A000720 -id:A000720 - cp's OEIS frontend

A291539 a(n) = PrimePi(n^3) - PrimePi(n) * PrimePi(n^2), where PrimePi = A000720.

0, 2, 1, 6, 3, 14, 8, 25, 41, 68, 67, 99, 93, 136, 188, 240, 229, 303, 306, 383, 467, 562, 566, 688, 795, 922, 1066, 1227, 1247, 1421, 1446, 1620, 1826, 2036, 2283, 2511, 2566, 2843, 3115, 3401, 3431, 3746, 3827, 4163, 4526, 4895, 4981, 5369, 5743, 6229, 6712, 7165, 7202, 7743, 8258, 8835, 9453, 9999, 10132, 10736

Offset: 1

Jonathan Sondow, Aug 25 2017

nonn

All terms are positive except a(1) = 0, by the PNT with error term for large n and computation for smaller n. In particular, PrimePi(n^3) > PrimePi(n) * PrimePi(n)^2 for n > 1.
For PrimePi(n) * PrimePi(n^2) - PrimePi(n)^3, see A291540.
For PrimePi(n^3) - PrimePi(n)^3, see A291538.
For prime(n) * prime(n^2) - prime(n^3), see A291541.

			a(2) = PrimePi(2^3) - PrimePi(2) * PrimePi(2^2) = 4 - 1 * 2 = 2.

Cf. A000720, A123914, A262199, A291440, A291538, A291540, A291541, A291542.

Table[ PrimePi[n^3] - PrimePi[n]*PrimePi[n^2], {n, 60}]

a(n) = primepi(n^3) - primepi(n) * primepi(n^2); \\ Michel Marcus, Sep 10 2017

a(n) = A000720(n^3) - A000720(n) * A000720(n)^2.
a(n) = A291538(n) - A291540(n).
a(n) ~ (n^3 / log(n))*(1/3  - 1/(2*log(n)^2)) as n tends to infinity.

A291540 a(n) = PrimePi(n) * PrimePi(n^2) - PrimePi(n)^3, where PrimePi = A000720.

Original entry on oeis.org

0, 1, 0, 4, 0, 6, -4, 8, 24, 36, 25, 45, 18, 48, 72, 108, 84, 119, 64, 112, 168, 224, 162, 216, 297, 369, 432, 504, 460, 540, 451, 561, 660, 770, 869, 979, 900, 1008, 1152, 1284, 1222, 1365, 1218, 1386, 1540, 1722, 1560, 1755, 1980, 2130, 2295, 2520, 2448, 2640, 2848, 3024, 3216, 3488, 3366, 3638, 3510, 3744, 4050, 4320, 4572

Offset: 1

Jonathan Sondow, Aug 25 2017

sign

All terms are positive except a(1) = a(3) = a(5) = 0 and a(7) = -4, by the PNT with error term for large n and computation for smaller n. In particular, PrimePi(n) * PrimePi(n^2) > PrimePi(n)^3, for n > 7.
For PrimePi(n^3) - PrimePi(n) * PrimePi(n^2), see A291539.
For PrimePi(n^3) - PrimePi(n)^3, see A291538.
For prime(n)^3 - prime(n) * prime(n^2), see A291542.

			a(7) = PrimePi(7) * PrimePi(7^2) - PrimePi(7)^3 = 4 * 15 - 4^3 = -4.

Cf. A000720, A123914, A262199, A291440, A291538, A291539, A291541, A291542.

Table[ PrimePi[n] * PrimePi[n^2] - PrimePi[n]^3, {n, 65}]

a(n) = primepi(n) * primepi(n^2) - primepi(n)^3; \\ Michel Marcus, Sep 10 2017

A291539(n) + a(n) = A291538(n).

A293447 Fully additive with a(p^e) = e * A000225(PrimePi(p)), where PrimePi(n) = A000720(n) and A000225(n) = (2^n)-1.

Original entry on oeis.org

0, 1, 3, 2, 7, 4, 15, 3, 6, 8, 31, 5, 63, 16, 10, 4, 127, 7, 255, 9, 18, 32, 511, 6, 14, 64, 9, 17, 1023, 11, 2047, 5, 34, 128, 22, 8, 4095, 256, 66, 10, 8191, 19, 16383, 33, 13, 512, 32767, 7, 30, 15, 130, 65, 65535, 10, 38, 18, 258, 1024, 131071, 12, 262143, 2048, 21, 6, 70, 35, 524287, 129, 514, 23, 1048575, 9, 2097151, 4096, 17, 257, 46, 67, 4194303, 11, 12

Offset: 1

Antti Karttunen, Nov 09 2017

nonn

Original, equal definition: totally additive with a(p^e) = e * A005187(2^(PrimePi(p)-1)), where PrimePi(n) = A000720(n).

