A080696 -id:A080696 - cp's OEIS frontend

A090886 Primes in the products of consecutive prime-indexed primes + 2 or A080696(n) + 2.

5, 17, 167, 14093057717, 16192486429745837, 114681479899746991802547357477494807, 12994174855450524638613469509299054674461271855442034674644855689971858721497

Offset: 1

Cino Hilliard, Feb 12 2004

nonn

These numbers, except for the first one, always end in 7.

The next term (a(8)) has 129 digits. - Harvey P. Dale, Apr 05 2019

Amiram Eldar, Table of n, a(n) for n = 1..12

Cf. A080696, A087400.

Select[FoldList[Times,Table[Prime[n],{n,Prime[Range[40]]}]]+2,PrimeQ] (* Harvey P. Dale, Apr 05 2019 *)

piptorial2(n) = { y=1; for(x=1,n, v=prime(prime(x)); y*=v; if(isprime(y+2),print1(y+2",")) ) }

A087400 Primes p such that p+2 is a piptorial number. Also numbers such that A080696(n)- 2 is prime.

Original entry on oeis.org

13, 163, 2803, 3565153, 210344143, 86915972211813115391953, 4419764102942908730796303703, 114681479899746991802547357477494803

Offset: 1

Cino Hilliard, Oct 21 2003

nonn

Piptorial numbers are the partial products of prime-indexed primes.
Sum of reciprocals = 0.08341509210884323904648676616...
a(9) = A080696(1111) - 2 = 1.0954...*10^4885. - Amiram Eldar, Jul 05 2024

			(Product of first four pips) - 2 = 3*5*11*17 - 2 = 2805 - 2 = 2803, which is prime, so 2803 is in the sequence.

Cf. A080696, A090886.

seq[kmax_] := Module[{r = 1, p = 1, s = {}}, Do[p = NextPrime[p]; r *= Prime[p]; If[PrimeQ[r - 2], AppendTo[s, r - 2]], {k, 1, kmax}]; s]; seq[20] (* Amiram Eldar, Jul 05 2024 *)

piptorial(n) = { s=0; p=1; for(x=1,n, p=p*prime(prime(x)); if(isprime(p-2),print1(p-2","); s+=1.0/(p-2)) ); print(); print(s) }

A327406 Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum divisor that is 1 or whose prime indices have a common divisor > 1 (A327405, A327656).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 21 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. Numbers whose prime indices have a common divisor > 1 are listed in A318978.

Note that A318978 includes also all odd primes and their powers, thus the only numbers for which a maximum such divisor is 1 are the powers of 2. Therefore A000079 gives the indices of zeros in this sequence. - Antti Karttunen, Dec 06 2021

			We have 5115 -> 165 -> 15 -> 3 -> 1, so a(5115) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Gus Wiseman, Sequences counting and encoding certain classes of multisets
Index entries for sequences computed from indices in prime factorization

First appearance of n is A080696(n).
See link for additional cross-references.
Cf. A000005, A000079 (positions of 0's), A056239, A112798, A281116, A289509, A302569, A318978.

Table[Length[FixedPointList[#/Max[Select[Divisors[#],GCD@@PrimePi/@First/@FactorInteger[#]!=1&]]&,n]]-2,{n,100}]

A327405(n) = (n / vecmax(select(d -> (1==d)||(gcd(apply(primepi,factor(d)[, 1]~))>1), divisors(n))));
A327406(n) = { my(u = A327405(n), k=0); while(u!=n, k++; n = u; u = A327405(n)); (k); }; \\ Antti Karttunen, Dec 06 2021

Data section extended up to 105 terms by Antti Karttunen, Dec 06 2021

A304117 If n = Product (p_j^k_j) then a(n) = Product (pi(p_j)*k_j), where pi() = A000720.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 3, 4, 3, 5, 4, 6, 4, 6, 4, 7, 4, 8, 6, 8, 5, 9, 6, 6, 6, 6, 8, 10, 6, 11, 5, 10, 7, 12, 8, 12, 8, 12, 9, 13, 8, 14, 10, 12, 9, 15, 8, 8, 6, 14, 12, 16, 6, 15, 12, 16, 10, 17, 12, 18, 11, 16, 6, 18, 10, 19, 14, 18, 12, 20, 12, 21, 12, 12, 16, 20, 12, 22, 12

Offset: 1

Ilya Gutkovskiy, May 06 2018

			a(36) = 8 because 36 = 2^2*3^2 = prime(1)^2*prime(2)^2 and 1*2*2*2 = 8.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Ilya Gutkovskiy, Extended graphical example
Index entries for sequences computed from indices in prime factorization
Index entries for sequences computed from exponents in factorization of n

Cf. A000026, A000040, A000720, A001414, A002110, A003963, A005361, A056239, A156061, A304037.

a[n_] := Times @@ (PrimePi[#[[1]]] #[[2]] & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 1, 80}]

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = primepi(f[k,1])*f[k,2]; f[k, 2] = 1); factorback(f); \\ Michel Marcus, May 06 2018

a(n) = A005361(n)*A156061(n).
a(p^k) = A000720(p)*k where p is a prime.
a(A002110(m)^k) = k^m*m!.
As an example:
a(A000040(k)) = k.
a(A006450(k)) = A000040(k).
a(A001248(k)) = a(A031215(k)) = A005843(k).
a(A030078(k)) = a(A031336(k)) = A008585(k)
a(A061742(k)) = A000165(k).
a(A115964(k)) = A032031(k).
a(A002110(k)) = A000142(k).
a(A080696(k)) = A002110(k).

