A108951 -id:A108951 - cp's OEIS frontend

A330690 Number of ways to factor A108951(n) into "Fermi-Dirac primes" (A050376), where A108951 is fully multiplicative with a(prime(k)) = k-th primorial.

1, 1, 1, 2, 1, 2, 1, 2, 4, 2, 1, 2, 1, 2, 4, 4, 1, 4, 1, 2, 4, 2, 1, 4, 8, 2, 4, 2, 1, 4, 1, 4, 4, 2, 8, 8, 1, 2, 4, 4, 1, 4, 1, 2, 4, 2, 1, 4, 16, 8, 4, 2, 1, 8, 8, 4, 4, 2, 1, 8, 1, 2, 4, 6, 8, 4, 1, 2, 4, 8, 1, 8, 1, 2, 8, 2, 16, 4, 1, 4, 16, 2, 1, 8, 8, 2, 4, 4, 1, 8, 16, 2, 4, 2, 8, 6, 1, 16, 4, 16, 1, 4, 1, 4, 8

Offset: 1

Antti Karttunen, Dec 28 2019

nonn

a(64) = 6 is the first term which is not a power of 2.

Antti Karttunen, Table of n, a(n) for n = 1..10201
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A034386, A018819, A050376, A050377, A050378, A108951, A329901.

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
A018819(n) = if( n<1, n==0, if( n%2, A018819(n-1), A018819(n/2)+A018819(n-1))); \\ From A018819
A050377(n) = factorback(apply(e -> A018819(e), factor(n)[, 2]));
A330690(n) = A050377(A108951(n));

a(n) = A050377(A108951(n)).

a(n) = A050378(A329901(n)).

A331284 Number of values of k, 1 <= k <= n, with A329605(k) = A329605(n), where A329605 is the number of divisors of primorial inflation of n (A108951).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 14 2020

nonn

Ordinal transform of A329605, or equally, of A329606.

Cf. A000005, A108951, A329605, A329606, A331285 (positions of the first occurrences of each n, also positions of records).

Cf. also A067004.

c[n_] := c[n] = If[n == 1, 1, Module[{f = FactorInteger[n], p, e}, If[Length[f] > 1, Times @@ c /@ Power @@@ f, {{p, e}} = f; Times @@ (Prime[Range[PrimePi[p]]]^e)]]];
A329605[n_] := DivisorSigma[0, c[n]];
Module[{b}, b[_] = 0;
a[n_] := With[{t = A329605[n]}, b[t] = b[t] + 1]];
Array[a, 105] (* Jean-François Alcover, Jan 12 2022 *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A329605(n) = if(1==n,1,my(f=factor(n),e=1,m=1); forstep(i=#f~,1,-1, e += f[i,2]; m *= e^(primepi(f[i,1])-if(1==i,0,primepi(f[i-1,1])))); (m));
v331284 = ordinal_transform(vector(up_to, n, A329605(n)));
A331284(n) = v331284[n];

a(A331285(n)) = n for all n.

A347379 Möbius transform of A108951, the primorial inflation of n.

Original entry on oeis.org

1, 1, 5, 2, 29, 5, 209, 4, 30, 29, 2309, 10, 30029, 209, 145, 8, 510509, 30, 9699689, 58, 1045, 2309, 223092869, 20, 870, 30029, 180, 418, 6469693229, 145, 200560490129, 16, 11545, 510509, 6061, 60, 7420738134809, 9699689, 150145, 116, 304250263527209, 1045, 13082761331670029, 4618, 870, 223092869, 614889782588491409

Offset: 1

Antti Karttunen, Sep 01 2021

Multiplicative because A108951 is.

Index entries for sequences related to primorial numbers.

