Jahangeer Kholdi - cp's OEIS frontend

A251360 Numbers k such that k is the concatenation of prime factors of pi(k), in increasing order.

1117, 2163, 2537, 5137, 222926801

Offset: 1

Jahangeer Kholdi, Dec 01 2014

Numbers k such that k = A037276(A000720(k)).
Conjecture: numbers k such that k = A084317(A000720(k)). - Chai Wah Wu, Apr 04 2018
a(6) > 10^12 if exists. - Max Alekseyev, May 16 2025

			1117 is in the sequence since pi(1117) = 11*17,
2163 is in the sequence since pi(2163) = 2*163,
2537 is in the sequence since pi(2537) = 2*5*37,
and 5137 is in the sequence since pi(5137) = 5*137.

Chris Caldwell, G. L. Honaker and Lewis, 1117

Cf. A000040, A000720, A251361, A251362.

a251360[n_Integer] := Select[Range[n], # == FromDigits[Flatten@IntegerDigits[First@ Transpose@ FactorInteger[PrimePi[#]]]] &]; a251360[10^5] (* Michael De Vlieger, Dec 03 2014 *)

from sympy import prime, factorint
A251360_list, p = [], 3
for n in range(2,10**6):
    q, fn = prime(n+1), factorint(n)
    m = int(''.join(str(d)*fn[d] for d in sorted(fn)))
    if p <= m < q:
        A251360_list.append(m)
    p = q # Chai Wah Wu, Dec 10 2014, corrected Apr 04 2018

a(5) from Chai Wah Wu, Dec 10 2014

A251363 Numbers n such that n is the concatenation of distinct prime factors of phi(n), in decreasing order.

Original entry on oeis.org

237532, 832332, 82953292, 423238803752

Offset: 1

Jahangeer Kholdi, Dec 03 2014

Numbers n such that n = A085307(A000010(n)). - Michel Marcus, Dec 06 2014

			237532 is in the sequence since phi(237532)=23*7*5*3^2*2^4,
832332 is in the sequence since phi(832332)=83*23*3^2*2^4, and
82953292 is in the sequence since phi(82953292)=829*53*29*2^5.

Cf. A000010, A000040, A085307, A251360, A251361, A251362.

a251363[n_Integer] :=
Rest@ Select[Range[n], # == FromDigits[Flatten@ IntegerDigits[
Sort[First@ Transpose@ FactorInteger[EulerPhi[#]], Greater]]] &]; a251363[10^6] (* Michael De Vlieger, Dec 03 2014 *)

a(4) from Max Alekseyev, Feb 10 2025

A251362 Numbers n such that n is the concatenation of distinct prime factors of phi(n), in increasing order.

Original entry on oeis.org

25, 235741, 23517131, 274873357929, 2357131984859

Offset: 1

Jahangeer Kholdi, Dec 03 2014

Numbers n such that n = A084317(A000010(n)). - Michel Marcus, Dec 06 2014

			25 is in the sequence since phi(25)=2^2*5,
235741 is in the sequence since phi(235741)=2^4*3^2*5*7*41,
23517131 is in the sequence since phi(23517131)=2^7*3*5^2*17*131.

Cf. A000010, A000040, A084317, A251360, A251361, A251363, A329026.

a251362[n_Integer] := Rest@ Select[Range[n], # ==
FromDigits[Flatten@IntegerDigits[First@Transpose@FactorInteger[EulerPhi[#]]]] &]; a251362[10^6] (* Michael De Vlieger, Dec 03 2014 *)

a(4)-a(5) from Max Alekseyev, Feb 11 2025

A251361 Numbers k such that pi(k) is the concatenation of distinct prime factors of k, in increasing order.

Original entry on oeis.org

4, 100, 31509, 7560625

Offset: 1

Jahangeer Kholdi, Dec 02 2014

Next term must be greater than 4*10^8.

Numbers k such that A000720(k) = A084317(k). - Michel Marcus, Dec 06 2014

			4 is in the sequence since 4=2^2 and pi(4)=2,
100 is in the sequence since 100=2^2*5^2 and pi(100)=25,
31509 is in the sequence since 31509=3^4*389 and pi(31509)=3389, and
7560625 is in the sequence since 7560625=5^4*12097 and pi(7560625)=512097.