A304037 If n = Product (p_j^k_j) then a(n) = Sum (pi(p_j)^k_j), where pi() = A000720.

Original entry on oeis.org

0, 1, 2, 1, 3, 3, 4, 1, 4, 4, 5, 3, 6, 5, 5, 1, 7, 5, 8, 4, 6, 6, 9, 3, 9, 7, 8, 5, 10, 6, 11, 1, 7, 8, 7, 5, 12, 9, 8, 4, 13, 7, 14, 6, 7, 10, 15, 3, 16, 10, 9, 7, 16, 9, 8, 5, 10, 11, 17, 6, 18, 12, 8, 1, 9, 8, 19, 8, 11, 8, 20, 5, 21, 13, 11, 9, 9, 9, 22, 4, 16, 14, 23, 7, 10, 15, 12, 6

Offset: 1

Ilya Gutkovskiy, May 05 2018

nonn

			a(72) = 5 because 72 = 2^3*3^2 = prime(1)^3*prime(2)^2 and 1^3 + 2^2 = 5.

Ilya Gutkovskiy, Extended graphical example
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A000720, A002110, A003963, A008475, A056239, A066328, A156061, A222416.

a[n_] := Plus @@ (PrimePi[#[[1]]]^#[[2]]& /@ FactorInteger[n]); a[1] = 0; Table[a[n], {n, 1, 88}]

If gcd(u,v) = 1 then a(u*v) = a(u) + a(v).
a(p^k) = A000720(p)^k where p is a prime.
a(A002110(m)^k) = 1^k + 2^k + ... + m^k.
As an example:
a(A000040(k)) = k.
a(A006450(k)) = A000040(k).
a(A038580(k)) = A006450(k).
a(A001248(k)) = a(A011757(k)) = A000290(k).
a(A030078(k)) = a(A055875(k)) = A000578(k).
a(A002110(k)) = a(A011756(k)) = A000217(k).
a(A061742(k)) = A000330(k).
a(A115964(k)) = A000537(k).
a(A080696(k)) = A007504(k).
a(A076954(k)) = A001923(k).

A328512 Number of distinct connected components of the multiset of multisets with MM-number n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 3, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2

Offset: 1

Gus Wiseman, Oct 20 2019

nonn

For n > 1, the first appearance of n is 2 * A080696(n - 1).

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The multiset of multisets with MM-number 1508 is {{},{},{1,2},{1,3}}, which has the 3 connected components {{}}, {{}}, and {{1,2},{1,3}}, two of which are distinct, so a(1508) = 2.
The multiset of multisets with MM-number 12818 is {{},{1,2},{4},{1,3}}, which has the 3 connected components {{}}, {{1,2},{1,3}}, and {{4}}, so a(12818) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to prime indices in the factorization of n.

Positions of 0's and 1's are A305078 together with all powers of 2.
Connected numbers are A305078.
Connected components are A305079.
Cf. A007814, A056239, A112798, A286518, A302242, A304714, A304716, A322389, A327076, A328513.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Table[Length[Union[zsm[primeMS[n]]]],{n,100}]

zero_first_elem_and_connected_elems(ys) = { my(cs = List([ys[1]]), i=1); ys[1] = 0; while(i<=#cs, for(j=2, #ys, if(ys[j]&&(1!=gcd(cs[i], ys[j])), listput(cs, ys[j]); ys[j] = 0)); i++); (ys); };
A007814(n) = valuation(n, 2);
A000265(n) = (n/2^A007814(n));
A328512(n) = if(!(n%2), 1+A328512(A000265(n)), my(cs = apply(p -> primepi(p), factor(n)[, 1]~), s=0); while(#cs, cs = select(c -> c, zero_first_elem_and_connected_elems(cs)); s++); (s)); \\ Antti Karttunen, Jan 28 2025

If n is even, a(n) = A305079(n) - A007814(n) + 1; otherwise, a(n) = A305079(n).

Data section extended to a(105) by Antti Karttunen, Jan 28 2025

cp's OEIS Frontend

A090886 Primes in the products of consecutive prime-indexed primes + 2 or A080696(n) + 2.

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A087400 Primes p such that p+2 is a piptorial number. Also numbers such that A080696(n)- 2 is prime.

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A327406 Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum divisor that is 1 or whose prime indices have a common divisor > 1 (A327405, A327656).

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A304117 If n = Product (p_j^k_j) then a(n) = Product (pi(p_j)*k_j), where pi() = A000720.

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A304037 If n = Product (p_j^k_j) then a(n) = Sum (pi(p_j)^k_j), where pi() = A000720.

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A328512 Number of distinct connected components of the multiset of multisets with MM-number n.

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