A276156 -id:A276156 - cp's OEIS frontend

A329029 a(n) = A069359(A276086(n)), where A276086 is the primorial base exp-function and A069359(n) = n * Sum_{p|n} 1/p.

0, 1, 1, 5, 3, 15, 1, 7, 8, 31, 24, 93, 5, 35, 40, 155, 120, 465, 25, 175, 200, 775, 600, 2325, 125, 875, 1000, 3875, 3000, 11625, 1, 9, 10, 41, 30, 123, 12, 59, 71, 247, 213, 741, 60, 295, 355, 1235, 1065, 3705, 300, 1475, 1775, 6175, 5325, 18525, 1500, 7375, 8875, 30875, 26625, 92625, 7, 63, 70, 287, 210, 861, 84, 413, 497, 1729

Offset: 0

Antti Karttunen, Nov 07 2019

nonn

A380535 gives the indices n where a(n) is a multiple of A053669(n). This can be seen from the formula a(n) = A003557(A276086(n)) * A069359(A328571(n)). The left hand side of the product is a multiple of A053669(n) if and only if A276088(n) > 1, while the right hand side is never a multiple of A053669(n), as it is equal to A329031(n) = A003415(A007947(A276086(n))). - Antti Karttunen, Feb 11 2025

Antti Karttunen, Table of n, a(n) for n = 0..2310
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..32768
Index entries for sequences related to primorial base

Cf. A003415, A003557, A053669, A069359, A276086, A276088, A328571, A328572, A329031, A380535.

Coincides with A327860 on the positions given by A276156.

A329029(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); if(e, m *= (p^e); s += (1/p)); n = n\p; p = nextprime(1+p)); (s*m); };

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A069359(n) = (n*sumdiv(n, d, isprime(d)/d));
A329029(n) = A069359(A276086(n));

a(n) = A069359(A276086(n)).

a(n) = A328572(n) * A329031(n) = A003557(A276086(n)) * A069359(A328571(n)). - Antti Karttunen, Feb 11 2025

A328110 Fixed points of A327860: numbers k for which A003415(A276086(k)) = k, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 7, 8, 2556

Offset: 1

Antti Karttunen, Oct 08 2019

Applying A276086 to these terms gives the fixed points of A327859: 1, 2, 10, 15, 5005, ..., i.e., A369650 without any of the terms of A100716.
No more terms below <= 2550136832.
From Antti Karttunen, Feb 09 2024: (Start)
The known five terms are all members of A276156, which is equal to the claim that the intersection of A048103 and A369650 is squarefree. See the example, and also comments in A351088 and in A380527.
Even terms here must be multiples of 4, see comment in A327860.
No terms of A047257 may occur in this sequence, which is equal to the claim that A276086(a(n)) is never a multiple of 9. See comment in A327859.
(End)

			Computing A327860(2556) is easy, because it is a member of A276156, as 2556 = 6 + 30 + 210 + 2310. Therefore A327860(2556) = A003415(5*7*11*13) = (5*7*11) + (5*7*13) + (5*11*13) + (7*11*13) = 2556, and 2556 is included in this sequence. - _Antti Karttunen_, Feb 04 2024

Cf. A002110, A002620, A003415, A047257, A048103, A100716, A276085, A276086, A327859, A327860, A369650.
After 0, the intersection of A351087 and A380527, thus like the latter, also this is conjectured to be a subsequence of A276156.
After two initial terms (0 & 1), a subsequence of A328118. Subsequence of A351088.

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); };
isA328110(n) = (A327860(n) == n);

A328841 Substitute ones for all nonzero digits in primorial base expansion of n, then convert back to decimal.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 6, 7, 8, 9, 8, 9, 6, 7, 8, 9, 8, 9, 6, 7, 8, 9, 8, 9, 6, 7, 8, 9, 8, 9, 30, 31, 32, 33, 32, 33, 36, 37, 38, 39, 38, 39, 36, 37, 38, 39, 38, 39, 36, 37, 38, 39, 38, 39, 36, 37, 38, 39, 38, 39, 30, 31, 32, 33, 32, 33, 36, 37, 38, 39, 38, 39, 36, 37, 38, 39, 38, 39, 36, 37, 38, 39, 38, 39, 36, 37, 38, 39, 38, 39, 30

Offset: 0

Antti Karttunen, Oct 30 2019

Antti Karttunen, Table of n, a(n) for n = 0..30029
Index entries for sequences related to primorial base

Cf. A002110, A007947, A257993, A267263, A276085, A276086, A328570, A328571, A328620, A328842, A328843.
Cf. A276156 (fixed points).
Cf. A276008 for analogous sequence.

A328841(n) = { my(p=2, r=1, s=0); while(n, s += ((!!(n%p))*r); r *= p; n = n\p; p = nextprime(1+p)); (s); };

a(n) = n - A328842(n).
a(n) = A276085(A328571(n)) = A276085(A007947(A276086(n))).
For all n>= 0, a(A276086(n)) = A328843(n).
For all n >= 1, A257993(a(n)) = A257993(n).
For all n >= 0, A328570(a(n)) = A328570(n), A328620(a(n)) = A328620(n), and A267263(a(n)) = A267263(n).