Cf. A000040, A000720, A084317, A251360.

a251361[n_Integer] := Select[Range[n], PrimePi[#] == FromDigits[
Flatten@ IntegerDigits[First@ Transpose@ FactorInteger[#]]] &]; a251361[10^6] (* Michael De Vlieger, Dec 03 2014 *)

is(n)=eval(fold((x,y)->Str(x,y),factor(n)[,1]))==primepi(n) \\ Charles R Greathouse IV, Dec 06 2014

Definition corrected by Max Alekseyev, Feb 12 2025

A248902 Numbers k such that A248891(k) = 2.

Original entry on oeis.org

6, 7, 8, 10, 19, 21, 23, 24, 28, 33, 72, 80, 270, 271, 323, 404, 2058, 4118, 5986, 7118, 13387, 16041, 20164, 30024, 30025, 76955, 151818, 622213, 1012549, 2482211, 2482212, 6330859, 9988608, 14984623, 105181797, 180589455

Offset: 1

Jahangeer Kholdi and Farideh Firoozbakht, Nov 22 2014

If k is in {6, 7, 23, 270, 30024, 2482211, ...} then both numbers k and k+1 are in the sequence. It seems that this set has more members.

Cf. A001359, A248891, A248901.

a(27)-a(36) from Jinyuan Wang, Nov 28 2020

A248894 Numbers k such that both numbers k and k+1 are in the sequence A248903.

Original entry on oeis.org

17, 69, 70, 241, 405, 2606, 3339, 4116, 7116, 15148, 15149, 20162, 137633, 324410, 332504, 439298, 1012547, 1121608, 2482209, 5028662, 6330857, 7180864, 7180865, 9569168, 14452770, 17021632, 110229972, 110229973, 193329301

Offset: 1

Jahangeer Kholdi and Farideh Firoozbakht, Nov 22 2014

If k is in the set {69, 15148, 7180864, 110229972, ...} then both numbers k and k+1 are in the sequence. This means that k, k+1 and k+2 are in the sequence A248903. It seems that there exist more such numbers k.

Cf. A001359, A248891, A248903.

a(13)-a(29) from Jinyuan Wang, Nov 28 2020

A248855 a(n) is the smallest positive integer m such that if k >= m then a(k+1,n)^(1/(k+1)) <= a(k,n)^(1/k), where a(k,n) is the k-th term of the sequence {p | p and p+2n are primes}.

Original entry on oeis.org

1, 1, 1, 1, 3556, 1, 34, 3, 4, 1, 2, 1, 11285, 5, 2, 124, 569, 1, 290, 3, 1, 165, 2, 1, 1, 2, 1, 316, 1, 2, 58957, 1, 3, 58617, 522, 2, 1, 1, 4, 1, 2, 1, 1, 2, 1, 7932, 4, 1, 5875, 1679, 4, 4, 3, 3, 1, 2, 307, 1, 1, 1, 1, 1, 4, 3206, 2, 1, 1, 3, 2, 1, 1, 1, 1, 5, 2, 11170, 1, 2, 4245, 1, 1, 81, 2, 1, 1, 2, 58, 1, 3, 4, 7303, 1, 1, 5, 1, 3, 3, 3, 383, 111408, 1

Offset: 0

Farideh Firoozbakht and Jahangeer Kholdi, Nov 25 2014

nonn

All terms conjecturally are found. Note that according to the definition a(k,0) is the k-th term of the sequence {p | p is prime} namely for every positive integer k, a(k,0) = prime(k). Hence if Firoozbakht's conjecture is true then a(0)=1.

			a(0)=a(1)=a(2)=a(3)=1 conjecturally states that the four sequences A000040, A001359, A023200 and A023201 have this property: For every positive integer n, b(n) exists and b(n+1) < b(n)^(1+1/n). Namely b(n)^(1/n) is a strictly decreasing function of n.
If in the definition instead of the sequence {p | p and p+2n are primes} we set {p | p is prime and nextprime(p)=p+2n} then it seems that except for n=3 all terms of the new sequence {c(n)} are equal to 1 and for n=3, c(3)=7746. Note that c(3)=7746 means that the sequence {p | p is prime and nextprime(p)=p+6} = A031924 has this property: For all k >= 7746, A031924(k+1)^(1/(k+1)) < A031924(k)^(1/k).

Wikipedia, Firoozbakht's conjecture

Cf. A000040, A001359, A023200, A023201, A023202, A023203, A031924.

