A324886 -id:A324886 - cp's OEIS frontend

A344591 Numbers k such that the primorial inflation of k is a sum of distinct primorial numbers.

1, 2, 3, 5, 7, 8, 9, 11, 13, 17, 19, 23, 27, 29, 31, 32, 37, 40, 41, 42, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 115, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 228, 229, 233, 239, 241, 251, 252, 257, 263, 269, 271, 277, 281, 283, 293

Offset: 1

Antti Karttunen, May 26 2021

nonn

Numbers k such that A108951(k) is in A276156.

			A108951(40) = 240 and 240 is in A276156 because 240 = A002110(4) + A002110(3) = 210 + 30, therefore 40 is included in this sequence.

Positions of ones in A329344, in A344592 and in A344593.
Positions of squarefree terms in A324886.
Union of A008578 and A351959.
Cf. A002110, A108951, A276156, A351957 (characteristic function).
Cf. also A351958.

Name changed by Antti Karttunen, Apr 04 2022

A345941 a(n) = gcd(n, A329044(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5, 4, 17, 9, 19, 5, 7, 11, 23, 3, 25, 13, 3, 7, 29, 5, 31, 2, 11, 17, 7, 9, 37, 19, 13, 5, 41, 7, 43, 11, 5, 23, 47, 3, 7, 25, 17, 13, 53, 9, 11, 7, 19, 29, 59, 5, 61, 31, 7, 4, 13, 11, 67, 17, 23, 7, 71, 9, 73, 37, 25, 19, 11, 13, 79, 5, 3, 41, 83, 7, 17, 43, 29, 11, 89

Offset: 1

Antti Karttunen, Jul 03 2021

nonn

Only powers of primes (A000961) occur as terms. A346087 gives the exponents. - Antti Karttunen, Jul 07 2021

Cf. A000961, A006530, A020639, A071178, A108951, A324886, A329348, A329044, A345942, A345943, A346087, A346097.

A345941(n) = gcd(n,A329044(n)); \\ Rest of program given in A329044.

a(n) = gcd(n, A329044(n)).
a(n) = n / A345942(n).
a(n) = A329044(n) / A345943(n).
a(p) = p for all primes p.
From Antti Karttunen, Jul 07 2021: (Start)
a(n) = A006530(n)^A346087(n) = A006530(n)^min(A071178(n), A329348(n)).
a(n) = gcd(n, A346097(n)).
A006530(a(n)) = A020639(A329044(n)) = A006530(n).
(End)

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).

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).

A329618 a(n) = gcd(A001222(n), A324888(n)), where A324888(n) is the minimal number of primorials (A002110) that add to A108951(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 18 2019

nonn

Cf. A001222, A034386, A108951, A276086, A324886, A324888, A329619, A329620, A329621.

With[{b = Reverse@ Prime@ Range@ 120}, Array[GCD[PrimeOmega@ #1, Total@ IntegerDigits[#2, 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
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));
A329618(n) = gcd(bigomega(n), bigomega(A324886(n)));

a(n) = gcd(A001222(n), A324888(n)) = gcd(A001222(n), A001222(A324886(n))).

A331292 The next more significant digit after A329348(n) in the primorial base expansion of A108951(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 1, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 4, 0, 0, 0, 1, 0, 1, 0, 0, 5, 0, 0, 3, 6, 8, 0, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 3, 0, 1, 0, 0, 0, 0, 5, 0, 2, 0, 0, 3, 0, 16, 0, 0, 2, 1, 0, 0, 2, 1, 0, 0, 0, 0, 10, 12, 0, 0, 0, 1, 6, 0, 12, 2, 6, 0, 0, 0, 0, 3

Offset: 1

Antti Karttunen, Jan 17 2020

nonn

Cf. A007949, A061395, A108951, A117366, A151800, A246277, A324886, A329348, A331188,

A331293.