A328828 Index of the least significant digit larger than one in the primorial base expansion of n, 0 if no such digit exists.

Original entry on oeis.org

0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 4, 4, 4, 4, 2, 2, 4, 4, 4, 4, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 4, 4, 4, 4, 2, 2, 4, 4, 4, 4, 2, 2, 3, 3, 3, 3

Offset: 0

Antti Karttunen, Oct 29 2019

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to primorial base.

Cf. A000720, A049345, A055396, A276086, A276156 (positions of 0's), A277885, A328572, A328829 [= a(A276086(n))], A328832, A381032 [= A008578(1+a(n))].

a[n_] := Module[{k = n, p = 2, s = {}, r, i}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, AppendTo[s, r]; p = NextPrime[p]]; i = FirstPosition[s, ?(# > 1 &)]; If[MissingQ[i], 0, i[[1]]]]; Array[a, 100] (* _Amiram Eldar, Mar 13 2024 *)

A328828(n) = { my(i=1, p=2); while(n, if((n%p)>1, return(i)); i++; n = n\p; p = nextprime(1+p)); (0); };

a(n) = A277885(A276086(n)) = A055396(A328572(n)).

a(n) = A000720(A381032(n)). - Antti Karttunen, Feb 23 2025

A328842 Decrement each nonzero digit by one in primorial base representation of n, then convert back to decimal.

Original entry on oeis.org

0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 2, 6, 6, 6, 6, 8, 8, 12, 12, 12, 12, 14, 14, 18, 18, 18, 18, 20, 20, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 2, 6, 6, 6, 6, 8, 8, 12, 12, 12, 12, 14, 14, 18, 18, 18, 18, 20, 20, 30, 30, 30, 30, 32, 32, 30, 30, 30, 30, 32, 32, 36, 36, 36, 36, 38, 38, 42, 42, 42, 42, 44, 44, 48, 48, 48, 48, 50, 50, 60, 60

Offset: 0

Antti Karttunen, Oct 30 2019

Antti Karttunen, Table of n, a(n) for n = 0..30030
Index entries for sequences related to primorial base

Cf. A002110, A003557, A276085, A276086, A328114, A328572, A328841, A328844.
Cf. A276156 (positions of zeros).
Cf. A276009 for analogous sequence.

A328842(n) = { my(p=2, r=1, s=0); while(n, if((n%p)>0, s += ((n%p)-1)*r); r *= p; n = n\p; p = nextprime(1+p)); (s); };

a(n) = n - A328841(n).
a(n) = A276085(A328572(n)) = A276085(A003557(A276086(n))).
For all n >= 0, a(A276086(n)) = A328844(n).
For all n >= 1, A328114(a(n)) = A328114(n) - 1.

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));

A351576 Factorial base expansion of n reinterpreted as a primorial base expansion, then converted back to decimal.

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, 100

Offset: 0

Antti Karttunen, Apr 01 2022

			n = 313 has factorial base representation (see A007623) "23001" because 2*5! + 3*4! + 1*1! = 240+72+1 = 313. When this is reinterpreted as a primorial base expansion (see A049345), we obtain 2*A002110(4) + 3*A002110(3) + 1*A002110(0) = 511, therefore a(313) = 511.

Cf. A000142, A002110, A007623, A049345, A276076, A276085.

Cf. also A276156.

a[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; FromDigits[Reverse[s], MixedRadix[Reverse@ Prime@ Range@ Length[s]]]]; Array[a, 100, 0] (* Amiram Eldar, Feb 07 2024 *)

A002110(n) = prod(i=1,n,prime(i));
A276076(n) = { my(i=0,m=1,f=1,nextf); while((n>0),i=i+1; nextf = (i+1)*f; if((n%nextf),m*=(prime(i)^((n%nextf)/f));n-=(n%nextf));f=nextf); m; };
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
A351576(n) = A276085(A276076(n));

a(n) = A276085(A276076(n)).

A328832 Numbers that are sums of distinct primorial numbers, A002110, and do not have a factor of the form p^p.

Original entry on oeis.org

1, 2, 3, 6, 7, 9, 30, 31, 33, 37, 38, 39, 210, 211, 213, 217, 218, 219, 241, 242, 246, 247, 249, 2310, 2311, 2313, 2317, 2318, 2319, 2341, 2342, 2343, 2346, 2347, 2521, 2522, 2523, 2526, 2527, 2529, 2550, 2551, 2553, 2557, 2558, 2559, 30030, 30031, 30033, 30037, 30038, 30039, 30061, 30062, 30063, 30066, 30067, 30069, 30241

Offset: 1

Antti Karttunen, Oct 30 2019

nonn

Numbers n such that A129251(n) = 0 and A328828(n) = 0 (or equally, A328114(n) = 1).

Terms k in A276156 for which A276086(A276085(k)) = k, i.e., those terms of A276156 which are in the range of A276086.

