A108951 -id:A108951 - cp's OEIS frontend

A346109 a(n) = A276085(A108951(A346097(n))), where A346097(n) gives the denominator of the primorial deflation of A276086(A108951(n)).

0, 1, 3, 2, 9, 6, 39, 1, 3, 18, 249, 12, 2559, 78, 54, 2, 32589, 6, 543099, 36, 234, 498, 10242789, 9, 96, 5118, 42, 156, 233335659, 45, 6703028889, 10, 1494, 65178, 312, 12, 207263519019, 1086198, 15354, 9, 7628001653829, 39, 311878265181039, 996, 165, 20485578, 13394639596851069, 21, 1284, 192, 195534, 10236, 628284422185342479

Offset: 1

Antti Karttunen, Jul 08 2021

nonn

Cf. A002110, A064989, A108951, A276085, A324886, A319627, A346097, A346105, A346107, A346108.

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A319627(n) = (A064989(n) / gcd(n, A064989(n)));
A346097(n) = A319627(A324886(n));
A346107(n) = A108951(A346097(n));
A002110(n) = prod(i=1,n,prime(i));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
A346109(n) = A276085(A346107(n)); \\ Rest of program given in A324886.

a(n) = A346105(A346097(n)) = A276085(A346107(n)) = A276085(A108951(A346097(n))).

a(n) = A346108(n) - A108951(n).

A373985 a(n) = gcd(A108951(n), A373158(n)), where 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, 2, 6, 4, 30, 4, 210, 2, 12, 4, 2310, 2, 30030, 4, 36, 8, 510510, 2, 9699690, 2, 36, 4, 223092870, 12, 60, 4, 18, 2, 6469693230, 2, 200560490130, 2, 12, 4, 60, 16, 7420738134810, 4, 12, 12, 304250263527210, 2, 13082761331670030, 2, 6, 4, 614889782588491410, 2, 420, 2, 36, 2, 32589158477190044730, 4, 180, 24, 36, 4

Offset: 1

Antti Karttunen, Jun 25 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2048
Index entries for sequences related to primorial numbers

Cf. A000040, A002110, A034386, A108951, A373158, A373984, A373986, A373987.

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); };

a(n) = gcd(A373158(n), A373984(n)).
a(n) = A108951(n) / A373987(n).
For n >= 2, a(n) = A373158(n) / A373986(n).
For n >= 1, a(A000040(n)) = A002110(n).

A329619 Difference between the maximal digit value used when A108951(n) is written in primorial base and its 2-adic valuation.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, -2, -1, 0, 0, 1, 0, 0, 4, -2, 0, -1, 0, 1, 4, 0, 0, -1, 2, 0, -2, 1, 0, 2, 0, -4, 4, 0, 6, 0, 0, 0, 4, -3, 0, -2, 0, 1, 2, 0, 0, -2, 4, 5, 4, 1, 0, -2, 2, 4, 4, 0, 0, -1, 0, 0, 0, -4, 11, 9, 0, 1, 4, 2, 0, -2, 0, 0, 2, 1, 14, 9, 0, -3, 2, 0, 0, -2, 9, 0, 4, 4, 0, 6, 10, 1, 4, 0, 5, 0, 0, 9, 7, 2, 0, 9, 0, 4, 1

Offset: 1

Antti Karttunen, Nov 18 2019

sign

Cf. A001222, A007814, A034386, A108951, A276086, A324886, A328114, A329344, A329618, A329620, A329622.

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}]] - PrimeOmega[#] &, 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));
A329619(n) = (A329344(n) - bigomega(n));

a(n) = A329344(n) - A001222(n).
a(n) = A328114(A108951(n)) - A007814(A108951(n)).
a(p) = 0 for all primes p.

A346093 a(n) = A276085(A328571(A108951(n))).

Original entry on oeis.org

1, 2, 6, 2, 30, 6, 210, 8, 36, 30, 2310, 6, 30030, 210, 30, 8, 510510, 36, 9699690, 30, 210, 2310, 223092870, 36, 240, 30030, 216, 210, 6469693230, 240, 200560490130, 32, 2310, 510510, 2520, 36, 7420738134810, 9699690, 30030, 240, 304250263527210, 2520, 13082761331670030, 2310, 240, 223092870, 614889782588491410, 36, 32550

Offset: 1

Antti Karttunen, Jul 09 2021

nonn

Cf. A002110, A108951, A276085, A328571, A346091, A346092.

A002110(n) = prod(i=1,n,prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A002110(primepi(f[i, 1]))^f[i, 2]) };
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
A328571(n) = { my(m=1, p=2); while(n, m *= (p^!!(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A346093(n) = A276085(A328571(A108951(n)));

a(n) = A276085(A346091(n)) = A276085(A328571(A108951(n))).

a(n) = A108951(n) - A346092(n).

A346105 a(n) = A276085(A108951(n)).

Original entry on oeis.org

0, 1, 3, 2, 9, 4, 39, 3, 6, 10, 249, 5, 2559, 40, 12, 4, 32589, 7, 543099, 11, 42, 250, 10242789, 6, 18, 2560, 9, 41, 233335659, 13, 6703028889, 5, 252, 32590, 48, 8, 207263519019, 543100, 2562, 12, 7628001653829, 43, 311878265181039, 251, 15, 10242790, 13394639596851069, 7, 78, 19, 32592, 2561, 628284422185342479, 10, 258, 42

Offset: 1

Antti Karttunen, Jul 08 2021

nonn

Additive with a(p^e) = e * A143293(A000720(p)-1), where A143293 is the partial sums of primorials, A002110. (Compare to the formula of A276085).