Cf. A000040, A002110, A008683, A034386, A108951.

prim[p_] := Product[Prime[i], {i, 1, PrimePi[p]}]; f[p_, e_] := (pr = prim[p])^e - pr^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Sep 16 2023 *)

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]) };
A347379(n) = sumdiv(n,d,moebius(n/d)*A108951(d));

a(n) = Sum_{d|n} A008683(n/d) * A108951(d).
a(A000040(n)) = A002110(n) - 1.
From Amiram Eldar, Sep 16 2023: (Start)
Multiplicative with a(p^e) = A034386(p)^e - A034386(p)^(e-1).
Sum_{n>=1} 1/a(n) = Product_{n>=1} (1 + A002110(n)/(A002110(n)-1)^2) = 3.8730356211898760903... . (End)

A354352 Sum of primorial inflation (A108951) and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 4, 0, 24, 0, 8, 36, 120, 0, 24, 0, 840, 360, 16, 0, 72, 0, 120, 2520, 9240, 0, 48, 900, 120120, 216, 840, 0, 0, 0, 32, 27720, 2042040, 12600, 144, 0, 38798760, 360360, 240, 0, 0, 0, 9240, 1080, 892371480, 0, 96, 44100, 1800, 6126120, 120120, 0, 432, 138600, 1680, 116396280, 25878772920, 0, 720, 0, 802241960520

Offset: 1

Antti Karttunen, Jun 05 2022

nonn

Cf. A001248, A002110, A061742, A108951, A354351.

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
memoA354351 = Map();
A354351(n) = if(1==n,1,my(v); if(mapisdefined(memoA354351,n,&v), v, v = -sumdiv(n,d,if(dA108951(n/d)*A354351(d),0)); mapput(memoA354351,n,v); (v)));
A354352(n) = (A354351(n)+A108951(n));

a(n) = A108951(n) + A354351(n).
a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A108951(d) * A354351(n/d).
For all n >= 1, a(A001248(n)) = A061742(n).

A373984 a(n) = 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, 0, 0, 0, 0, 4, 0, 2, 24, 28, 0, 14, 0, 208, 144, 8, 0, 58, 0, 86, 1044, 2308, 0, 36, 840, 30028, 198, 626, 0, 322, 0, 22, 11544, 510508, 6060, 128, 0, 9699688, 150144, 204, 0, 2302, 0, 6926, 1038, 223092868, 0, 82, 43680, 1738, 2552544, 90086, 0, 412, 66960, 1464, 48498444, 6469693228, 0, 680, 0, 200560490128

Offset: 1

Antti Karttunen, Jun 25 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4096
Index entries for sequences related to primorial numbers

Cf. A002110, A034386, A108951, A373158, A373985.

A373984(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]); (m-s); };

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)).

A329889 a(n) is the unique integer k such that A108951(k) = A260633(n).

Original entry on oeis.org

1, 3, 6, 12, 5, 10, 20, 15, 30, 60, 28, 45, 21, 56, 90, 42, 180, 84, 63, 168, 70, 126, 140, 252, 189, 280, 504, 210, 378, 264, 1008, 420, 315, 220, 840, 630, 1680, 792, 330, 1260, 1584, 945, 1400, 660, 2520, 495, 1890, 882, 1320, 2100, 990, 1764, 2640, 4200, 1980, 3528, 1485, 2200, 8400, 2646, 3960, 6300, 2970, 5292, 7920, 3300, 5940, 2772

Offset: 1

Antti Karttunen, Dec 28 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2011 (computed from the b-file of A260633)

Cf. A008480, A050382, A108951, A260633, A329900.

Cf. also A329902, A330686.

a(n) = A329900(A260633(n)).

A330685 Primorial deflation of highly factorable numbers: a(n) is the unique integer x such that A108951(x) = A033833(n).

Original entry on oeis.org

1, 4, 8, 6, 16, 12, 9, 24, 18, 48, 20, 36, 96, 27, 40, 72, 30, 54, 80, 144, 60, 160, 45, 288, 120, 90, 240, 180, 84, 480, 200, 360, 168, 960, 270, 400, 126, 720, 336, 540, 800, 252, 1440, 672, 280, 1080, 504, 1344, 378, 560, 1008, 420, 2688, 2400, 756, 1120, 2016, 840, 1512, 630, 4032, 1680, 3024, 1260, 2268, 3360, 6048, 2520, 4536, 6720

Offset: 1

Antti Karttunen, Dec 28 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..235 (computed from the b-file of A033833)

Cf. A001055, A028422, A033833, A108951, A329900.

a(n) = A329900(A033833(n)).

