A324886 -id:A324886 - cp's OEIS frontend

A329345 Lexicographically earliest infinite sequence such that a(i) = a(j) => A246277(A329044(i)) = A246277(A329044(j)) for all i, j.

1, 2, 2, 3, 2, 3, 2, 4, 4, 3, 2, 5, 2, 3, 6, 7, 2, 7, 2, 5, 6, 3, 2, 8, 9, 3, 10, 5, 2, 11, 2, 10, 6, 3, 12, 13, 2, 3, 6, 4, 2, 4, 2, 5, 14, 3, 2, 15, 16, 17, 6, 5, 2, 18, 19, 20, 6, 3, 2, 21, 2, 3, 21, 18, 22, 23, 2, 5, 6, 24, 2, 25, 2, 3, 26, 5, 27, 23, 2, 7, 28, 3, 2, 7, 29, 3, 6, 20, 2, 30, 31, 5, 6, 3, 32, 33, 2, 34, 35, 16, 2, 23, 2, 20, 36

Offset: 1

Antti Karttunen, Nov 11 2019

nonn

Restricted growth sequence transform of function f(n) = A246277(A329044(n)).
For all i, j:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A329045(i) = A329045(j),
  a(i) = a(j) => A329343(i) = A329343(j),
  a(i) = a(j) => A329348(i) = A329348(j),
  a(i) = a(j) => A329349(i) = A329349(j).

Cf. A034386, A064989, A108951, A246277, A276086, A305800, A324886, A329044, A329045, A329343, A329348, A329349.

Cf. also A329048.

up_to = 1024;
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
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));
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)};
A329044(n) = A064989(A324886(n));
A246277(n) = if(1==n, 0, my(f = factor(n), k = primepi(f[1,1])-1); for (i=1, #f~, f[i,1] = prime(primepi(f[i,1])-k)); factorback(f)/2);
v329345 = rgs_transform(vector(up_to, n, A246277(A329044(n))));
A329345(n) = v329345[n];

A344592 a(n) = A003557(A276086(A108951(n))).

Original entry on oeis.org

1, 1, 1, 3, 1, 5, 1, 1, 1, 7, 1, 125, 1, 11, 16807, 15, 1, 35, 1, 343, 161051, 13, 1, 25, 9317, 17, 1, 1331, 1, 2401, 1, 1, 371293, 19, 253333223, 42875, 1, 23, 1419857, 1, 1, 1, 1, 2197, 14641, 29, 1, 49, 371293, 6684099653, 2476099, 4913, 1, 55, 37349, 19487171, 6436343, 31, 1, 5929, 1, 37, 20449, 21, 582622237229761, 1792160394037

Offset: 1

Antti Karttunen, May 26 2021

nonn

Cf. A003557, A085731, A108951, A276086, A324886, A328572, A329047, A342002, A342920, A346091, A346092 [= A276085(a(n))].

Cf. A344591 (positions of ones), A344593 (rgs-transform).

Block[{b = MixedRadix[Reverse@ Prime@ Range@ 20]}, Array[#/(Times @@ FactorInteger[#][[All, 1]]) &@ Apply[Times, Power @@@ #] &@ Transpose@ {Prime@ Range@ Length@ #, Reverse@ #} &@ IntegerDigits[#, b] &@ Apply[Times, Map[(Times @@ Prime@ Range@ PrimePi@ #1)^#2 & @@ # &, FactorInteger[#]]] &, 66]] (* Michael De Vlieger, Jul 14 2021 *)

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));

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A344592(n) = A003557(A276086(A108951(n)));

a(n) = A328572(A108951(n)) = A003557(A324886(n)) = A003557(A276086(A108951(n))).
a(n) = A329047(n) / A342920(n).
a(n) = A085731(A324886(n)) = gcd(A324886(n), A329047(n)) = A324886(n) / A346091(n). - Antti Karttunen, Jul 09 2021

A329343 Difference between the indices of the smallest and the largest primorial in the greedy sum of primorials adding to A108951(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 1, 0, 2, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 2, 1, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 1, 0, 0, 1, 2, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 2, 1, 2, 0, 0, 0, 0, 2

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.