A248901 Numbers k such that A248891(k) = 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 20, 29, 30, 2059, 5987, 7119, 20165, 151819, 14984624, 105181798

Offset: 1

Jahangeer Kholdi and Farideh Firoozbakht, Nov 22 2014

If k is in {1, 2, 3, 4, 29, ...} then both numbers k and k+1 are in the sequence.

Cf. A001359, A248891, A248902, A248903.

lista(nn) = {my(c=0, q=r=5, s=3, t); forprime(p=9, nn, if(p-2==q, if((t=s^(1+1/c++))>r && tJinyuan Wang, Nov 28 2020

a(13)-a(15) from Jinyuan Wang, Nov 28 2020

A248891 Number of primes p such that p+2 is prime and A001359(n) < p < A001359(n)^(1+1/n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 3, 2, 4, 3, 4, 4, 3, 4, 3, 3, 2, 1, 2, 3, 2, 2, 5, 4, 3, 2, 1, 1, 4, 3, 2, 3, 4, 6, 5, 4, 3, 4, 5, 6, 7, 6, 5, 4, 5, 5, 5, 4, 5, 4, 5, 5, 5, 6, 9, 8, 8, 7, 7, 7, 8, 7, 6, 6, 5, 4, 3, 3, 3, 2, 7, 6, 5, 5, 5, 4, 3, 2, 5, 5, 8, 9, 11, 10, 10, 9, 9, 8, 7, 7, 6, 6, 6, 5, 4, 5, 8, 8

Offset: 1

Jahangeer Kholdi and Farideh Firoozbakht, Nov 22 2014

nonn

Conjecture: For every positive integer n, A001359(n+1)^(1/(n+1)) < A001359(n)^(1/n). Note that this conjecture is equivalent to " A001359 is infinite and for every n, A001359(n+1) < A001359(n)^(1+1/n). This implies for every n, a(n) is positive. See comment lines of the sequence A001359.

			Take n=1, A001359(1)=3, 3 < 5 < 3^(1+1/1)=9 hence a(1)=1.
Take n=6, A001359(6)=41, 41 < 59 < 71 < 41^(1+1/6)~76.13 hence a(6)=2.

Cf. A001359, A248901, A248902, A248903.

A248903 Numbers k such that A248891(k) = 3.

Original entry on oeis.org

9, 12, 15, 17, 18, 22, 27, 32, 34, 39, 69, 70, 71, 79, 128, 143, 172, 226, 241, 242, 248, 269, 322, 325, 403, 405, 406, 420, 745, 2057, 2272, 2606, 2607, 3339, 3340, 3562, 4116, 4117, 4446, 5985, 6834, 7116, 7117, 7490, 7669, 13386, 13388, 15148, 15149, 15150

Offset: 1

Jahangeer Kholdi and Farideh Firoozbakht, Nov 22 2014

nonn

Sequence A248894 gives terms a(n) such that a(n+1)=a(n)+1.

Jinyuan Wang, Table of n, a(n) for n = 1..162

Cf. A001359, A248891, A248894, A248901, A248902.

lista(nn) = {my(c=q=0, r, v=List([3, 5, 11, 17])); forprime(p=19, nn, if(p-2==q, listput(v,q); if((t=v[c++]^(1+1/c))>v[c+3] && tJinyuan Wang, Nov 28 2020

cp's OEIS Frontend

User: Jahangeer Kholdi

A251360 Numbers k such that k is the concatenation of prime factors of pi(k), in increasing order.

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A251363 Numbers n such that n is the concatenation of distinct prime factors of phi(n), in decreasing order.

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A251362 Numbers n such that n is the concatenation of distinct prime factors of phi(n), in increasing order.

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A251361 Numbers k such that pi(k) is the concatenation of distinct prime factors of k, in increasing order.

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A248902 Numbers k such that A248891(k) = 2.

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A248894 Numbers k such that both numbers k and k+1 are in the sequence A248903.

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A248855 a(n) is the smallest positive integer m such that if k >= m then a(k+1,n)^(1/(k+1)) <= a(k,n)^(1/k), where a(k,n) is the k-th term of the sequence {p | p and p+2n are primes}.

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A248901 Numbers k such that A248891(k) = 1.

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A248891 Number of primes p such that p+2 is prime and A001359(n) < p < A001359(n)^(1+1/n).

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