A373844 -id:A373844 - cp's OEIS frontend

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.

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)));

A373845 Triangle read by rows: T(n,k) = arithmetic derivative of (1 + A002110(n) + A002110(k)), 1 <= k <= n, where A002110(n) is the n-th primorial number.

Original entry on oeis.org

1, 6, 1, 14, 1, 1, 74, 38, 1, 1, 1551, 338, 1, 1, 1, 21084, 8631, 1330, 1, 1, 3550, 172655, 72938, 1970, 3410, 1, 1, 5822, 3233234, 4157356, 421750, 228491, 10190, 13610, 537398, 289610, 297753138, 32805527, 5188250, 8698439, 761710, 1, 18344100, 1, 6954431, 2156564414, 929540471, 68769335, 335525472, 4283242, 21900155, 348965439, 109820278, 185002, 32593310

Offset: 1

Antti Karttunen, Jun 21 2024

Arithmetic derivatives of the sums of three primorials, of which one is 1 [= A002110(0)], and two are > 1.
Ones occur in positions where 1 + A002110(n) + A002110(k) is a prime.
See also comments in A373844, and in A373848.

			Triangle begins as:
        1,
        6,        1,
       14,        1,       1,
       74,       38,       1,       1,
     1551,      338,       1,       1,      1,
    21084,     8631,    1330,       1,      1,  3550,
   172655,    72938,    1970,    3410,      1,     1,     5822,
  3233234,  4157356,  421750,  228491,  10190, 13610,   537398, 289610,
297753138, 32805527, 5188250, 8698439, 761710,     1, 18344100,      1, 6954431,
etc.

Cf. A002110, A003415, A370121, A373844, A373848.

Cf. also A024451, A370129, A370138 (arithmetic derivative applied to the sums of a constant number of primorials).

A002110(n) = prod(i=1,n,prime(i));
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A373845(n) = { n--; my(c = (sqrtint(8*n + 1) - 1) \ 2, x=A002110(1+n - binomial(c + 1, 2))); A003415(1+(A002110(1+c)+x)); };

For n, k >= 1, T(n, k) = A003415(1+A370121(n, k)).

cp's OEIS Frontend

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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A373845 Triangle read by rows: T(n,k) = arithmetic derivative of (1 + A002110(n) + A002110(k)), 1 <= k <= n, where A002110(n) is the n-th primorial number.

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