A351228 -id:A351228 - cp's OEIS frontend

A369650 Numbers k such that A003415(k) = A276085(k), where A003415 is the arithmetic derivative, and A276085 is the primorial base log-function.

1, 2, 10, 15, 28, 5005

Offset: 1

Antti Karttunen, Feb 05 2024

Intersection with A048103 gives the fixed points (1, 2, 10, 15, 5005, ...) of A327859. Question: Does that set preclude nonsquarefree numbers? Certainly it does not contain any multiples of 9. See also comments in A328110.
If k == 2 (mod 4), then both A003415(k) and A276085(k) are odd, and the latter is of the form 4m+1 (if k has an odd number of prime factors), or of the form 4m+3 (if k has an even number of prime factors). Therefore, for k of the form 4m+2 to be included in this sequence, a necessary condition is that it must be either in the intersection of A026424 and A358772 (like, for example, 2 is) or in A369668 (the intersection of A028260 and A358774), like for example, 10 is.
If k is odd, then A276085(k) is even, and for A003415(k) to be even with k odd, then k has to be in A046337 (odd numbers with an even number of prime factors, counted with multiplicity). But A276085(A046337(n)) == 0 (mod 4) for all n, so also A003415(k) has to be a multiple of 4, so k has to be in A360110 (itself a subsequence of A369002), like for example k=15 and k=5005 are.
If it exists, a(7) > 2^19.

			As 5005 = 5*7*11*13, A003415(5*7*11*13) = (5*7*11) + (5*7*13) + (5*11*13) + (7*11*13) = 2556 = 2^2 * 3^2 * 71 = A276085(5005) = A002110(2) + A002110(3) + A002110(4) + A002110(5) [as 5, 7, 11 and 13 are prime(3) .. prime(6)], therefore 5005 is included in this sequence.

Index entries for sequences related to primorial base

Cf. A003415, A046337, A048103, A276085, A276086, A327859, A328110, A369002, A369668, A373144.
Positions of 0's in A373146.
Intersection of A373487 and A373490.
Cf. also A351228.

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))); };
isA369650(n) = (A003415(n) == A276085(n));

A351088 Numbers k such that A327860(k) is reachable from k by iterating the arithmetic derivative (A003415) and there are no terms with p^p-factors on the path there.

Original entry on oeis.org

0, 1, 2, 6, 7, 8, 30, 2310, 2556, 30030, 223092870

Offset: 1

Antti Karttunen, Feb 05 2022

Sequence includes also the terms for which no iterations are needed (when k is already equal to A327860(k)), thus A328110 is a subsequence. The other terms (and also 1) seem to be the intersection of primorials (A002110) with sequence A099308. This includes terms A002110(A109628(n)), whose arithmetic derivatives are in A244622.
The numbers k for which A276086(k) is reachable from k by iterating A003415 form a subsequence of this sequence, but so far only one term is known: 6, for which A276086(6) = A003415(6) = 5. (See A351228). It would be interesting to know whether there are more such terms, especially terms that require more than one iteration of A003415.
Question: The eleven known terms are all sums of distinct primorials (in A276156), i.e., contain only digits 0's and 1's in primorial base. Is this a necessary property for the terms of this sequence (and also for A328110)? - Antti Karttunen, Feb 04 2024, corrected May 11 2024.

Cf. A002110, A003415, A099308, A109628, A244622, A276086, A276156, A328110 (subsequence), A327860.

Cf. also A327969, A351089, A351228, A358215.

A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i,2]>=f[i,1],return(0), s += f[i, 2]/f[i, 1])); (n*s)); \\ Like A003415, but return zero also for n that have p^p-factor(s).
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); };
\\ This simple program doesn't check for any hypothetical p^p-free A003415-loops (they are so rare that they are conjectured not to exist at all):
isA351088(n) = if(!n, 1, my(g=A327860(n)); while(n>0, if(n==g, return(1)); n = A003415checked(n)); (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)));

A369663 Numbers k of the form 4m+3, whose arithmetic derivative k' is of the form 4u+2, and k' has an even number of prime factors.

