G. L. Honaker, Jr. - cp's OEIS frontend

A358527 Position of p in the factorization (without multiplicity) of 2^(p-1)-1, where p is the n-th odd prime.

1, 2, 2, 2, 4, 3, 3, 2, 3, 4, 6, 6, 3, 2, 3, 2, 8, 4, 5, 8, 3, 2, 5, 6, 6, 3, 2, 8, 6, 6, 4, 4, 4, 3, 5, 7, 5, 2, 3, 2, 14, 4, 7, 7, 8, 9, 3, 2, 5, 5, 4, 12, 4, 4, 2, 3, 8, 7, 12, 3, 3, 6, 4, 10, 3, 9, 13, 2, 7, 7, 2, 3, 5, 8, 2, 3, 13, 10, 10, 4, 19, 4, 13, 3

Offset: 1

G. L. Honaker, Jr., Nov 20 2022

nonn

			a(19) = 5 because the 19th odd prime is 71 and 71 is the 5th largest distinct prime factor of 2^(71-1)-1 = 1180591620717411303423 = 3 * 11 * 31 * 43 * 71 * 127 * 281 * 86171 * 122921.

Amiram Eldar, Table of n, a(n) for n = 1..196

Cf. A065091, A098102, A358699.

Array[FirstPosition[FactorInteger[2^(# - 1) - 1], #][[1]] &[Prime[# + 1]] &, 50] (* Michael De Vlieger, Nov 27 2022 *)

a(n) = my(p=prime(n+1), v=factor(2^(p-1)-1)[,1]); vecsearch(v, p); \\ Michel Marcus, Nov 28 2022

More terms from Amiram Eldar, Nov 23 2022

A358401 Difference in number of 0's in first n terms of Van Eck's sequence and number of primes less than or equal to n.

Original entry on oeis.org

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

Offset: 1

G. L. Honaker, Jr., Nov 13 2022

sign

			a(41) = -2 because there are 11 0's in the first 41 terms of Van Eck's sequence and 13 prime numbers less than or equal to 41, and 11 - 13 = -2.

Cf. A000040, A000720, A171896.

a(n) = A171896(n) - A000720(n).

A357908 Index of the first occurrence of n-th prime in Van Eck's sequence (A181391), or 0 if n-th prime never appears.

Original entry on oeis.org

5, 20, 12, 66, 44, 121, 41, 89, 101, 225, 72, 92, 548, 199, 297, 1486, 490, 1001, 735, 455, 420, 611, 772, 673, 187, 1612, 3690, 581, 417, 2584, 7574, 162, 1483, 1048, 689, 330, 1320, 4007, 3739, 2884, 528, 3376, 3045, 3658, 2869, 411, 935, 303, 1751, 1122, 376, 5506, 599, 13191, 494

Offset: 1

G. L. Honaker, Jr., Nov 08 2022

nonn

The first 85669653 terms exist and are less than 10^12, see A181391. - Charles R Greathouse IV, Nov 09 2022

			a(3) = 12 because position 12 of Van Eck's sequence (A181391) is the first occurrence of 3rd prime number, i.e., 5.

Cf. A181391.

a(n) > prime(n). - Charles R Greathouse IV, Nov 09 2022

A358180 Indices for A358168.

Original entry on oeis.org

1, 30, 162, 1150, 11603, 104511, 1041245, 10226995, 101514698, 1008495923, 10060201866

Offset: 1

G. L. Honaker, Jr., Nov 02 2022

a(6)-a(7) from Chuck Gaydos.

Cf. A181391, A358168.

nn = 2^20; q[] = False; q[0] = True; a[] = 0; c[] = -1; c[0] = 2; m = 1; {1}~Join~Reap[Do[j = c[m]; a[n] = m; c[m] = n; m = 0; If[j > 0, m = n - j]; If[! q[#2], Sow[n]; q[#2] = True] & @@ {a[n], IntegerLength[a[n]]}, {n, 3, nn}] ][[-1, -1]] (* _Michael De Vlieger, Nov 05 2022 *)

from itertools import count
def A358180(n):
    b, bdict, k = 0, {0:(1,)},10**(n-1) if n > 1 else 0
    for m in count(2):
        if b >= k:
            return m-1
        if len(l := bdict[b]) > 1:
            b = m-1-l[-2]
            if b in bdict:
                bdict[b] = (bdict[b][-1],m)
            else:
                bdict[b] = (m,)
        else:
            b = 0
            bdict[0] = (bdict[0][-1],m) # Chai Wah Wu, Nov 05 2022

a(8) from Michael De Vlieger, Nov 05 2022
a(9)-a(10) from Chai Wah Wu, Nov 05 2022
a(11) from Martin Ehrenstein, Nov 05 2022

A358168 First n-digit number to occur in Van Eck's Sequence (A181391).