A331283 a(n) = gcd(n, A329605(n)), where A329605(n) gives the number of divisors of primorial inflation of n, A108951(n).

Original entry on oeis.org

1, 2, 1, 1, 1, 6, 1, 4, 9, 2, 1, 4, 1, 2, 3, 1, 1, 6, 1, 4, 3, 2, 1, 2, 1, 2, 1, 4, 1, 6, 1, 2, 3, 2, 1, 3, 1, 2, 3, 20, 1, 6, 1, 4, 1, 2, 1, 12, 1, 2, 3, 4, 1, 2, 1, 8, 3, 2, 1, 30, 1, 2, 1, 1, 1, 6, 1, 4, 3, 2, 1, 18, 1, 2, 3, 4, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 3, 8, 1, 10, 1, 4, 3, 2, 1, 2, 1, 2, 1, 5, 1, 6, 1, 8, 3

Offset: 1

Antti Karttunen, Jan 14 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..11025
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to primorial numbers

Cf. A000142, A002110, A034386, A108951, A329605.

Cf. also A009191, A329612.

A331283(n) = if(1==n,1,my(f=factor(n),e=1,m=1); forstep(i=#f~,1,-1, e += f[i,2]; m *= e^(primepi(f[i,1])-if(1==i,0,primepi(f[i-1,1])))); gcd(n,m));

a(n) = gcd(n, A329605(n)).

a(A002110(n)) = gcd(A002110(n), A000142(1+n)) = A034386(1+n), for n >= 0.

A337471 Primorial inflation of n prime shifted once: a(n) = A003961(A108951(n)).

Original entry on oeis.org

1, 3, 15, 9, 105, 45, 1155, 27, 225, 315, 15015, 135, 255255, 3465, 1575, 81, 4849845, 675, 111546435, 945, 17325, 45045, 3234846615, 405, 11025, 765765, 3375, 10395, 100280245065, 4725, 3710369067405, 243, 225225, 14549535, 121275, 2025, 152125131763605, 334639305, 3828825, 2835, 6541380665835015, 51975, 307444891294245705, 135135

Offset: 1

Antti Karttunen, Aug 28 2020

Row 1 of A337470.

Cf. A000265, A002110, A003961, A034386, A070826, A108951.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f);
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
A337471(n) = A003961(A108951(n));

a(n) = A003961(A108951(n)).
a(n) = A000265(A108951(A003961(n))).
Completely multiplicative with a(prime(i)) = A003961(A002110(i)) = A070826(1+i). - Antti Karttunen, Nov 17 2020
Sum_{n>=1} 1/a(n) = 1 / Product_{k>=2} (1 - 1/A070826(k)) = 1.6241170949... . - Amiram Eldar, Dec 08 2022

cp's OEIS Frontend

A330690 Number of ways to factor A108951(n) into "Fermi-Dirac primes" (A050376), where A108951 is fully multiplicative with a(prime(k)) = k-th primorial.

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A331284 Number of values of k, 1 <= k <= n, with A329605(k) = A329605(n), where A329605 is the number of divisors of primorial inflation of n (A108951).

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A347379 Möbius transform of A108951, the primorial inflation of n.

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A354352 Sum of primorial inflation (A108951) and its Dirichlet inverse.

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A373984 a(n) = 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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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).

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A329889 a(n) is the unique integer k such that A108951(k) = A260633(n).

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A330685 Primorial deflation of highly factorable numbers: a(n) is the unique integer x such that A108951(x) = A033833(n).

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A331283 a(n) = gcd(n, A329605(n)), where A329605(n) gives the number of divisors of primorial inflation of n, A108951(n).

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