A324888 -id:A324888 - cp's OEIS frontend

A329344 Number of times most frequent primorial is present in the greedy sum of primorials adding to A108951(n).

1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 6, 2, 1, 2, 1, 4, 6, 2, 1, 3, 4, 2, 1, 4, 1, 5, 1, 1, 6, 2, 8, 4, 1, 2, 6, 1, 1, 1, 1, 4, 5, 2, 1, 3, 6, 8, 6, 4, 1, 2, 4, 8, 6, 2, 1, 3, 1, 2, 3, 2, 13, 12, 1, 4, 6, 5, 1, 3, 1, 2, 5, 4, 16, 12, 1, 2, 6, 2, 1, 2, 11, 2, 6, 8, 1, 10, 12, 4, 6, 2, 7, 6, 1, 12, 10, 6, 1, 12, 1, 8, 4

Offset: 1

Antti Karttunen, Nov 11 2019

nonn

The greedy sum is also the sum with the minimal number of primorials, used for example in the primorial base representation.

			For n = 24 = 2^3 * 3, A108951(24) = A034386(2)^3 * A034386(3) = 2^3 * 6 = 48 = 30 + 6 + 6 + 6, and as the most frequent primorial in the sum is 6 = A002110(2), we have a(24) = 3.

Cf. A002110, A034386, A051903, A108951, A276086, A324886, A324888, A328114, A329040, A329045, A329343, A329348, A329349.

With[{b = Reverse@ Prime@ Range@ 120}, Array[Max@ IntegerDigits[#, MixedRadix[b]] &@ Apply[Times, Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]] &, 105] ] (* Michael De Vlieger, Nov 18 2019 *)

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A328114(n) = { my(s=0, p=2); while(n, s = max(s,(n%p)); n = n\p; p = nextprime(1+p)); (s); };
A329344(n) = A328114(A108951(n));

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324886(n) = A276086(A108951(n));
A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A329344(n) = A051903(A324886(n));

a(n) = A328114(A108951(n)) = A051903(A324886(n)).

A328771 Minimal number of primorials (A002110) that add to A328768(n), where A328768 is the first primorial based variant of arithmetic derivative.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 2, 1, 2, 2, 5, 1, 4, 1, 4, 6, 2, 1, 3, 1, 4, 6, 6, 1, 6, 2, 5, 5, 10, 1, 6, 1, 6, 8, 7, 8, 6, 1, 6, 8, 6, 1, 5, 1, 8, 7, 8, 1, 6, 2, 9, 6, 10, 1, 8, 8, 8, 6, 9, 1, 10, 1, 4, 9, 8, 10, 13, 1, 8, 8, 14, 1, 10, 1, 5, 5, 10, 12, 10, 1, 6, 2, 7, 1, 8, 10, 6, 10, 14, 1, 5, 14, 8, 6, 8, 12, 6, 1, 9, 15, 8, 1, 16, 1, 14, 7

Offset: 0

Antti Karttunen, Oct 28 2019

nonn

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

Cf. A002110, A276150, A324888, A328768, A328772.

A002110(n) = prod(i=1,n,prime(i));
A328768(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i,1])-1)/f[i, 1]));
A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
A328771(n) = A276150(A328768(n));

a(n) = A276150(A328768(n)).

A342462 Sum of digits when A329886(n) is written in primorial base, where A329886 is the primorial inflation of Doudna-tree.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 6, 4, 6, 4, 2, 4, 1, 2, 6, 4, 10, 6, 6, 4, 8, 12, 10, 8, 22, 4, 8, 2, 1, 2, 6, 4, 6, 2, 6, 2, 18, 10, 8, 6, 18, 12, 16, 4, 26, 16, 24, 8, 20, 14, 4, 6, 26, 16, 14, 8, 30, 6, 8, 4, 1, 2, 6, 4, 14, 12, 12, 8, 18, 12, 24, 4, 8, 12, 14, 4, 24, 20, 28, 20, 26, 16, 16, 12, 32, 26, 24, 14, 28, 16

Offset: 0

Antti Karttunen, Mar 15 2021

nonn

From David A. Corneth's Feb 27 2019 comment in A276150 follows that the only odd terms in this sequence are 1's occurring at 0 and at two's powers.

Subsequences starting at each n = 2^k are slowly converging towards A329886: 1, 2, 6, 4, 30, 12, 36, 8, 210, 60, 180, 24, etc.. Compare also to the behaviors of A324342 and A342463.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to primorial base

Cf. A001222, A005940, A108951, A276150, A329886, A342456, A342457, A342461, A342463, A342464.

Cf. also A324342, A324383, A324387, A324888.

A342462(n) = bigomega(A342456(n)); \\ Other code as in A342456.