Original entry on oeis.org

0, 14, 131, 1024, 10381, 100881, 1014748, 10001558, 100246289, 1000943528, 10010107437

Offset: 1

G. L. Honaker, Jr., Nov 01 2022

a(6)-a(7) from Chuck Gaydos.

			a(2) = 14 because 14 is the first 2-digit number occurring in A181391.

Cf. A181391, A358180.

nn = 2^20; q[] = False; q[0] = True; a[] = 0; c[] = -1; c[0] = 2; m = 1; {0}~Join~Rest@ Reap[Do[j = c[m]; a[n] = m; c[m] = n; m = 0; If[j > 0, m = n - j]; If[! q[#2], Sow[#1]; q[#2] = True] & @@ {a[n], IntegerLength[a[n]]}, {n, 3, nn}] ][[-1, -1]] (* _Michael De Vlieger, Nov 05 2022 *)

from itertools import count
def A358168(n):
    b, bdict, k = 0, {0:(1,)},10**(n-1) if n > 1 else 0
    for m in count(2):
        if b >= k:
            return b
        if len(l := bdict[b]) > 1:
            b = m-1-l[-2]
            if b in bdict:
                bdict[b] = (bdict[b][-1],m)
            else:
                bdict[b] = (m,)
        else:
            b = 0
            bdict[0] = (bdict[0][-1],m) # Chai Wah Wu, Nov 05 2022

a(8)-a(10) from Chai Wah Wu, Nov 05 2022

a(11) from Martin Ehrenstein, Nov 05 2022

A347294 Primes that become semiprimes when turned upside down.

Original entry on oeis.org

191, 691, 811, 991, 1009, 1069, 1619, 1801, 1861, 1889, 6089, 6869, 6911, 6961, 8101, 8191, 8609, 8669, 8689, 9001, 9811, 10009, 10099, 10111, 10169, 10181, 10601, 10889, 10891, 11119, 11161, 11689, 11699, 11801, 11969, 11981, 16061, 16691, 16699, 18089, 18119

Offset: 1

G. L. Honaker, Jr., Jan 22 2022

			811 is a term because when 811 is turned upside down (rotated 180 degrees) it becomes 118=2*59, a semiprime.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
C. K. Caldwell, and G. L. Honaker, Jr., Prime Curio for 191.

Cf. A000040, A001358, A048889.

semiQ[n_] := PrimeOmega[n] == 2; q[n_] := PrimeQ[n] && Module[{d = IntegerDigits[n]}, AllTrue[d, MemberQ[{0, 1, 6, 8, 9}, #] &] && semiQ[FromDigits[Reverse[d] /. {6 -> 9, 9 -> 6}]]]; Select[Range[20000], q] (* Amiram Eldar, Jan 23 2022 *)

from sympy import isprime, factorint
from itertools import count, islice, product
def f(s): return s[::-1].translate({ord("6"):ord("9"), ord("9"):ord("6")})
def agen():
    for digits in count(3):
        for first in "1689":
            for mid in product("01689", repeat=digits-2):
                for last in "19":
                    s = first + "".join(mid) + last
                    t = int(s)
                    if isprime(t):
                        flip = f(s)
                        if sum(factorint(int(flip)).values()) == 2:
                            yield t
print(list(islice(agen(), 41))) # Michael S. Branicky, Feb 16 2024

More terms from Amiram Eldar, Jan 23 2022

A348185 Smallest number k in a set of three consecutive triangular numbers that are sphenic.

Original entry on oeis.org

406, 861, 39621, 2166321, 3924201, 11146281, 14804961, 19198306, 73951041, 83417986, 97951006, 209643526, 310415986, 522339681, 526225461, 583333246, 611153241, 801460666, 1601581906, 2520251506, 2690954841, 4455349606, 6681853401, 9895642221, 13878029901

Offset: 1

G. L. Honaker, Jr., Oct 05 2021

nonn

a(2)-a(9) from Chuck Gaydos.

			a(1)=406 because 406 is the smallest number in the first set of three consecutive triangular numbers that are sphenic, i.e., {406=2*7*29, 435=3*5*29, 465=3*5*31}.

Chris K. Caldwell and G. L. Honaker, Jr., Prime Curio for 406

Cf. A000217 (triangular numbers), A007304 (sphenic numbers), A128896 (sphenic triangular numbers). Subsequence of A349696.

t[n_] := n*(n+1)/2; spQ[n_] := FactorInteger[n][[;;,2]] == {1,1,1}; Select[Partition[t /@ Range[170000], 3, 1], AllTrue[#, spQ] &][[;; , 1]] (* Amiram Eldar, Oct 06 2021 *)

a(10)-a(25) from Alois P. Heinz, Oct 05 2021

A339498 Number of zeroless strictly pandigital numbers divisible by the n-th prime.