Antti Karttunen, Table of n, a(n) for n = 1..23330
Index entries for sequences related to primorial base

Intersection of A048103 and A276156.

Cf. A002110, A129251, A276085, A276086, A328114, A328828, A328831, A328833, A328836.

A129251(n) = { my(f = factor(n)); sum(k=1, #f~, (f[k, 2]>=f[k, 1])); };
A328828(n) = { my(i=1, p=2); while(n, if((n%p)>1, return(i)); i++; n = n\p; p = nextprime(1+p)); (0); };
isA328832(n) = ((0==A129251(n)) && (0==A328828(n)));

A276156(n) = { my(p=2,pr=1,s=0); while(n,if(n%2,s += pr); n >>= 1; pr *= p; p = nextprime(1+p)); (s); };
k=0; for(n=1,(2^15)-1, if(!A129251(u=A276156(n)), k++; write("b328832.txt", k, " ", u, " ")));

a(n) = A276086(A328833(n)).

A328849 Numbers in whose primorial base expansion only even digits appear.

Original entry on oeis.org

0, 4, 12, 16, 24, 28, 60, 64, 72, 76, 84, 88, 120, 124, 132, 136, 144, 148, 180, 184, 192, 196, 204, 208, 420, 424, 432, 436, 444, 448, 480, 484, 492, 496, 504, 508, 540, 544, 552, 556, 564, 568, 600, 604, 612, 616, 624, 628, 840, 844, 852, 856, 864, 868, 900, 904, 912, 916, 924, 928, 960, 964, 972, 976, 984, 988, 1020, 1024

Offset: 1

Antti Karttunen, Oct 30 2019

Numbers for which the prime factor form (A276086) of their primorial base expansion is a square, A000290.

			144 is written as "4400" in primorial base (A049345), because 4*A002110(3) + 4*A002110(2) + 0*A002110(1) + 0*A002110(0) = 4*30 + 4*6 = 144, thus all the digits are even and 144 is included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..9072
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..90720
Index entries for sequences related to primorial base.

Cf. A000290, A002110, A010052, A049345, A276086, A276156, A328770.
Cf. A328834, A328850 (squares in this sequence).
Similar sequences: A005823 (ternary), A014263 (decimal), A062880 (quaternary), A351893 (factorial base).

With[{max = 5}, bases = Prime@ Range[max, 1, -1]; nmax = Times @@ bases - 1; prmBaseDigits[n_] := IntegerDigits[n, MixedRadix[bases]]; Select[Range[0, nmax, 2], AllTrue[prmBaseDigits[#], EvenQ] &]] (* Amiram Eldar, May 23 2023 *)

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA328849(n) = issquare(A276086(n));

a(n) = 2*A328770(n).

A000196(A276086(a(n))) = A276086(a(n)/2) = A328834(n).

A341518 Numbers k such that the primorial base representation of their arithmetic derivative does not contain digits larger than 1.

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 9, 10, 11, 13, 14, 15, 16, 17, 19, 23, 28, 29, 30, 31, 37, 41, 43, 45, 47, 53, 58, 59, 61, 62, 67, 71, 73, 74, 79, 83, 87, 89, 97, 101, 103, 107, 108, 109, 112, 113, 127, 131, 136, 137, 139, 149, 151, 155, 157, 161, 163, 167, 173, 179, 181, 189, 191, 193, 197, 198, 199, 203, 209, 210, 211, 212, 217

Offset: 1

Antti Karttunen, Feb 28 2021

nonn

Numbers k for which A328390(k) <= 1, numbers k such that A003415(k) is in A276156.

Numbers k such that A327859(k) = A276086(A003415(k)) is squarefree.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to primorial base

Cf. A003415, A005117, A049345, A276086, A276156, A327859, A328114, A328390.
Positions of nonzero terms in A341517.
Subsequences: A000040, A327978, A328232, A369647 (terms k where A051903(k) obtains novel values).
Cf. also A327969.

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); };
isA341518(n) = ismaxprimobasedigit_at_most(A003415(n),1); \\ Antti Karttunen, Feb 03 2024

For all n > 2, A328390(a(n)) = A328114(A003415(a(n))) = 1.

cp's OEIS Frontend

A329029 a(n) = A069359(A276086(n)), where A276086 is the primorial base exp-function and A069359(n) = n * Sum_{p|n} 1/p.

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A328110 Fixed points of A327860: numbers k for which A003415(A276086(k)) = k, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

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A328841 Substitute ones for all nonzero digits in primorial base expansion of n, then convert back to decimal.

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A328828 Index of the least significant digit larger than one in the primorial base expansion of n, 0 if no such digit exists.

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A328842 Decrement each nonzero digit by one in primorial base representation of n, then convert back to decimal.

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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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A351576 Factorial base expansion of n reinterpreted as a primorial base expansion, then converted back to decimal.

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A328832 Numbers that are sums of distinct primorial numbers, A002110, and do not have a factor of the form p^p.

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A328849 Numbers in whose primorial base expansion only even digits appear.