A369639 -id:A369639 - cp's OEIS frontend

A369642 Composite numbers k, not squarefree semiprimes, such that k' is a sum of distinct primorial numbers, where k' stands for the arithmetic derivative of k, A003415.

9, 16, 28, 30, 45, 108, 112, 136, 189, 198, 210, 212, 225, 236, 244, 246, 282, 290, 361, 374, 399, 435, 507, 1480, 1940, 2132, 2212, 2308, 2356, 2524, 2655, 2766, 2802, 3018, 3054, 3501, 3590, 3771, 3938, 4225, 4454, 4755, 4809, 5005, 5763, 6123, 6771, 9024, 9936, 10295, 11881, 12221, 16296, 22491, 24389, 26865

Offset: 1

Antti Karttunen, Jan 31 2024

nonn

Composite numbers k, not squarefree semiprimes, such that A327859(k) = A276086(A003415(k)) is squarefree number, or equally, k' is in A276156.

Squares that appear in this sequence: 9, 16, 225, 361, 4225, 11881, 1371241, 1635841, 225930961, 228644641, 229189321, 262083721, ...

Antti Karttunen, Table of n, a(n) for n = 1..2450
Index entries for sequences related to primorial base

Sequence A369641 without any terms of A006881.
Cf. A003415, A276086, A276156, A327859, A369647 (subsequence after its two initial terms).
Nonsquarefree terms all occur in A369639.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
ismaxprimobasedigit_at_most(n,k) = { my(s=0, p=2); while(n, if((n%p)>k, return(0)); n = n\p; p = nextprime(1+p)); (1); };
A369640(n) = if(n<2 || isprime(n), 0, ismaxprimobasedigit_at_most(A003415(n),1));
isA369642(n) = (((bigomega(n)>2)||(bigomega(n)>omega(n))) && A369640(n));

A370133 Numbers with no digit larger than 3 in primorial base, A049345.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99

Offset: 1

Antti Karttunen, Feb 11 2024

Numbers k for which A328114(k) <= 3.

Numbers k such that A276086(k) is biquadratefree, A046100.

Antti Karttunen, Table of n, a(n) for n = 1..24576
Index entries for sequences related to primorial base.

Cf. A046100, A049345, A276086, A328114.
Cf. A369639 (nonsquarefree numbers whose arithmetic derivative is in this sequence).
Cf. A370132, A276156 (subsequences).
Subsequence of A351576: a(n) differs from A351576(n-1) for the first time at n=97, where a(97) = 210, while A351576(96) = 120, a term not present here.

q[n_] := Module[{k = n, p = 2, s = {}, r}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, AppendTo[s, r]; p = NextPrime[p]]; Count[s, ?(# > 3 &)] == 0]; Select[Range[0, 100], q] (* _Amiram Eldar, Mar 06 2024 *)

ismaxprimobasedigit_at_most(n,k) = { my(s=0, p=2); while(n, if((n%p)>k, return(0)); n = n\p; p = nextprime(1+p)); (1); };
isA370133(n) = ismaxprimobasedigit_at_most(n,3);

cp's OEIS Frontend

A369642 Composite numbers k, not squarefree semiprimes, such that k' is a sum of distinct primorial numbers, where k' stands for the arithmetic derivative of k, A003415.

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