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences computed from indices in prime factorization

Cf. A000225, A000720, A003557, A005187, A087207.

Cf. also A046645, A048675, A293442, A331591, A331740.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; }; \\ This function from Charles R Greathouse IV
A293447(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2] * A005187(2^(primepi(f[k, 1])-1))); }

(define (A293447 n) (cond ((= 1 n) 0) (else (+ (A005187 (A087207 n)) (A293447 (A003557 n))))))
;; Alternatively:
(define (A293447 n) (if (= 1 n) 0 (+ (A005187 (A000079 (+ -1 (A061395 n)))) (A293447 (/ n (A006530 n))))))

Totally additive with a(p^e) = e * A005187(2^(PrimePi(p)-1)), where PrimePi(n) = A000720(n).
a(1) = 0, and for n > 1, a(n) = A005187(A087207(n)) + a(A003557(n)).
Other identities:
For all n >= 1, a(A293442(n)) = A046645(n).
For all n >= 2 and all k >= 0, a(n^k) = k*a(n).
For all n >= 1, a(n) >= A048675(n) >= A331740(n) >= A331591(n).

Definition simplified by Antti Karttunen, Feb 05 2020

A332212 Fully multiplicative with a(p) = A332211(A000720(p)).

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 5, 8, 9, 14, 31, 12, 11, 10, 21, 16, 127, 18, 13, 28, 15, 62, 17, 24, 49, 22, 27, 20, 19, 42, 23, 32, 93, 254, 35, 36, 29, 26, 33, 56, 8191, 30, 37, 124, 63, 34, 41, 48, 25, 98, 381, 44, 43, 54, 217, 40, 39, 38, 131071, 84, 47, 46, 45, 64, 77, 186, 524287, 508, 51, 70, 53, 72, 59, 58, 147, 52, 155, 66, 61, 112, 81, 16382, 67, 60

Offset: 1

Antti Karttunen, Feb 09 2020

Antti Karttunen, Table of n, a(n) for n = 1..29662
Index entries for sequences that are permutations of the natural numbers

Cf. A000043, A000668, A000720, A332211, A332213 (inverse permutation), A332214.

\\ Needs also code from A332211:
A332212(n) = { my(f=factor(n)); f[,1] = apply(A332211,apply(primepi,f[,1])); factorback(f); };

a(1) = 1, a(p^e) = A332211(A000720(p))^e, a(m*n) = a(m)*a(n).

A333365 T(n,k) is the number of times that prime(k) is the least part in a partition of n into prime parts; triangle T(n,k), n >= 0, 1 <= k <= max(1,A000720(A331634(n))), read by rows.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 2, 0, 0, 1, 2, 1, 3, 1, 3, 1, 1, 4, 1, 0, 0, 1, 5, 1, 1, 6, 2, 0, 0, 0, 1, 7, 2, 0, 1, 9, 2, 1, 10, 3, 1, 12, 3, 1, 0, 0, 0, 1, 14, 3, 1, 1, 17, 4, 1, 0, 0, 0, 0, 1, 19, 5, 1, 1, 23, 5, 1, 1, 26, 6, 2, 0, 1, 30, 7, 2, 0, 0, 0, 0, 0, 1

Offset: 0

Alois P. Heinz, Mar 16 2020

			In the A000607(11) = 6 partitions of 11 into prime parts, (11), 335, 227, 2225, 2333, 22223 the least parts are 11 = prime(5) (once), 3 = prime(2)(once), and 2 = prime(1) (four times), whereas 5 and 7 (prime(3) and prime(4)) do not occur. Thus row 11 is [4,1,0,0,1].
Triangle T(n,k) begins:
   0    ;
   0    ;
   1    ;
   0, 1    ;
   1       ;
   1, 0, 1    ;
   1, 1       ;
   2, 0, 0, 1    ;
   2, 1          ;
   3, 1          ;
   3, 1, 1       ;
   4, 1, 0, 0, 1    ;
   5, 1, 1          ;
   6, 2, 0, 0, 0, 1    ;
   7, 2, 0, 1          ;
   9, 2, 1             ;
  10, 3, 1             ;
  12, 3, 1, 0, 0, 0, 1    ;
  14, 3, 1, 1             ;
  17, 4, 1, 0, 0, 0, 0, 1    ;
  19, 5, 1, 1                ;
  ...

