequences are: A055932 ⊃ A025487 ⊃ A002182, and also A002110.
Subsequence of A356316.
Positions of 1's in A324198, positions 0's in A351254, A356302 and A356303, positions of fixed points in A351250 and in A356309.
Cf. also A355821, A356311.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324198(n) = gcd(n,A276086(n));
for(n=0,oo,if(1==A324198(n),print1(n,", ")));

Initial 0 prepended by Antti Karttunen, Nov 03 2022

A351251 Denominator of n / A276086(n).

Original entry on oeis.org

1, 2, 3, 2, 9, 18, 5, 10, 15, 10, 9, 90, 25, 50, 75, 10, 225, 450, 125, 250, 75, 250, 1125, 2250, 625, 50, 1875, 1250, 5625, 11250, 7, 14, 21, 14, 63, 18, 35, 70, 105, 70, 63, 630, 25, 350, 525, 70, 1575, 3150, 875, 250, 105, 1750, 7875, 15750, 4375, 1750, 1875, 8750, 39375, 78750, 49, 98, 147, 14, 441, 882, 245, 490

Offset: 0

Antti Karttunen, Feb 05 2022

Antti Karttunen, Table of n, a(n) for n = 0..11550

Cf. A276086, A324198, A351250 (numerators), A351253.

Cf. also A351231.

Array[Block[{i, m, n = #, p}, m = i = 1; While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; Denominator[#/m]] &, 68, 0] (* Michael De Vlieger, Feb 06 2022 *)

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A351251(n) = denominator(n/A276086(n));

a(n) = A276086(n) / gcd(n,A276086(n)) = A276086(n) / A324198(n).

a(n) = A276086(A351253(n)).

A328387 Numbers k such that A276086(k) is a multiple of k.

Original entry on oeis.org

1, 3, 15, 25, 75, 105, 147, 175, 343, 385, 525, 539, 625, 735, 825, 1029, 1155, 1225, 1331, 1375, 1617, 1815, 2695, 3003, 3025, 3675, 3773, 3993, 4375, 5005, 5145, 5577, 5775, 6655, 6825, 8085, 8125, 8281, 8575, 9075, 9555, 9625, 10725, 11011, 11319, 12675, 12705, 13013, 13377, 15015, 15379, 15925, 17303, 17745, 17787, 17875

Offset: 1

Antti Karttunen, Oct 15 2019

nonn

All terms are odd. Of the first 3003 terms, 1709 are multiples of five.

Antti Karttunen, Table of n, a(n) for n = 1..3003

Cf. A276086, A324198.
Indices of 0's in A328386. Indices of 1's in A351250.
Subsequence of A048103 and of A358226.
Cf. also A370114, A358231.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA328387(n) = (0==(A276086(n)%n));

A369038 Numerator of ratio A003415(n) / A003415(A276086(n)), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 3, 6, 7, 1, 8, 1, 9, 8, 2, 1, 7, 1, 12, 2, 13, 1, 11, 2, 3, 27, 16, 1, 31, 1, 8, 14, 19, 4, 5, 1, 21, 16, 34, 1, 41, 1, 12, 39, 5, 1, 56, 14, 9, 4, 14, 1, 27, 16, 46, 22, 31, 1, 46, 1, 33, 51, 16, 6, 61, 1, 36, 26, 59, 1, 13, 1, 39, 1, 8, 6, 71, 1, 11, 108, 43, 1, 62, 22, 9, 32, 1, 1, 41, 20

Offset: 1

Antti Karttunen, Jan 20 2024

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A003415, A276086, A327860, A345000, A369039 (denominators).

Cf. also A351230, A351250.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
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); };
A369038(n) = { my(u=A003415(n)); (u/gcd(u,A327860(n))); };

a(n) = A003415(n) / A345000(n) = A003415(n) / gcd(A003415(n), A327860(n)).

cp's OEIS Frontend

A324583 Numbers k such that k and A276086(k) are coprime, where A276086 is the primorial base exp-function.

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A351251 Denominator of n / A276086(n).

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A328387 Numbers k such that A276086(k) is a multiple of k.

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A369038 Numerator of ratio A003415(n) / A003415(A276086(n)), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

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