A331292(n) = { my(h=A061395(n), u = A108951(n) - (A002110(h)*((A331188(n)%prime(1+h))))); ((u/A002110(1+h))%prime(2+h)); };

a(n) = A007949(A246277(A324886(n))).

a(n) = A331293(n) modulo A000040(2+A061395(n)).

A344593 Lexicographically earliest infinite sequence such that a(i) = a(j) => A344592(i) = A344592(j), for all i, j >= 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 1, 1, 4, 1, 5, 1, 6, 7, 8, 1, 9, 1, 10, 11, 12, 1, 13, 14, 15, 1, 16, 1, 17, 1, 1, 18, 19, 20, 21, 1, 22, 23, 1, 1, 1, 1, 24, 25, 26, 1, 27, 18, 28, 29, 30, 1, 31, 32, 33, 34, 35, 1, 36, 1, 37, 38, 39, 40, 41, 1, 42, 43, 44, 1, 45, 1, 46, 47, 48, 49, 50, 1, 51, 11, 52, 1, 53, 54, 55, 56, 57, 1, 58, 59, 60, 61, 62, 63, 64, 1, 65, 66, 11, 1

Offset: 1

Antti Karttunen, May 26 2021

nonn

Restricted growth sequence transform of A344592, where A344592(n) = A003557(A276086(A108951(n))).

For all i, j: a(i) = a(j) => A329344(i) = A329344(j).

			Both a(14) = 6 and a(32768) = 6, because A344592(14) = 11 is the sixth distinct value occurring in A344592, and A344592(32768) = A003557(A276086(A108951(32768))) = A003557(A276086(32768)) = A003557(401115) = A003557(3 * 5 * 11^2 * 13 * 17) = 11 also, which is the second time 11 occurs in A344592.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003557, A108951, A276086, A324886, A329344, A344591 (positions of ones), A344592.

Cf. also A329045, A329345, A344594.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
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
A328572(n) = { my(m=1, p=2); while(n, if(n%p, m *= p^((n%p)-1)); n = n\p; p = nextprime(1+p)); (m); };
A344592(n) = A328572(A108951(n));
v344593 = rgs_transform(vector(up_to, n, A344592(n)));
A344593(n) = v344593[n];

A346099 a(n) = gcd(n, A346098(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 1, 1, 5, 11, 3, 13, 7, 5, 1, 17, 1, 19, 5, 7, 11, 23, 3, 1, 13, 3, 7, 29, 5, 31, 2, 11, 17, 7, 1, 37, 19, 13, 1, 41, 1, 43, 11, 1, 23, 47, 1, 1, 1, 17, 13, 53, 9, 11, 7, 19, 29, 59, 1, 61, 31, 1, 4, 13, 11, 67, 17, 23, 1, 71, 3, 73, 37, 25, 19, 1, 13, 79, 1, 1, 41, 83, 1, 17, 43, 29, 11, 89

Offset: 1

Antti Karttunen, Jul 07 2021

nonn

Only powers of primes (A000961) occur as terms. A346100 lists the exponents.

Cf. A000961, A064989, A319626, A324886, A346095, A346096, A346097, A346098, A346100.

Cf. A346090 (positions of ones).

A064989(n) = { my(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)));
A346099(n) = gcd(n, A064989(A319626(A324886(n)))); \\ Rest of program in A324886.

a(n) = gcd(n, A346098(n)) = gcd(n, A064989(A319626(A324886(n)))).

cp's OEIS Frontend

A344591 Numbers k such that the primorial inflation of k is a sum of distinct primorial numbers.

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A345941 a(n) = gcd(n, A329044(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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A329349 Number of occurrences of the largest primorial present in the greedy sum of primorials adding to A108951(n).

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A329618 a(n) = gcd(A001222(n), A324888(n)), where A324888(n) is the minimal number of primorials (A002110) that add to A108951(n).

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A331292 The next more significant digit after A329348(n) in the primorial base expansion of A108951(n).

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