Alois P. Heinz, Rows n = 0..1000, flattened

Columns k=1-2 give: A000607(n-2) for n>1, A099773(n-3) for n>2.
Row sums give A000607 for n>0.
Length of n-th row is A000720(A331634(n)) for n>1.
Indices of rows without 1's: A330433.
Cf. A000040, A000720, A010051, A333129, A333238, A333259.

b:= proc(n, p, t) option remember; `if`(n=0, 1, `if`(p>n, 0, (q->
      add(b(n-p*j, q, 1), j=1..n/p)*t^p+b(n, q, t))(nextprime(p))))
    end:
T:= proc(n) option remember; (p-> seq(`if`(isprime(i),
      coeff(p, x, i), [][]), i=2..max(2,degree(p))))(b(n, 2, x))
    end:
seq(T(n), n=0..23);

b[n_, p_, t_] := b[n, p, t] = If[n == 0, 1, If[p > n, 0, With[{q = NextPrime[p]}, Sum[b[n - p*j, q, 1], {j, 1, n/p}]*t^p + b[n, q, t]]]];
T[n_] := If[n < 2, {0}, MapIndexed[If[PrimeQ[#2[[1]]], #1, Nothing]&, Rest @ CoefficientList[b[n, 2, x], x]]];
T /@ Range[0, 23] // Flatten (* Jean-François Alcover, Mar 30 2021, after Alois P. Heinz *)

T(n,pi(n)) = A010051(n) for n > 1.
T(p,pi(p)) = 1 if p is prime.
T(prime(k),k) = 1 for k >= 1.
Recursion: T(n,k) = Sum_{q=k..pi(n-p)} T(n-p, q) with p := prime(k) and T(n,k) = 0 if n < p, or 1 if n = p. - David James Sycamore, Mar 28 2020

A354203 Fully multiplicative with a(p^e) = A354201(A000720(p))^e.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 7, 1, 5, 3, 2, 1, 13, 1, 11, 2, 3, 7, 19, 1, 4, 5, 1, 3, 17, 2, 23, 1, 7, 13, 6, 1, 29, 11, 5, 2, 37, 3, 31, 7, 2, 19, 43, 1, 9, 4, 13, 5, 41, 1, 14, 3, 11, 17, 47, 2, 53, 23, 3, 1, 10, 7, 59, 13, 19, 6, 67, 1, 61, 29, 4, 11, 21, 5, 71, 2, 1, 37, 79, 3, 26, 31, 17, 7, 73, 2, 15, 19, 23

Offset: 1

Antti Karttunen, May 23 2022

Index entries for sequences computed from indices in prime factorization

Left inverse of A354202.

Cf. A354201, A354206.

A354201(n) = if(n<=3,(n+1)\2,my(m=prime(n)%4); forstep(i=n-1,0,-1,if(m==(prime(i)%4),return(prime(i)))));
A354203(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354201(primepi(f[k,1]))); factorback(f); };

A047885 Array a(m,n) = pi(m+n) - pi(m) - pi(n) read by antidiagonals, where pi() = A000720 (m,n >= 0).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

A212210-A212213 are the preferred versions of this array.

			Beginning of array is
  0  0  0  0  0  0  0  0 ...
  0  1  1  0  1  0  1  0 ...
  0  1  0  0  0  0 -1 ...
  0  0  0 -1  0 -1 -1 ...
  ...

Cf. A047886, A212210-A212213, A000720.

A066457 Numbers k such that product of factorials of digits of k equals pi(k) (A000720).

Original entry on oeis.org

13, 1512, 1520, 1521, 12016, 12035, 226130351, 209210612202, 209210612212, 209210612220, 209210612221, 13030323000581525

Offset: 1

Jason Earls, Jan 02 2002

The Caldwell/Honaker paper does not discuss this, only suggests further areas of investigation.
There are no other members of the sequence up to and including n=1000000. - Harvey P. Dale, Jan 07 2002
If 10n is in the sequence and 10n+1 is composite then 10n+1 is also in the sequence (the proof is easy). - Farideh Firoozbakht, Oct 24 2008
a(13) > 10^19 if it exists. - Chai Wah Wu, May 03 2018

			12016 is a term because there are exactly 1!*2!*0!*1!*6! (or 1440) prime numbers less than or equal to 12016.
pi(209210612202) = 8360755200 = 2!*0!*9!*2!*1!*0!*6!*1!*2!*2!*0!*2!. [Qu,Shun Liang (medie2006(AT)126.com), Nov 23 2008]

C. Caldwell and G. L. Honaker, Jr., Is pi(6521)=6!+5!+2!+1! unique?
A discussion about this topic: bbs.emath.ac.cn [From Qu,Shun Liang (medie2006(AT)126.com), Nov 23 2008]
Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.

