?(#===2||PrimeQ[PrimePi[#]]&), - cp's OEIS frontend

A360359 Numbers k such that A360331(k) = A360331(k+1).

69, 574, 713, 781, 2394, 2506, 5699, 5750, 6499, 6509, 8441, 19250, 26529, 32130, 36549, 38065, 41749, 41929, 43239, 48025, 50301, 53037, 53382, 59178, 59822, 61754, 66906, 67689, 70277, 71198, 81620, 94000, 100775, 119214, 124640, 127442, 134665, 153202, 154908

Offset: 1

Amiram Eldar, Feb 04 2023

nonn

			69 is a term since A360331(69) = A360331(70) = 24.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A360331.

Similar sequences: A002961, A064115, A064125, A293183, A306985, A360358.

f[p_, e_] := If[PrimeQ[PrimePi[p]], 1, (p^(e+1)-1)/(p-1)]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; seq = {}; s1 = s[1]; n = 2; c = 0; While[c < 40, s2 = s[n]; If[s1 == s2, c++; AppendTo[seq, n - 1]]; s1 = s2; n++]; seq

s(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), 1, (p[i]^(e[i]+1)-1)/(p[i]-1)));}
lista(nmax) = {my(s1 = s(1), s2); for(n=2, nmax, s2=s(n); if(s1 == s2, print1(n-1, ", ")); s1 = s2); }

A360358 Numbers k such that A360327(k) = A360327(k+1) > 1.

Original entry on oeis.org

714, 6603, 16115, 18920, 23154, 24530, 39984, 41360, 42789, 51204, 56814, 58190, 59619, 60995, 65229, 66605, 68034, 69410, 73644, 79304, 82059, 84249, 84864, 86240, 94655, 101375, 101694, 103070, 107304, 108680, 121374, 125510, 126125, 126939, 135128, 135354, 137329

Offset: 1

Amiram Eldar, Feb 04 2023

nonn

Numbers k such that A360327(k) = A360327(k+1) = 1 are terms of A360357.

			714 is a term since A360327(714) = A360327(715) = 72 > 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A360327, A360357.

Similar sequences: A002961, A064115, A064125, A293183, A306985, A360359.

f[p_, e_] := If[PrimeQ[PrimePi[p]], (p^(e+1)-1)/(p-1), 1]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; seq = {}; s1 = s[1]; n = 2; c = 0; While[c < 40, s2 = s[n]; If[s1 == s2 > 1, c++; AppendTo[seq, n - 1]]; s1 = s2; n++]; seq

s(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), (p[i]^(e[i]+1)-1)/(p[i]-1), 1));}
lista(nmax) = {my(s1 = s(1), s2); for(n=2, nmax, s2=s(n); if(s2 > 1 && s1 == s2, print1(n-1, ", ")); s1 = s2); }

A360357 Numbers k such that k and k+1 are both products of primes of nonprime index (A320628).

Original entry on oeis.org

1, 7, 13, 28, 37, 46, 52, 73, 91, 97, 103, 106, 112, 148, 151, 172, 181, 193, 196, 202, 223, 226, 232, 256, 262, 292, 298, 301, 316, 337, 343, 346, 361, 376, 388, 397, 427, 448, 457, 463, 466, 478, 487, 502, 511, 523, 541, 556, 568, 592, 601, 607, 613, 622, 631

Offset: 1

Amiram Eldar, Feb 04 2023

nonn

There are no 3 consecutive integers that are products of primes of nonprime index since 1 out of 3 consecutive integers is divisible by 3 which is a prime-indexed prime (A006450).

If a Mersenne prime (A000668) is a prime of nonprime index, then it is in this sequence. Of the first 10 Mersenne primes 6 are in this in sequence: A000668(k) for k = 2, 5, 7, 8, 9, 10 (see A059305).