Original entry on oeis.org

375, 459, 783, 819, 875, 1071, 1107, 1155, 1375, 1395, 1715, 1911, 1935, 1995, 2223, 2275, 2375, 2499, 2619, 2655, 2695, 2727, 2875, 2907, 2943, 3003, 3051, 3135, 3195, 3255, 3315, 3519, 3575, 3627, 3699, 3843, 3927, 3975, 4059, 4459, 4515, 4671, 4815, 4887, 4935, 4959, 5187, 5247, 5375, 5415, 5607, 5635, 5655

Offset: 1

Antti Karttunen, Feb 06 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Intersection of A004767 and A369661.
Subsequence of A369666.
Cf. A003415, A351228.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA369663(n) = if(3!=(n%4), 0, my(d=A003415(n)); (2==(d%4) && !(bigomega(d)%2)));

A369959 Numbers k such that A003415(k) >= A276086(k) and gcd(k, A003415(k)) = gcd(k, A276086(k)), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

6, 30, 210, 214, 2310, 2313, 2315, 2317, 2318, 2342, 2346, 2370, 2526, 4622, 4830, 30030, 30031, 30033, 30034, 30035, 30038, 30041, 30043, 30046, 30054, 30061, 30062, 30066, 30069, 30074, 30094, 30098, 30102, 30242, 30245, 30247, 30249, 30254, 30270, 30274, 30282, 32342, 32345, 32347, 32350, 32354, 32374, 32553, 60062

Offset: 1

Antti Karttunen, Feb 07 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..3195
Index entries for sequences related to primorial base

Intersection of A351228 and A369962.
Subsequence of A369958.
Cf. A003415, A083345, A085731, A276086, A324198, A351251, A369960 (subsequence).

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA369959(n) = { my(t=A003415(n), u=A276086(n), g=gcd(n,t), h=gcd(n,u)); ((g==h) && ((t/g) >= (u/h))); };

{k | A085731(n) == A324198(n) and A083345(k) >= A351251(k)}.

A369960 Numbers k such that A003415(k) >= A276086(k) and gcd(k, A003415(k)) = gcd(k, A276086(k)) > 1, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

2313, 30033, 30069, 30249, 30282, 32350, 32553, 60093, 60273, 510550, 510561, 510579, 510633, 510723, 510741, 513063, 540963, 542853, 570573, 572910, 1021023, 1021062, 1021239, 1023363, 1531539, 1561563, 9699741, 9699746, 9699759, 9699903, 9699942, 9699957, 9699965, 9700150, 9700353, 9702009, 9702027, 9702049, 9702121

Offset: 1

Antti Karttunen, Feb 07 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..350
Index entries for sequences related to primorial base

Intersection of A351228 and A369963.
Subsequence of the following sequences: A013929, A369958, A369959, A369962.
Cf. A003415, A060735, A083345, A085731, A276086, A324198, A351251.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA369960(n) = { my(t=A003415(n), u=A276086(n), g=gcd(n,t), h=gcd(n,u)); ((t >= u) && (g==h) && (g>1)); };
isA369960(n) = if(!n || issquarefree(n), 0, my(t=A003415(n), u=A276086(n), g=gcd(n,t), h=gcd(n,u)); ((t >= u) && (g==h)));

A085731(n) = { my(f=factor(n)); for(i=1, #f~, if (f[i, 2] % f[i, 1], f[i, 2]--); ); factorback(f); };
A324198(n) = { my(m=1, p=2, orgn=n); while(n, m *= (p^min(n%p, valuation(orgn, p))); n = n\p; p = nextprime(1+p)); (m); };
isA369960(n) = if(!n || issquarefree(n),0,((A085731(n) == A324198(n)) && (A003415(n) >= A276086(n))));

{k | A085731(n) > 1 and A085731(n) == A324198(n) and A083345(k) >= A351251(k)}.

A369970 Numbers k such that A003415(k) is a multiple of A276086(k), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 6, 2315, 510510

Offset: 1

Antti Karttunen, Feb 07 2024

For the general dynamics of this phenomenon, see the scatter plots of A351231 and A351233.
Question: Are the terms by necessity all squarefree?
As a subsequence this sequence includes all primorials with indices k such that A024451(k) is a multiple of A000040(1+k). See A369972 and A369973.
872415232 < a(6) <= 13082761331670030 [= A369973(4)].

			2315 is included as A003415(2315) = 5+463 = 468 = 2^2 * 3^2 * 13 (note that 2315 is a semiprime = 5*463, thus its arithmetic derivative is the sum of its two prime factors), and because that 468 is a multiple of A276086(2315) = 234 = 2 * 3^2 * 13 [the exponents of primes are here read from the primorial base expansion of 2315, A049345(2315) = 100021].
510510 is included because A003415(510510) = 19*37693, which is a multiple of A276086(510510) = 19.