A283980(n) = {my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 6, nextprime(p+1))^e)};
A329886(n) = if(n<2,1+n,if(!(n%2),A283980(A329886(n/2)),2*A329886(n\2)));
A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
A342462(n) = A276150(A329886(n));

a(n) = A001222(A342456(n)) = A001222(A342457(n)).
a(n) = A276150(A329886(n)) = A324888(A005940(1+n)).
a(n) >= A342461(n).
For n >= 0, a(2^n) = 1.

A329349 Number of occurrences of the largest primorial present in the greedy sum of primorials adding to A108951(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 6, 2, 1, 2, 1, 4, 6, 2, 1, 1, 4, 2, 1, 4, 1, 1, 1, 1, 6, 2, 2, 4, 1, 2, 6, 1, 1, 1, 1, 4, 5, 2, 1, 3, 1, 8, 6, 4, 1, 2, 2, 8, 6, 2, 1, 3, 1, 2, 3, 2, 1, 12, 1, 4, 6, 5, 1, 1, 1, 2, 2, 4, 16, 12, 1, 2, 6, 2, 1, 2, 1, 2, 6, 8, 1, 10, 12, 4, 6, 2, 1, 6, 1, 2, 2, 1, 1, 12, 1, 8, 1

Offset: 1

Antti Karttunen, Nov 11 2019

nonn

The greedy sum is also the sum with the minimal number of primorials, used for example in the primorial base representation.

			For n = 21 = 3 * 7, A108951(21) = A034386(3) * A034386(7) = 6 * 210, so the factor of the largest primorial present (210) in the greedy sum is 6 (as 1260 = 210 + 210 + 210 + 210 + 210 + 210), thus a(21) = 6.
For n = 24 = 2^3 * 3, A108951(24) = A034386(2)^3 * A034386(3) = 2^3 * 6 = 48 = 1*30 + 3*6, and as the factor of the largest primorial in the sum is 1, we have a(24) = 1.

Cf. A002110, A034386, A071178, A108951, A276086, A276153, A324886, A324888, A329040, A329343, A329344, A329345, A329348.

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A276153(n) = { my(e=0, p=2); while(n, e=n%p; n = n\p; p = nextprime(1+p)); (e); };
A329349(n) = A276153(A108951(n));

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324886(n) = A276086(A108951(n));
A071178(n) = if(1==n,0,my(es=factor(n)[,2]); es[#es]);
A329349(n) = A071178(A324886(n));

a(n) = A276153(A108951(n)) = A071178(A324886(n)).

a(n) <= A324888(n).

A324905 a(n) = A007895(A003965(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 3, 2, 2, 1, 2, 3, 2, 3, 3, 1, 3, 1, 3, 2, 2, 3, 2, 1, 2, 2, 3, 1, 3, 1, 3, 3, 2, 1, 3, 3, 3, 2, 3, 1, 4, 3, 3, 2, 2, 1, 2, 1, 2, 4, 3, 3, 3, 1, 3, 2, 4, 1, 4, 1, 2, 4, 3, 3, 3, 1, 4, 3, 2, 1, 3, 3, 2, 2, 3, 1, 2, 3, 3, 2, 2, 3, 3, 1, 3, 4, 3, 1, 3, 1, 3, 4

Offset: 1

Antti Karttunen, Apr 14 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..17711
Index entries for sequences computed from indices in prime factorization

Cf. A003965, A007895, A300837, A324888, A324907.

A003965(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = fibonacci(2+primepi(f[i, 1]))); factorback(f); };
A007895(n) = { my(s=0); while(n>0, s++; n -= fibonacci(1+A072649(n))); (s); }
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A324905(n) = A007895(A003965(n));

a(n) = A007895(A003965(n)).

A324907 a(n) = A007895(A113175(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 2, 1, 3, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 1, 4, 3, 2, 1, 1, 1, 3, 1, 2, 1, 1, 2, 1, 1, 3, 1, 3, 2, 1, 1, 2, 3, 1, 2, 1, 1, 3, 1, 4, 2, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1, 1, 3, 4, 1, 2, 1, 3, 1, 1, 4, 3, 3, 1, 2, 1, 1, 4

Offset: 1

Antti Karttunen, Apr 14 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..17711

Cf. A113175, A007895, A300837, A324888, A324905.

A113175(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = fibonacci(f[i, 1])); factorback(f); };
A007895(n) = { my(s=0); while(n>0, s++; n -= fibonacci(1+A072649(n))); (s); }
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A324907(n) = A007895(A113175(n));

a(n) = A007895(A113175(n)).

a(2n) = a(n).

A324887 a(n) = A108951(n) * A276086(A108951(n)).

Original entry on oeis.org

2, 6, 30, 36, 210, 300, 2310, 120, 1260, 2940, 30030, 15000, 510510, 50820, 21176820, 3600, 9699690, 88200, 223092870, 288120, 2232166860, 780780, 6469693230, 42000, 645668100, 17357340, 11880, 12298440, 200560490130, 66555720, 7420738134810, 672, 66899572740, 368588220, 228227900600700, 216090000, 304250263527210

Offset: 1

Antti Karttunen, Mar 30 2019

nonn

Cf. A034386, A108951, A276086, A324580, A324886, A324888.