Positions of the records (and conjecturally, the positions of the first occurrences of each n) begin as 1, 8, 27, 162, 289, 529, 841, 1369, 1681, 2209, 2809, 3481, 4489, 5041, 5329, 6889, ..., that after 162 all seem to be squares of certain primes. See also A329051.

			For n = 18 = 2 * 3^2, A108951(18) = A034386(2) * A034386(3)^2 = 2 * 6^2 = 72 = 30 + 30 + 6 + 6, and as the largest primorial in the sum is 30 = A002110(3), and the least primorial is 6 = A002110(2), we have a(18) = 3-2 = 1.

Cf. A002110, A034386, A108951, A243055, A276086, A324886, A324888, A329040, A329051.

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));
A243055(n) = if(1==n,0,my(f = factor(n), lpf = f[1, 1], gpf = f[#f~, 1]); (primepi(gpf)-primepi(lpf)));
A329343(n) = A243055(A324886(n));

a(n) = A243055(A324886(n)).

A329344 Number of times most frequent primorial is 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, 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)).

A342920 a(n) = A342002(A108951(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 8, 12, 2, 1, 4, 1, 2, 6, 16, 1, 24, 1, 4, 6, 2, 1, 26, 50, 2, 16, 4, 1, 62, 1, 10, 6, 2, 126, 48, 1, 2, 6, 18, 1, 24, 1, 4, 46, 2, 1, 22, 1486, 100, 6, 4, 1, 32, 94, 8, 6, 2, 1, 54, 1, 2, 72, 20, 264, 12, 1, 4, 6, 120, 1, 376, 1, 2, 1142, 4, 242, 12, 1, 36, 342, 2, 1, 48, 272, 2, 6, 8, 1, 92, 318

Offset: 1

Antti Karttunen, Apr 06 2021

nonn

Cf. A108951, A324886, A327860, A328572, A342002, A342463.

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]) };
A342920(n) = A342002(A108951(n)); \\ Needs also code from A342002.

a(n) = A342002(A108951(n)) = A327860(A108951(n)) / A328572(A108951(n)).

A342456 A276086 applied to the primorial inflation of Doudna-tree, where A276086(n) is the prime product form of primorial base expansion of n.

Original entry on oeis.org

2, 3, 5, 9, 7, 25, 35, 15, 11, 49, 117649, 625, 717409, 1225, 55, 225, 13, 121, 1771561, 2401, 36226650889, 184877, 1127357, 875, 902613283, 514675673281, 3780549773, 1500625, 83852850675321384784127, 3025, 62004635, 21, 17, 169, 4826809, 14641, 8254129, 143, 2924207, 77, 8223741426987700773289, 59797108943, 546826709

Offset: 0

Antti Karttunen and Michael De Vlieger, Mar 14 2021

nonn

This sequence (which could be viewed as a binary tree, like the underlying A005940 and A329886) is similar to A324289, but unlike its underlying tree A283477 that generates only numbers that are products of distinct primorial numbers (i.e., terms of A129912), here the underlying tree A329886 generates all possible products of primorial numbers, i.e., terms of A025487, but in different order.

Cf. A005940, A025487, A108951, A129912, A276086, A283980, A324886, A342457 [= 2*A246277(a(n))], A342461 [= A001221(a(n))], A342462 [= A001222(a(n))], A342463 [= A342001(a(n))], A342464 [= A051903(a(n))].

Cf. A324289 (a subset of these terms, in different order).

