A370129 -id:A370129 - cp's OEIS frontend

A370121 Triangle read by rows: T(n,k) = A002110(n) + A002110(k), 0 <= k <= n; sums of two primorials, not necessarily distinct.

2, 3, 4, 7, 8, 12, 31, 32, 36, 60, 211, 212, 216, 240, 420, 2311, 2312, 2316, 2340, 2520, 4620, 30031, 30032, 30036, 30060, 30240, 32340, 60060, 510511, 510512, 510516, 510540, 510720, 512820, 540540, 1021020, 9699691, 9699692, 9699696, 9699720, 9699900, 9702000, 9729720, 10210200, 19399380, 223092871, 223092872

Offset: 0

Antti Karttunen, Feb 29 2024

After the initial 2, numbers with either one 2 or two 1's in their primorial base representation (A049345), with all the other digits zeros.

			Triangle begins as:
        2;
        3,       4;
        7,       8,      12;
       31,      32,      36,      60;
      211,     212,     216,     240,     420;
     2311,    2312,    2316,    2340,    2520,    4620;
    30031,   30032,   30036,   30060,   30240,   32340,   60060;
   510511,  510512,  510516,  510540,  510720,  512820,  540540,  1021020;
  9699691, 9699692, 9699696, 9699720, 9699900, 9702000, 9729720, 10210200, 19399380;

Cf. A002110, A049345, A087112, A276086, A276150, A370129 (arithmetic derivative applied to this triangle).
Cf. A006862 (left edge), A088860 (right edge).
Cf. A177689 (same triangle without the right edge), A370134 (without the leftmost column).
Subsequence of A370132.
Cf. also A173786.

A002110(n) = prod(i=1,n,prime(i));
A370121(n) = { my(c = (sqrtint(8*n + 1) - 1) \ 2); (A002110(c) + A002110(n - binomial(c + 1, 2))); };

For n >= 1, A276150(a(n)) = 2.

For n >= 1, A276086(a(n)) = A087112(1+n).

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)));

A370138 Arithmetic derivatives of the sums of three primorials > 1.

Original entry on oeis.org

5, 7, 9, 21, 19, 21, 41, 33, 61, 123, 109, 111, 191, 165, 211, 459, 213, 361, 705, 951, 1361, 1319, 3537, 1173, 2195, 2479, 1481, 2111, 3295, 3421, 2313, 5415, 5885, 5891, 11091, 15019, 16371, 35067, 15033, 25061, 33373, 15123, 26057, 31309, 42955, 16691, 48573, 36329, 45845, 62385, 31167, 72201, 62123, 80969, 141399, 151113

Offset: 1

Antti Karttunen, Mar 09 2024

nonn

For n > 20, a(n) > A369979(n).

Antti Karttunen, Table of n, a(n) for n = 1..10660
Index entries for sequences related to primorial numbers

Cf. A000292, A002110, A003415, A369979, A370137.

Cf. also A024451, A370129.

up_to = 15180;
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]));
A370137list(up_to) = { my(v = vector(up_to), i=0); for(x=1,oo, for(y=1,x, for(z=1,y, i++; if(i > up_to, return(v)); v[i] = A002110(x)+A002110(y)+A002110(z)))); (v); };
v370137 = A370137list(up_to);
A370137(n) = v370137[n];
A370138(n) = A003415(A370137(n));

a(n) = A003415(A370137(n)).

A370136 Triangle read by rows: T(n,k) = arithmetic derivative of ((A002110(n) + A002110(k)) / A002110(k)), 1 <= k <= n.

Original entry on oeis.org

1, 4, 1, 32, 5, 1, 55, 60, 12, 1, 1292, 195, 71, 16, 1, 22532, 2505, 841, 384, 9, 1, 382892, 102723, 8897, 8640, 191, 21, 1, 2469635, 3502740, 323328, 34133, 9980, 756, 24, 1, 111738812, 18755325, 10308201, 1568312, 50621, 5211, 371, 44, 1, 4853127108, 2003156919, 107924801, 178347008, 2376149, 251367, 6339, 672, 31, 1

Offset: 1

Antti Karttunen, Mar 07 2024

			Triangle begins as:
          1;
          4,        1;
         32,        5,        1;
         55,       60,       12,       1;
       1292,      195,       71,      16,     1;
      22532,     2505,      841,     384,     9,    1;
     382892,   102723,     8897,    8640,   191,   21,   1;
    2469635,  3502740,   323328,   34133,  9980,  756,  24,  1;
  111738812, 18755325, 10308201, 1568312, 50621, 5211, 371, 44, 1;

Cf. A002110, A003415, A370129, A370135.

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]));
A370135(n) = { n--; my(c = (sqrtint(8*n + 1) - 1) \ 2, x=A002110(1+n - binomial(c + 1, 2))); ((A002110(1+c)+x)/x); };
A370136(n) = A003415(A370135(n));

a(n) = A003415(A370135(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

A370121 Triangle read by rows: T(n,k) = A002110(n) + A002110(k), 0 <= k <= n; sums of two primorials, not necessarily distinct.

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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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A370138 Arithmetic derivatives of the sums of three primorials > 1.

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A370136 Triangle read by rows: T(n,k) = arithmetic derivative of ((A002110(n) + A002110(k)) / A002110(k)), 1 <= k <= n.

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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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