Original entry on oeis.org

161280, 362880, 40320, 51752, 31680, 27776, 21271, 19138, 15788, 12613, 11707, 9072, 8832, 8423, 7725, 6822, 6241, 5937, 5454, 5113, 4796, 4629, 4310, 4122, 3744, 3168, 3528, 3410, 3305, 3160, 2826, 2827, 2778, 2619, 2316, 2297, 2297, 2173, 2163, 2094, 2077, 2027, 1879, 1915, 1836, 1780, 1773

Offset: 1

G. L. Honaker, Jr., Dec 07 2020

Calculated by Chuck Gaydos.

a(4620), for prime(4620) = 44449, is the first zero entry. The last nonzero entry is a(6289143) for prime 109739359 = 987654231 / 9. - Michael S. Branicky, Dec 07 2020

Cf. A000040, A050289.

from sympy import prime
from itertools import permutations
def zeroless_pans():
    for p in permutations("123456789"):
        yield int("".join(p))
def a(n):
    pn = prime(n)
    return sum(zlp%pn==0 for zlp in zeroless_pans())
print([a(n) for n in range(1, 43)]) # Michael S. Branicky, Dec 07 2020

A335889 a(n) is the number of Mersenne primes between consecutive perfect numbers.

Original entry on oeis.org

1, 2, 0, 3, 1, 0, 0, 3, 1, 0, 0, 2, 0, 3, 2, 1, 0, 0, 0, 3, 0, 2, 1, 0, 1, 1, 2, 1, 0, 0, 4, 0, 0, 0, 2, 0, 2, 3, 3

Offset: 1

G. L. Honaker, Jr., Jun 28 2020

			a(1) = 1 because there is exactly 1 Mersenne prime (7) between the first and second perfect numbers (6 and 28).
a(4) = 3 because there are exactly 3 Mersenne primes (8191, 131071, 524287) between the fourth and fifth perfect numbers (8128 and 33550336).

Chris K. Caldwell and G. L. Honaker, Jr., Prime Curio for 756839

Cf. A000043, A000396, A000668.

p = MersennePrimeExponent @ Range[47];mer[p_] := 2^p - 1; perf[p_] := mer[p] * 2^(p - 1); mers = mer /@ p; perfs = Select[perf /@ p, # < mers[[-1]] &]; BinCounts[mers, {perfs}] (* Amiram Eldar, Jun 29 2020 *)

a(5)-a(13) from Metin Sariyar, Jun 28 2020
a(14)-a(16) and a(20)-a(39) from Metin Sariyar, Jun 29 2020
a(17)-a(19) from Amiram Eldar, Jun 29 2020

A331858 a(n) = (2^p-1)(2^(p-1))((2^p-1)^2-2), where p is the n-th prime.

Original entry on oeis.org

42, 1316, 475664, 131080256, 8783210218496, 2250975213522944, 147570574898545885184, 37778715690312487141376, 2475879193127080196116054016, 41538374636164863806350357434466304, 10633823951424046514111736193740701696, 178405961584350762488394070192754827810832384

Offset: 1

G. L. Honaker, Jr., Jan 29 2020

Integers a(1), a(2), a(4), a(8) corresponding to p = 2, 3, 7, 19 are also terms of A331805. - Bernard Schott, Feb 04 2020

Cf. A000040 (primes), A000396 (perfect numbers), A093112 ((2^n-1)^2-2), A060286 (2^(p-1)*(2^p-1)), A331805.

f[p_] := (2^p-1)*(2^(p-1))*((2^p-1)^2-2); f @ Prime @ Range[12] (* Amiram Eldar, Jan 29 2020 *)

[(2^p-1)*((2^p-1)^2-2)<<(p-1) | p<-primes(12)] \\ or: a(n,p=prime(n))={...}. - M. F. Hasler, Jan 29 2020

a(n) = A060286(n)*A093112(prime(n)). - M. F. Hasler, Jan 31 2020

cp's OEIS Frontend

User: G. L. Honaker, Jr.

A358527 Position of p in the factorization (without multiplicity) of 2^(p-1)-1, where p is the n-th odd prime.

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A358401 Difference in number of 0's in first n terms of Van Eck's sequence and number of primes less than or equal to n.

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A357908 Index of the first occurrence of n-th prime in Van Eck's sequence (A181391), or 0 if n-th prime never appears.

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A358180 Indices for A358168.

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A358168 First n-digit number to occur in Van Eck's Sequence (A181391).

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A347294 Primes that become semiprimes when turned upside down.

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A348185 Smallest number k in a set of three consecutive triangular numbers that are sphenic.

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A339498 Number of zeroless strictly pandigital numbers divisible by the n-th prime.

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A335889 a(n) is the number of Mersenne primes between consecutive perfect numbers.

A331858 a(n) = (2^p-1)(2^(p-1))((2^p-1)^2-2), where p is the n-th prime.