Cf. A000720, A066459, A049529, A105327.

Select[Range[1000000], Times@@( # !&/@IntegerDigits[ # ])==PrimePi[ # ]&]

isok(n) = my(d = digits(n)); prod(k=1, #d, d[k]!) == primepi(n); \\ Michel Marcus, May 04 2018

a(7) from Farideh Firoozbakht, Apr 20 2005
a(8)-a(11) from Qu,Shun Liang (medie2006(AT)126.com), Nov 23 2008
a(12) from Chai Wah Wu, May 03 2018

A157190 Primes which produce records in A157188, at index i=pi(a(n)) (pi=A000720).

Original entry on oeis.org

2, 3, 59, 71, 1151, 2399, 7559, 42839, 110879, 181439, 241919, 262079, 453599, 665279, 1713599, 2827439, 6425999, 11309759, 12700799, 14137199, 16707599, 37837799, 45239039, 64864799, 82162079, 86486399, 93562559, 260124479, 410810399, 735134399, 950019839

Offset: 1

M. F. Hasler, Mar 11 2009

nonn

Primes that can be written in more ways as p*q-(p+q) (p,q prime) than any smaller prime.

M. F. Hasler, Table of n, a(n), for n=1..40.
Carlos Rivera, Puzzle 482: Two Bergot questions, The Prime Puzzles & Problems Connection.

Cf. A000720, A157187, A157188, A157189, A336296.

A157187(a(n)) = A157188(A000720(a(n))) > A157187(p) for all primes p < a(n).

a(28) corrected by M. F. Hasler, Mar 15 2009

A237615 a(n) = |{0 < k < n: k^2 + k - 1 and pi(k*n) are both prime}|, where pi(.) is given by A000720.

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 2, 1, 2, 1, 3, 2, 1, 4, 1, 3, 4, 4, 2, 4, 3, 6, 2, 2, 2, 3, 7, 4, 3, 4, 5, 6, 1, 3, 2, 3, 9, 3, 3, 4, 7, 5, 8, 5, 2, 2, 5, 5, 4, 5, 6, 4, 5, 6, 10, 6, 6, 10, 9, 9, 10, 12, 2, 8, 7, 3, 6, 6, 4, 6

Offset: 1

Zhi-Wei Sun, Feb 10 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 5.
(ii) For each n = 4, 5, ..., there is a positive integer k < n with k^2 + k - 1 and pi(k*n) + 1 both prime. Also, for any integer n > 6, there is a positive integer k < n with k^2 + k - 1 and pi(k*n) - 1 both prime.
(iii) For every integer n > 15, there is a positive integer k < n such that pi(k) - 1 and pi(k*n) are both prime.
Note that part (i) is a refinement of the first assertion in the comments in A237578.

			a(8) = 1 since 4^2 + 4 - 1 = 19 and pi(4*8) = 11 are both prime.
a(33) = 1 since 28^2 + 28 - 1 = 811 and pi(28*33) = 157 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
Zhi-Wei Sun, A combinatorial conjecture on primes, a message to Number Theory List, Feb. 9, 2014.

Cf. A000040, A000720, A002327, A045546, A237578, A237597, A237598.

p[k_,n_]:=PrimeQ[k^2+k-1]&&PrimeQ[PrimePi[k*n]]
a[n_]:=Sum[If[p[k,n],1,0],{k,1,n-1}]
Table[a[n],{n,1,70}]

cp's OEIS Frontend

A291539 a(n) = PrimePi(n^3) - PrimePi(n) * PrimePi(n^2), where PrimePi = A000720.

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A291540 a(n) = PrimePi(n) * PrimePi(n^2) - PrimePi(n)^3, where PrimePi = A000720.

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A293447 Fully additive with a(p^e) = e * A000225(PrimePi(p)), where PrimePi(n) = A000720(n) and A000225(n) = (2^n)-1.

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A332212 Fully multiplicative with a(p) = A332211(A000720(p)).

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A333365 T(n,k) is the number of times that prime(k) is the least part in a partition of n into prime parts; triangle T(n,k), n >= 0, 1 <= k <= max(1,A000720(A331634(n))), read by rows.

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A354203 Fully multiplicative with a(p^e) = A354201(A000720(p))^e.

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A047885 Array a(m,n) = pi(m+n) - pi(m) - pi(n) read by antidiagonals, where pi() = A000720 (m,n >= 0).

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A066457 Numbers k such that product of factorials of digits of k equals pi(k) (A000720).

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