			7 = prime(4) is a term since 4 is nonprime, 7 + 1 = 8 = prime(1)^3, and 1 is also nonprime.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000668, A006450, A059305, A320628.

q[n_] := AllTrue[FactorInteger[n][[;; , 1]], ! PrimeQ[PrimePi[#]] &]; seq = {}; q1 = q[1]; n = 2; c = 0; While[c < 55, q2 = q[n]; If[q1 && q2, c++; AppendTo[seq, n - 1]]; q1 = q2; n++]; seq

is(n) = {my(p = factor(n)[,1]); for(i = 1, #p, if(isprime(primepi(p[i])), return(0))); 1;}
lista(nmax) = {my(q1 = is(1), q2); for(n = 2, nmax, q2 = is(n); if(q1 && q2, print1(n-1, ", ")); q1 = q2); }

A360356 Primitive terms of A360332: terms of A360332 with no proper divisor in A360332.

Original entry on oeis.org

56, 104, 196, 304, 364, 368, 464, 532, 644, 812, 1036, 1184, 1204, 1316, 1376, 1484, 1504, 1696, 1708, 1952, 1988, 2044, 2212, 2492, 2716, 2828, 2884, 2996, 3164, 3496, 3668, 3836, 3892, 4172, 4228, 4408, 4544, 4564, 4672, 4676, 4844, 5056, 5068, 5336, 5404, 5516

Offset: 1

Amiram Eldar, Feb 04 2023

nonn

If m is a term then k*m is a term of A360332 for all k in A320628.

Analogous to primitive abundant numbers (A091191) with divisors that are restricted to numbers that have only nonprime-indexed prime factors.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A360332.
Cf. A320628.
Similar sequences: A006038, A091191, A249263, A302574, A360355.

f[p_, e_] := If[PrimeQ[PrimePi[p]], 1, (p^(e + 1) - 1)/(p - 1)]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; primQ[n_] := s[n] > 2*n && AllTrue[Divisors[n], # == n || s[#] <= 2*# &]; Select[Range[6000], primQ]

isab(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), 1, (p[i]^(e[i]+1)-1)/(p[i]-1))) > 2*n;}
is(n) = {if(!isab(n), return(0)); fordiv(n, d, if(d < n && isab(d), return(0))); return(1)};

A360355 Primitive terms of A360328: terms of A360328 with no proper divisor in A360328.

Original entry on oeis.org

7425, 8415, 46035, 76725, 101475, 182655, 355725, 669735, 1411425, 1606275, 2352375, 2891295, 3592215, 3650625, 4079295, 4861575, 5053455, 5870205, 6093225, 6636465, 6920595, 7732395, 8750835, 9120375, 9783675, 9850005, 9958905, 10155375, 11298375, 11532375, 12120075

Offset: 1

Amiram Eldar, Feb 04 2023

nonn

If m is a term then k*m is a term of A360328 for all k in A076610.

Analogous to primitive abundant numbers (A091191) with divisors that are restricted to numbers that have only prime-indexed prime factors.

Subsequence of A360328.
Cf. A076610.
Similar sequences: A006038, A091191, A249263, A302574, A360356.

f[p_, e_] := If[PrimeQ[PrimePi[p]], (p^(e + 1) - 1)/(p - 1), 1]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; primQ[n_] := s[n] > 2*n && AllTrue[Divisors[n], # == n || s[#] <= 2*# &]; Select[Range[10^6], primQ]

isab(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), (p[i]^(e[i]+1)-1)/(p[i]-1), 1)) > 2*n;}
is(n) = {if(!isab(n), return(0)); fordiv(n, d, if(d < n && isab(d), return(0))); return(1)};

A360332 Numbers k such that A360331(k) > 2*k.

Original entry on oeis.org

56, 104, 112, 196, 208, 224, 304, 364, 368, 392, 416, 448, 464, 532, 608, 644, 728, 736, 784, 812, 832, 896, 928, 1036, 1064, 1184, 1204, 1216, 1288, 1316, 1352, 1372, 1376, 1456, 1472, 1484, 1504, 1568, 1624, 1664, 1696, 1708, 1792, 1856, 1952, 1976, 1988, 2044

Offset: 1

Amiram Eldar, Feb 03 2023

nonn

Analogous to abundant numbers (A005101) with divisors that are restricted to numbers that have only nonprime-indexed prime factors.
The least odd term is 7^4 * (13*19)^3 * (29*...*71)^2 * (73*...*281) = 2.411... * 10^105 (where the dots are for consecutive terms in A007821).
Includes all the abundant (A005101) terms of A320628.
There are terms that are not in A320628, and the least of them is 3 * m, where m is a term of A320628 with sigma(m) > 6. Such a number exists, and it should be a positive multiple of Product_{i=1..k} A007821(k) = 2 * 7 * ... * 11443 = 9.164... * 10^4148, where k = 1160 is the least number such that Product_{i=1..k} A007821(k)/(A007821(k)-1) > 6.
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 1, 23, 215, 1997, 19231, 189457, 1873511, 18593697, ... . Apparently, the asymptotic density of this sequence equals 0.018... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A005101.

