A373846 -id:A373846 - cp's OEIS frontend

A373842 a(n) = A003415(A276085(n)), where A003415 is the arithmetic derivative and A276085 is the primorial base log-function.

0, 0, 1, 1, 5, 1, 31, 1, 4, 1, 247, 4, 2927, 1, 12, 4, 40361, 1, 716167, 12, 80, 1, 14117683, 1, 16, 1, 5, 80, 334406399, 6, 9920878441, 1, 216, 568, 60, 5, 314016924901, 33975, 3740, 6, 11819186711467, 14, 492007393304957, 216, 7, 28300, 21460568175640361, 5, 92, 1, 60080, 3740, 1021729465586766997, 1, 540, 14

Offset: 1

Antti Karttunen, Jun 20 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..481

Cf. A003415, A024451, A276085, A373843 [= gcd(n, a(n))], A373846 (positions of 1's), A373847 [k such that a(n)<=k], A373848.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1,primepi(f[k, 1]-1),prime(i))); };
A373842(n) = A003415(A276085(n));

For n >= 1, a(A000040(n)) = A024451(n-1).

A373848 Numbers k such that k is not divisible by p^p for any prime p, and for which 1 < A373842(k) <= k, where A373842 is the arithmetic derivative of the primorial base log-function.

Original entry on oeis.org

5, 9, 15, 25, 30, 42, 45, 63, 75, 105, 110, 125, 126, 147, 150, 165, 175, 198, 210, 225, 231, 245, 275, 294, 315, 330, 343, 363, 375, 385, 441, 462, 495, 525, 539, 605, 625, 650, 686, 693, 726, 735, 750, 770, 825, 847, 875, 882, 990, 1029, 1050, 1089, 1125, 1155, 1170, 1190, 1210, 1225, 1250, 1331, 1375, 1386, 1430

Offset: 1

Antti Karttunen, Jun 20 2024

nonn

The initial 5 is the only prime in this sequence (for a proof, consider Henry Bottomley's Sep 27 2006 formula for A024451), the next three terms 9, 15, 25 are only semiprimes (see A087112 and A370129), and there are 21 terms with three prime factors in total: 30, 42, 45, 63, 75, 105, 110, 125, 147, 165, 175, 231, 245, 275, 343, 363, 385, 539, 605, 847, 1331 (see A369979, A370138 and A373844). In general, there should be only a finite amount of terms x such that A001222(x) = k, for any k >= 1.

It is conjectured that 5 is the only fixed point of A373842, which would imply that x=6 is the only number for which A003415(x) = A276086(x). See A351228.

Antti Karttunen, Table of n, a(n) for n = 1..1565

Intersection of A048103 with the setwise difference A373847\(A373846 U {1, 2}).
Subsequence of A373847.
Cf. A001222, A003415, A024451, A087112, A276085, A276086, A359550, A369979, A370129, A370138, A373842, A373844, A373845.
Cf. also A351228, A373603.

\\ Uses the code from A373842, or its precomputed data:
A359550(n) = { my(f = factor(n)); prod(k=1, #f~, (f[k, 2]A373848(n) = if(!A359550(n), 0, my(u=A373842(n)); ((1

A002620(n) = ((n^2)>>2);
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1,primepi(f[k, 1]-1),prime(i))); };
\\ The following routine checks that n is not a prime larger than five, is in A048103, and in case n is odd, rules out cases that certainly cannot give A373842(n) <= n:
prefilter_for_A373848(n) = if(n < 3 || (isprime(n) && n > 5), 0, my(f=factor(n), k=#f~, lpf=f[1,1], p=f[k,1], m=f[k,2]); for(i=1, k, if(f[i, 2]>=f[i, 1], return(0))); if(2==lpf, return(1)); while(p>lpf, p = precprime(p-1); m *= p; if(m>n, return(0))); (1));
isA373848(n) = if(!prefilter_for_A373848(n), 0, my(x=A276085(n)); if(x>A002620(n), 0, (!isprime(x) && A003415(x)<=n)));

A373847 Numbers k for which A373842(k) <= k, where A373842 is the arithmetic derivative of the primorial base log-function.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 22, 24, 25, 26, 27, 30, 32, 36, 40, 42, 45, 48, 50, 54, 56, 60, 63, 64, 70, 72, 74, 75, 80, 81, 84, 88, 90, 96, 98, 100, 105, 108, 110, 112, 120, 125, 126, 128, 132, 135, 140, 144, 147, 150, 154, 160, 162, 165, 168, 175, 176, 180, 182, 189, 192, 196, 198, 200, 210

Offset: 1

Antti Karttunen, Jun 20 2024

nonn

Cf. A373842.

Subsequences: A373846, A373848.

PARI
```
isA373847(n) = (A373842(n)<=n);
```

cp's OEIS Frontend

A373842 a(n) = A003415(A276085(n)), where A003415 is the arithmetic derivative and A276085 is the primorial base log-function.

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A373848 Numbers k such that k is not divisible by p^p for any prime p, and for which 1 < A373842(k) <= k, where A373842 is the arithmetic derivative of the primorial base log-function.

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A373847 Numbers k for which A373842(k) <= k, where A373842 is the arithmetic derivative of the primorial base log-function.

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