Cf. A002110, A108951, A143293, A276085, A346108, A346109.

Cf. also A324886.

A002110(n) = prod(i=1,n,prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A002110(primepi(f[i, 1]))^f[i, 2]) };
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
A346105(n) = A276085(A108951(n));

A143293(n) = { if(n==0, return(1)); my(P=1, s=1); forprime(p=2, prime(n), s+=P*=p); s; }; \\ This function from A143293
A346105(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A143293(primepi(f[k, 1])-1)); };

a(n) = A276085(A108951(n)).

A346106 a(n) = A108951(A346096(n)), where A346096(n) gives the numerator of the primorial deflation of A276086(A108951(n)).

Original entry on oeis.org

2, 6, 30, 36, 210, 900, 2310, 30, 210, 44100, 30030, 810000, 510510, 5336100, 85766121000000, 900, 9699690, 44100, 223092870, 1944810000, 151939915084881000000, 901800900, 6469693230, 189000, 28473963210000, 260620460100, 69300, 28473963210000, 200560490130, 4492511100000, 7420738134810, 1260, 733384949590939374729000000

Offset: 1

Antti Karttunen, Jul 08 2021

nonn

Cf. A064989, A108951, A319626, A324886, A346096, A346107, A346108.

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A319626(n) = (n / gcd(n, A064989(n)));
A346096(n) = A319626(A324886(n));
A346106(n) = A108951(A346096(n)); \\ Rest of program in A324886.

a(n) = A108951(A346096(n)) = A108951(A319626(A324886(n))).

a(n) = A324886(n) * A346107(n).

A346107 a(n) = A108951(A346097(n)), where A346097(n) gives the denominator of the primorial deflation of A276086(A108951(n)).

Original entry on oeis.org

1, 2, 6, 4, 30, 36, 210, 2, 6, 900, 2310, 1296, 30030, 44100, 729000000, 4, 510510, 36, 9699690, 810000, 85766121000000, 5336100, 223092870, 216, 39690000, 901800900, 1260, 1944810000, 6469693230, 24300000, 200560490130, 60, 151939915084881000000, 260620460100, 3782285936100000000, 1296, 7420738134810, 94083986096100

Offset: 1

Antti Karttunen, Jul 08 2021

nonn

Cf. A064989, A108951, A276086, A319627, A324886, A346097, A346106, A346109.

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A319627(n) = (A064989(n) / gcd(n, A064989(n)));
A346097(n) = A319627(A324886(n));
A346107(n) = A108951(A346097(n)); \\ Rest of program given in A324886.

a(n) = A108951(A346097(n)) = A108951(A319627(A324886(n))).

a(n) = A346106(n) / A324886(n).

A354351 Dirichlet inverse of A108951, primorial inflation of n.

Original entry on oeis.org

1, -2, -6, 0, -30, 12, -210, 0, 0, 60, -2310, 0, -30030, 420, 180, 0, -510510, 0, -9699690, 0, 1260, 4620, -223092870, 0, 0, 60060, 0, 0, -6469693230, -360, -200560490130, 0, 13860, 1021020, 6300, 0, -7420738134810, 19399380, 180180, 0, -304250263527210, -2520, -13082761331670030, 0, 0, 446185740, -614889782588491410

Offset: 1

Antti Karttunen, Jun 05 2022

Multiplicative with a(p^e) = 0 if e > 1, and -A034386(p) otherwise.

Cf. A002110, A008683, A013929 (positions of 0's), A034386, A108951, A354352.

Cf. also A347379, A354186, A354349, A354359, A354365, A354366.

A002110(n) = prod(i=1,n,prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A002110(primepi(f[i, 1]))^f[i, 2]) }; \\ From A108951
A354351(n) = (moebius(n)*A108951(n));

a(n) = A008683(n) * A108951(n).
a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d < n} A108951(n/d) * a(d).
a(n) = A354352(n) - A108951(n).

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).

A329378 Least common multiple of exponents of prime factors of A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 17 2019

nonn

Cf. A019565, A034386, A072411, A108951, A284002, A329382, A329600, A329605.

Differs from related A329617 for the first time at n=36.

A034386(n) = prod(i=1, primepi(n), prime(i));
A072411(n) = lcm(factor(n)[, 2]); \\ From A072411
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A329378(n) = A072411(A108951(n));

a(n) = A072411(A108951(n)) = A072411(A329600(n)).
a(n) <= A329617(n) <= A329382(n) <= A329605(n).
a(A019565(n)) = A284002(n).

cp's OEIS Frontend

A346109 a(n) = A276085(A108951(A346097(n))), where A346097(n) gives the denominator of the primorial deflation of A276086(A108951(n)).

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A373985 a(n) = gcd(A108951(n), A373158(n)), where A108951 is fully multiplicative and A373158 is fully additive with a(p) = p# for prime p, where x# is the primorial A034386(x).

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A329619 Difference between the maximal digit value used when A108951(n) is written in primorial base and its 2-adic valuation.

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Mathematica

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A346093 a(n) = A276085(A328571(A108951(n))).

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A346105 a(n) = A276085(A108951(n)).

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A346106 a(n) = A108951(A346096(n)), where A346096(n) gives the numerator of the primorial deflation of A276086(A108951(n)).

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A346107 a(n) = A108951(A346097(n)), where A346097(n) gives the denominator of the primorial deflation of A276086(A108951(n)).

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A354351 Dirichlet inverse of A108951, primorial inflation of n.

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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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