Cf. A007821, A320628, A360331.

q:= n-> is(mul(`if`(isprime(numtheory[pi](i[1])), 1,
   (i[1]^(i[2]+1)-1)/(i[1]-1)), i=ifactors(n)[2])>2*n):
select(q, [$1..2050])[];  # Alois P. Heinz, Feb 03 2023

f[p_, e_] := If[PrimeQ[PrimePi[p]], 1, (p^(e+1)-1)/(p-1)]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[2000], s[#] > 2*# &]

is(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), 1, (p[i]^(e[i]+1)-1)/(p[i]-1))) > 2*n;}

A360331 a(n) is the sum of divisors of n that have only prime factors that are not prime-indexed primes.

Original entry on oeis.org

1, 3, 1, 7, 1, 3, 8, 15, 1, 3, 1, 7, 14, 24, 1, 31, 1, 3, 20, 7, 8, 3, 24, 15, 1, 42, 1, 56, 30, 3, 1, 63, 1, 3, 8, 7, 38, 60, 14, 15, 1, 24, 44, 7, 1, 72, 48, 31, 57, 3, 1, 98, 54, 3, 1, 120, 20, 90, 1, 7, 62, 3, 8, 127, 14, 3, 1, 7, 24, 24, 72, 15, 74, 114, 1

Offset: 1

Amiram Eldar, Feb 03 2023

Equivalently, a(n) is the sum of divisors of the largest divisor of n that has only prime factors that are not prime-indexed primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A006450, A007821, A076610, A320628, A360329, A360330.

a:= n-> mul(`if`(isprime(numtheory[pi](i[1])), 1,
   (i[1]^(i[2]+1)-1)/(i[1]-1)), i=ifactors(n)[2]):
seq(a(n), n=1..75);  # Alois P. Heinz, Feb 03 2023

f[p_, e_] := If[PrimeQ[PrimePi[p]], 1, (p^(e+1)-1)/(p-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), 1, (p[i]^(e[i]+1)-1)/(p[i]-1)));}

a(n) = 1 if and only if n is in A076610.
a(n) = A000203(n) if and only if n is in A320628.
a(n) = A000203(A360329(n)).
Multiplicative with a(p^e) = 1 if p is a prime-indexed prime (A006450), and (p^(e+1)-1)/(p-1) otherwise (A007821).

A360330 a(n) is the number of divisors of n that have only prime factors that are not prime-indexed primes.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 2, 4, 1, 2, 1, 3, 2, 4, 1, 5, 1, 2, 2, 3, 2, 2, 2, 4, 1, 4, 1, 6, 2, 2, 1, 6, 1, 2, 2, 3, 2, 4, 2, 4, 1, 4, 2, 3, 1, 4, 2, 5, 3, 2, 1, 6, 2, 2, 1, 8, 2, 4, 1, 3, 2, 2, 2, 7, 2, 2, 1, 3, 2, 4, 2, 4, 2, 4, 1, 6, 2, 4, 2, 5, 1, 2, 1, 6, 1, 4, 2

Offset: 1

Amiram Eldar, Feb 03 2023

Equivalently, a(n) is the number of divisors of the largest divisor of n that has only prime factors that are not prime-indexed primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A006450, A007821, A076610, A320628, A360329, A360331.

a:= n-> mul(`if`(isprime(numtheory[pi](i[1])), 1, i[2]+1), i=ifactors(n)[2]):
seq(a(n), n=1..87);  # Alois P. Heinz, Feb 03 2023

f[p_, e_] := If[PrimeQ[PrimePi[p]], 1, e+1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), 1, e[i]+1));}

a(n) = 1 if and only if n is in A076610.
a(n) = A000005(n) if and only if n is in A320628.
a(n) = A000005(A360329(n)).
Multiplicative with a(p^e) = 1 if p is a prime-indexed prime (A006450), and e+1 otherwise (A007821).