Permutation of A324577.

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A324886(n) = A276086(A108951(n));
A324887(n) = (A324886(n)*A108951(n));

a(n) = A324580(A108951(n)) = A108951(n) * A324886(n).

A328772 Minimal number of primorials (A002110) that add to A328769(n), where A328769 is the second primorial based variant of arithmetic derivative.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 4, 1, 5, 4, 4, 1, 3, 1, 8, 6, 7, 1, 4, 4, 7, 7, 10, 1, 3, 1, 8, 6, 5, 6, 2, 1, 5, 8, 6, 1, 11, 1, 12, 3, 7, 1, 6, 4, 7, 8, 10, 1, 9, 10, 16, 10, 9, 1, 8, 1, 5, 13, 10, 12, 15, 1, 12, 10, 7, 1, 8, 1, 7, 5, 10, 8, 11, 1, 12, 6, 9, 1, 8, 10, 9, 12, 14, 1, 9, 10, 12, 10, 7, 12, 6, 1, 13, 17, 14, 1, 11, 1, 14, 9

Offset: 0

Antti Karttunen, Oct 28 2019

nonn

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

Cf. A002110, A276150, A324888, A328769, A328771.

A002110(n) = prod(i=1,n,prime(i));
A328769(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i,1]))/f[i, 1]));
A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
A328772(n) = A276150(A328769(n));

a(n) = A276150(A328769(n)).

A373989 a(n) = A276150(gcd(A108951(n), A373158(n))), where A276150 is the digit sum in primorial base, A108951 is fully multiplicative and A373158 is fully additive with a(p) = p# for prime p, where x# is the primorial A034386(x).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 3, 1, 1, 1, 1, 1, 2, 2, 2, 4, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 6, 4, 2, 2, 1, 4, 1, 2, 1, 2, 6, 1, 1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 6, 1, 1, 1, 4, 2, 1, 4, 2, 2, 2, 2, 1, 2, 6, 1, 2, 2, 2, 4, 1, 1, 5, 4, 1, 1, 1, 2, 1

Offset: 1

Antti Karttunen, Jun 26 2024

nonn

Cf. A034386, A108951, A276150, A373158, A373985.

Cf. also A324888.

A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
A373985(n) = { my(f=factor(n),m=1,s=0); for(i=1, #f~, my(x=prod(i=1,primepi(f[i, 1]),prime(i))); s += f[i, 2]*x; m *= x^f[i, 2]); gcd(m,s); };
A373989(n) = A276150(A373985(n));

a(n) = A276150(A373985(n)).

A365461 Sum of digits when A181821(n) is written in primorial base (A049345).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 4, 1, 2, 4, 2, 2, 4, 4, 4, 1, 6, 6, 6, 4, 8, 4, 6, 2, 2, 8, 6, 2, 12, 6, 16, 1, 6, 10, 4, 6, 16, 10, 8, 4, 10, 6, 10, 4, 12, 10, 6, 2, 8, 6, 10, 8, 12, 10, 8, 8, 20, 10, 18, 2, 26, 14, 8, 1, 12, 12, 22, 10, 20, 12, 28, 6, 28, 14, 10, 10, 6, 8, 34, 4, 8, 18, 38, 4, 10, 14, 20, 6, 52, 10, 12, 10, 24

Offset: 1

Antti Karttunen, Sep 15 2023

nonn

Minimal number of primorials (A002110) that add to A181821(n).

Cf. A002110, A049345, A122111, A181821, A276150, A324888, A365460.

A181821(n) = { my(f=factor(n),p=0,m=1); forstep(i=#f~,1,-1,while(f[i,2], f[i,2]--; m *= (p=nextprime(p+1))^primepi(f[i,1]))); (m); };
A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
A365461(n) = A276150(A181821(n));

a(n) = A276150(A181821(n)).
a(n) = A324888(A122111(n)).
a(n) >= A365460(n).

cp's OEIS Frontend

A329344 Number of times most frequent primorial is present in the greedy sum of primorials adding to A108951(n).

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A328771 Minimal number of primorials (A002110) that add to A328768(n), where A328768 is the first primorial based variant of arithmetic derivative.

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A342462 Sum of digits when A329886(n) is written in primorial base, where A329886 is the primorial inflation of Doudna-tree.

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A329349 Number of occurrences of the largest primorial present in the greedy sum of primorials adding to A108951(n).

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A324905 a(n) = A007895(A003965(n)).

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A324907 a(n) = A007895(A113175(n)).

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A324887 a(n) = A108951(n) * A276086(A108951(n)).

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A328772 Minimal number of primorials (A002110) that add to A328769(n), where A328769 is the second primorial based variant of arithmetic derivative.

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