Index entries for sequences related to primorial base

Cf. A000040, A003415, A024451, A276086, A369972, A369973 (subsequence).
Positions of 1's in A351231, positions of 0's in A351233 and in A369971.
After the two initial terms, a subsequence of A351228.
Cf. also A358221.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA369970(n) = !(A003415(n)%A276086(n));

A328118 Numbers k for which A276086(k) <= A002620(k), where A276086 is the primorial base exp-function and A002620(k) = floor(k^2/4).

Original entry on oeis.org

6, 7, 8, 12, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 60, 61, 62, 63, 64, 65, 66, 67, 68, 72, 90, 91, 92, 93, 96, 120, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 234, 235, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255

Offset: 1

Antti Karttunen, Oct 08 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A002620, A276086, A328119 (complement).

Subsequences: A328110 (after its two initial terms), A351228, A370127.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A002620(n) = ((n^2)>>2);
isA328118(n) = (A276086(n) <= A002620(n));

A351232 a(n) = floor(A276086(n) / A003415(n)), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

Original entry on oeis.org

3, 6, 2, 18, 1, 10, 1, 5, 6, 90, 1, 50, 8, 18, 7, 450, 5, 250, 15, 75, 86, 2250, 14, 125, 125, 138, 175, 11250, 0, 14, 0, 3, 3, 10, 0, 70, 5, 13, 4, 630, 4, 350, 10, 26, 63, 3150, 7, 125, 58, 262, 140, 15750, 54, 546, 142, 1193, 1270, 78750, 0, 98, 4, 5, 2, 49, 4, 490, 10, 56, 37, 4410, 7, 2450, 94, 133, 137, 1225

Offset: 2

Antti Karttunen, Feb 05 2022

Antti Karttunen, Table of n, a(n) for n = 2..11050
Index entries for sequences related to primorial base

Cf. A003415, A276086, A351230, A351231.

Cf. A351228 (conjectured to give the positions of zeros from its second term onward).

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A351232(n) = floor(A276086(n) / A003415(n));

a(n) = floor(A276086(n) / A003415(n)) = floor(A351231(n) / A351230(n)).

A369664 Numbers k of the form 4m+1, whose arithmetic derivative k' is of the form 4u+2, and k' has an odd number of prime factors.

Original entry on oeis.org

65, 77, 141, 161, 185, 201, 209, 221, 301, 305, 321, 341, 365, 377, 381, 413, 437, 453, 481, 485, 497, 501, 537, 545, 589, 649, 681, 689, 717, 721, 729, 737, 745, 749, 785, 789, 849, 893, 901, 905, 917, 921, 989, 1037, 1073, 1081, 1101, 1121, 1133, 1141, 1157, 1165, 1169, 1189, 1205, 1253, 1261, 1293, 1313, 1317

Offset: 1

Antti Karttunen, Feb 06 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Intersection of A016813 and A369662.
Subsequence of A369666.
Cf. A003415, A351228.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA369664(n) = if(1!=(n%4), 0, my(d=A003415(n)); (2==(d%4) && (bigomega(d)%2)));

cp's OEIS Frontend

A369650 Numbers k such that A003415(k) = A276085(k), where A003415 is the arithmetic derivative, and A276085 is the primorial base log-function.

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A351088 Numbers k such that A327860(k) is reachable from k by iterating the arithmetic derivative (A003415) and there are no terms with p^p-factors on the path there.

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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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A369663 Numbers k of the form 4m+3, whose arithmetic derivative k' is of the form 4u+2, and k' has an even number of prime factors.

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A369959 Numbers k such that A003415(k) >= A276086(k) and gcd(k, A003415(k)) = gcd(k, A276086(k)), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

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A369960 Numbers k such that A003415(k) >= A276086(k) and gcd(k, A003415(k)) = gcd(k, A276086(k)) > 1, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

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A369970 Numbers k such that A003415(k) is a multiple of A276086(k), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

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A328118 Numbers k for which A276086(k) <= A002620(k), where A276086 is the primorial base exp-function and A002620(k) = floor(k^2/4).

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A351232 a(n) = floor(A276086(n) / A003415(n)), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

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