A360329 a(n) is the largest divisor of n that has only prime factors that are not prime-indexed primes.

Original entry on oeis.org

1, 2, 1, 4, 1, 2, 7, 8, 1, 2, 1, 4, 13, 14, 1, 16, 1, 2, 19, 4, 7, 2, 23, 8, 1, 26, 1, 28, 29, 2, 1, 32, 1, 2, 7, 4, 37, 38, 13, 8, 1, 14, 43, 4, 1, 46, 47, 16, 49, 2, 1, 52, 53, 2, 1, 56, 19, 58, 1, 4, 61, 2, 7, 64, 13, 2, 1, 4, 23, 14, 71, 8, 73, 74, 1, 76, 7

Offset: 1

Amiram Eldar, Feb 03 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006450, A007821, A076610, A302590, A320628, A360325, A360330, A360331.

f[p_, e_] := If[PrimeQ[PrimePi[p]], 1, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); for(i = 1, #f~, if(isprime(primepi(f[i,1])), f[i,1]=1)); factorback(f);}

a(n) = 1 if and only if n is in A076610.
a(n) = n if and only if n is in A320628.
a(n) = n/A360325(n).
Multiplicative with a(p^e) = 1 if p is a prime-indexed prime (A006450), and p^e otherwise (A007821).
Sum_{k=1..n} a(k) ~ (1/2) * c * n^2, where c = Product_{p in A006450} p/(p+1) < 0.4 (see A302590 for an estimate of 1/c).

A360328 Numbers k such that A360327(k) > 2*k.

Original entry on oeis.org

7425, 8415, 22275, 25245, 37125, 42075, 46035, 66825, 75735, 76725, 81675, 92565, 101475, 111375, 126225, 138105, 143055, 182655, 185625, 200475, 210375, 227205, 230175, 245025, 260865, 277695, 304425, 334125, 345015, 355725, 378675, 383625, 408375, 414315, 429165

Offset: 1

Amiram Eldar, Feb 03 2023

nonn

Analogous to abundant numbers (A005101) with divisors that are restricted to numbers that have only prime-indexed prime factors.
The abundancy index of numbers in A076610 (i.e., numbers whose prime factors are only prime-indexed primes) is bounded by P = Product_{p in A006450} p/(p-1) which seems to be less than 4 (see A076610). Therefore, there are no terms k of A076610 with sigma(k) >= 4*k, or equivalently, no even terms in this sequence, and all the terms of this sequence are in A076610. Also, assuming that P < 15/4 = 3.75, there are no terms in this sequence that are coprime to 15.
Since P > 3 there are terms that are not divisible by 3. The least of them must be larger than Product_{k=1..21826870} A006450(k) = 3 * 5 * 11 * ... * 8958801613 > 10^206662375, because Product_{k=2..m} A006450(k)/(A006450(k)-1) > 2 only for m >= 21826870.
The least term that is not divisible by 5 is 789909738655399955305165431.
The least term that is not divisible by 11 is a(30) = 355725.
The least squarefree term is 14093057715.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Intersection of A005101 (or A005231) and A076610.

Cf. A006450, A076610, A360327.

f[p_, e_] := If[PrimeQ[PrimePi[p]], (p^(e + 1) - 1)/(p - 1), 1]; s[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^6], s[#] > 2*# &]

is(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), (p[i]^(e[i]+1)-1)/(p[i]-1), 1)) > 2*n;}

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User: ?(#===2||PrimeQ[PrimePi[#]]&),

A360359 Numbers k such that A360331(k) = A360331(k+1).

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A360358 Numbers k such that A360327(k) = A360327(k+1) > 1.

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A360357 Numbers k such that k and k+1 are both products of primes of nonprime index (A320628).

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A360356 Primitive terms of A360332: terms of A360332 with no proper divisor in A360332.

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A360355 Primitive terms of A360328: terms of A360328 with no proper divisor in A360328.

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A360332 Numbers k such that A360331(k) > 2*k.

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A360331 a(n) is the sum of divisors of n that have only prime factors that are not prime-indexed primes.

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A360330 a(n) is the number of divisors of n that have only prime factors that are not prime-indexed primes.

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