Block[{a, f, r = MixedRadix[Reverse@ Prime@ Range@ 24]}, f[n_] :=
Times @@ MapIndexed[Prime[First[#2]]^#1 &, Reverse@ IntegerDigits[n, r]]; a[0] = 1; a[1] = 2; a[n_] := a[n] = If[EvenQ@ n, (Times @@ Map[Prime[PrimePi@ #1 + 1]^#2 & @@ # &, FactorInteger[#]] - Boole[# == 1])*2^IntegerExponent[#, 2] &[a[n/2]], 2 a[(n - 1)/2]]; Array[f@ a[#] &, 43, 0]] (* Michael De Vlieger, Mar 17 2021 *)

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
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)));
A342456(n) = A276086(A329886(n));

a(n) = A276086(A329886(n)) = A324886(A005940(1+n)).
For all n >= 0, gcd(a(n), A329886(n)) = 1.
For all n >= 1, A055396(a(n))-1 = A061395(A329886(n)) = A290251(n) = 1+A080791(n).
For all n >= 0, a(2^n) = A000040(2+n).

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

Original entry on oeis.org

2, 3, 5, 3, 7, 5, 11, 15, 35, 7, 13, 5, 17, 11, 7, 15, 19, 35, 23, 7, 11, 13, 29, 35, 77, 17, 55, 11, 31, 77, 37, 21, 13, 19, 143, 35, 41, 23, 17, 77, 43, 143, 47, 13, 77, 29, 53, 35, 2431, 77, 19, 17, 59, 55, 221, 11, 23, 31, 61, 77, 67, 37, 143, 21, 323, 13, 71, 19, 29, 143, 73, 385, 79, 41, 1001, 23, 221, 17, 83, 77, 385

Offset: 1

Antti Karttunen, Jul 09 2021

nonn

All terms are squarefree (in A005117). - Antti Karttunen, Apr 03 2022

Cf. A000040, A002110, A005117, A007947, A108951, A276086, A324886, A328571, A344592, A346093 [= A276085(a(n))], A351955 (rgs-transform).

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]) };
A328571(n) = { my(m=1, p=2); while(n, m *= (p^!!(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A346091(n) = A328571(A108951(n));

a(n) = A328571(A108951(n)) = A007947(A276086(A108951(n))).
a(n) = A324886(n) / A344592(n).
For all n >= 1, a(A000040(n)) = A000040(1+n).

A346108 a(n) = A276085(A108951(A346096(n))), where A346096(n) gives the numerator of the primorial deflation of A276086(A108951(n)).

Original entry on oeis.org

1, 3, 9, 6, 39, 18, 249, 9, 39, 78, 2559, 36, 32589, 498, 234, 18, 543099, 78, 10242789, 156, 1494, 5118, 233335659, 57, 996, 65178, 258, 996, 6703028889, 405, 207263519019, 42, 15354, 1086198, 6612, 156, 7628001653829, 20485578, 195534, 249, 311878265181039, 2559, 13394639596851069, 10236, 1245, 466671318, 628284422185342479

Offset: 1

Antti Karttunen, Jul 08 2021

nonn

Cf. A002110, A064989, A108951, A276085, A276086, A324886, A319626, A346096, A346105, 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)};
A319626(n) = (n / gcd(n, A064989(n)));
A346096(n) = A319626(A324886(n));
A346106(n) = A108951(A346096(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)); };
A346108(n) = A276085(A346106(n)); \\ Rest of program given in A324886.

a(n) = A346105(A346096(n)) = A276085(A346106(n)) = A276085(A108951(A346096(n))).

a(n) = A108951(n) + A346109(n).

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

Original entry on oeis.org

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).

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.

cp's OEIS Frontend

A329345 Lexicographically earliest infinite sequence such that a(i) = a(j) => A246277(A329044(i)) = A246277(A329044(j)) for all i, j.

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A344592 a(n) = A003557(A276086(A108951(n))).

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A329343 Difference between the indices of the smallest and the largest primorial in the greedy sum of primorials adding to A108951(n).

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

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A342920 a(n) = A342002(A108951(n)).

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A342456 A276086 applied to the primorial inflation of Doudna-tree, where A276086(n) is the prime product form of primorial base expansion of n.

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

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A346108 a(n) = A276085(A108951(A346096(n))), where A346096(n) gives the numerator of the primorial deflation of A276086(A108951(n)).

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